Topological Identification of Vortical Flow Structures in the Left Ventricle of the Heart

Abstract

Vortical blood flow structures inside the heart's left ventricle (LV) play a crucial role in an efficient blood supply from the heart to organs. Recent medical imaging and computational technology progress have brought us blood flow visualization tools in echocardiography and cardiac MRI. However, there are still few tools to precisely capture the vortical flow structures since the flow is highly unsteady and turbulent. Because of the importance of vortex flow power force on the prognosis of cardiac functions in heart diseases, identifying the vortex flow structure without ambiguity is essential in medical science. In this paper, we propose a mathematical method to describe the topological features of two-dimensional (2D) flows with symbolic graph expressions, called COT representations. Since the heart contracts and relaxes repeatedly in a short time range, the instantaneous blood flow pattern along this moving boundary would appear as a source/sink structure. This means that the flow does not satisfy the slip-boundary condition that is assumed in the preceding topological classification theory for 2D flows [T. Sakajo and T. Yokoyama, IMA J. Appl. Math., 83 (2018), pp. 380--411], [T. Sakajo and Y. Yokoyama, Discrete Math. Algorithms Appl., 15 (2023), 2250143]. We thus establish a new topological classification theory and an algorithm suitable for blood flow with the moving boundary condition by introducing a degenerate singular point named n-bundled ss-saddle. Applying the theory to 2D blood flow patterns obtained by the visualization tools, we successfully identify vortical flow structures as topological vortex structures. This realizes a new image processing characterizing healthy blood flow patterns as well as inefficient patterns in diseased hearts.