Physical and Mathematical Modeling of the Cardiovascular System: An Overview of Aortic Hemodynamics, 3D Heart Simulation, and Fluid-Structure Interaction

Abstract

A comprehensive overview of cardiovascular mathematics, a multidisciplinary field at the intersection of biomedical engineering, fluid dynamics, and computational mechanics is provided for analyzing multicomplex systems of blood circulation and the cardiovascular system. By transitioning from qualitative observation to rigorous mathematical modeling, researchers can generate highly accurate, patient-specific simulations of the circulatory system. This treatise explores the foundational Navier-Stokes equations governing blood flow, advanced non-Newtonian rheological models, and the critical role of hemodynamics in pathological remodeling, such as aneurysm formation and rupture. It further delves into the complexities of Fluid-Structure Interaction (FSI) utilizing the Arbitrary Lagrangian-Eulerian (ALE) framework, and the necessity of geometrical multiscale modeling (3D-1D-0D) to simulate the global cardiovascular network. The later sections address whole-heart electromechanics, passive myocardial hyperelasticity via the Holzapfel-Ogden model, and innovative valve modeling techniques. Finally, the report highlights the transformative impact of Artificial Intelligence and Scientific Machine Learning in accelerating these computationally intensive simulations to create real-time, personalized digital twins for clinical application.