Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.

中文翻译：

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心血管系统：数学建模、数值算法和临床应用

心血管系统的数学和数值建模是一个研究课题，由于其内在的数学难度和全球心血管疾病的影响越来越大，引起了数学界的极大兴趣。在这篇评论文章中，我们将讨论心血管系统的两个主要组成部分：动脉循环和心脏功能。我们将系统地描述该问题的各个方面，从数据成像采集、说明基本物理原理、分析包含 PDE 和 ODE 系统的相关数学模型、提出合理有效的逼近数值方法，以及模拟基准问题和临床启发的问题。数学建模本身带来了巨大的挑战，由于心脏循环系统的惊人复杂性，所涉及的生理过程的多尺度性质，以及需要设计稳定、可靠和高效的计算方法。关键问题包括过滤数据、确定数学模型的参数、设计最佳处理方法和考虑不确定性。出于这个原因，我们将把论文的最后一部分用于控制和反问题，包括参数估计、不确定性量化和降阶模型的开发，这在解决高复杂度问题时至关重要，否则这些问题将被排除在外。触手可及。可靠和高效。关键问题包括过滤数据、确定数学模型的参数、设计最佳处理方法和考虑不确定性。出于这个原因，我们将把论文的最后一部分用于控制和反问题，包括参数估计、不确定性量化和降阶模型的开发，这在解决高复杂度问题时至关重要，否则这些问题将被排除在外。触手可及。可靠和高效。关键问题包括过滤数据、确定数学模型的参数、设计最佳处理方法和考虑不确定性。出于这个原因，我们将把论文的最后一部分用于控制和反问题，包括参数估计、不确定性量化和降阶模型的开发，这在解决高复杂度问题时至关重要，否则这些问题将被排除在外。触手可及。