SIAM Journal on Imaging Sciences, Volume 16, Issue 3, Page 1491-1519, September 2023.
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 [math]-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.

中文翻译：

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心脏左心室涡流结构的拓扑识别

SIAM 影像科学杂志，第 16 卷，第 3 期，第 1491-1519 页，2023 年 9 月。
抽象的。心脏左心室 (LV) 内的涡旋血流结构在从心脏到器官的有效血液供应中发挥着至关重要的作用。最近的医学成像和计算技术的进步为我们带来了超声心动图和心脏 MRI 中的血流可视化工具。然而，由于流动是高度不稳定和湍流的，目前仍然很少有工具能够精确捕获涡流结构。由于涡流动力对心脏病中心脏功能的预后具有重要意义，因此明确地识别涡流结构在医学中至关重要。在本文中，我们提出了一种用符号图表达式（称为 COT 表示）来描述二维（2D）流的拓扑特征的数学方法。由于心脏在短时间内反复收缩和舒张，沿该移动边界的瞬时血流模式将表现为源/汇结构。这意味着流不满足前面的二维流拓扑分类理论中假设的滑移边界条件[T. Sakajo 和 T. Yokoyama，IMA J. Appl。数学，83（2018），第380–411页]，[T。Sakajo 和 Y. Yokoyama，《离散数学》。算法应用, 15 (2023), 2250143]. 因此，我们通过引入一个名为[math]-bundled ss-saddle的简并奇异点，建立了一种新的拓扑分类理论和适合具有移动边界条件的血流的算法。将该理论应用于可视化工具获得的二维血流模式，我们成功地将涡流结构识别为拓扑涡流结构。