The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications

Abstract

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.

A transmurally heterogeneous orthotropic activation model for ventricular contraction and its numerical validation

“…Computational models can significantly help to increase the understanding of the heart function and dysfunction. As the computing power increases, electromechanical and electro‐fluid‐mechanical models of the heart have been developed and numerically solved . Unfortunately, the real predictive capabilities of such models are not always clear.…”

“…As the computing power increases, electromechanical and electro-fluid-mechanical models of the heart have been developed and numerically solved. 1,[3][4][5][6][7][8][9][10] Unfortunately, the real predictive capabilities of such models are not always clear. In fact, many models fail in capturing the most obvious characteristics of the heart deformation.…”

Numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network

“…Computational modeling of the electromechanical coupling in the heart can be used to better understand the complex interplay between the chemical, electrical, and mechanical fields that are involved in the cardiac cycle . For instance, one may be interested in studying how a pathological condition of the electrical conduction system affects the overall contraction in the ventricles .…”

“…Computational modeling of the electromechanical coupling in the heart can be used to better understand the complex interplay between the chemical, electrical, and mechanical fields that are involved in the cardiac cycle. [1][2][3][4][5][6][7] For instance, one may be interested in studying how a pathological condition of the electrical conduction system affects the overall contraction in the ventricles. 8,9 The underlying motivation here is that outputs of computer-based simulations in patient-specific geometries can be used by the physicians to enhance diagnosis and therapy planning.…”

Detection of arterial wall abnormalities via Bayesian model selection

“…Mathematical models for hemodynamics trace back to the work of Euler, who described a onedimensional treatment of blood flow through an arterial network with rigid tubes [11,33]; more sophisticated one-dimensional models are still used to study a variety of physio-pathological phenomena [1,2,13,21,23,29,30,37]. Computational advances have also allowed for the development of computationally intensive three-dimensional models [12,14,32,34,39,40], which have been used to accurately simulate specific human arteries (e.g., the carotid arteries [18]) and model their material properties (e.g., of cerebral arterial walls [38]). There also exist multicomponent models [10], which are amenable to applications such as modeling oxygen transport to solid tumors [6] and surgical tissue flaps [24,25].…”