Abstract
This work concerns the mathematical and numerical modeling of the heart. The aim is to enhance the understanding of the cardiac function in both physiological and pathological conditions. Along this road, a challenge arises from the multi-scale and multi-physics nature of the mathematical problem at hand. In this paper, we propose an electromechanical model that, in bi-ventricle geometries, combines the monodomain equation, the Bueno-Orovio minimal ionic model, and the Holzapfel-Ogden strain energy function for the passive myocardial tissue modelling together with the active strain approach combined with a model for the transmurally heterogeneous thickening of the myocardium. Since the distribution of the electric signal is dependent on the fibres orientation of the ventricles, we use a Laplace-Dirichlet Rule-Based algorithm to determine the myocardial fibres and sheets configuration in the whole bi-ventricle. In this paper, we study the influence of different fibre directions and incompressibility constraint and penalization on the compressibility of the material (bulk modulus) on the pressure-volume relation simulating a full heart beat. The coupled electromechanical problem is addressed by means of a fully segregated scheme. The numerical discretization is based on the Finite Element Method for the spatial discretization and on Backward Differentiation Formulas for the time discretization. The arising non-linear algebraic system coming from application of the implicit scheme is solved through the Newton method. Numerical simulations are carried out in a patient-specific biventricle geometry to highlight the most relevant results of both electrophysiology and mechanics and to compare them with physiological data and measurements. We show how various fibre configurations and bulk modulus modify relevant clinical quantities such as stroke volume, ejection fraction and ventricle contractility.
Highlights
The heart has the role of pumping deoxigenated blood to the lungs to get oxygen and, simultaneously, delivers blood rich of all sort of vital substances to tissues and organs through the arterial circulatory system [32]
We propose an electromechanical model that, in bi-ventricle geometries, combines the monodomain equation, the Bueno-Orovio minimal ionic model, and the Holzapfel-Ogden strain energy function for the passive myocardial tissue modelling together with the active strain approach combined with a model for the transmurally heterogeneous thickening of the myocardium
We successfully implemented a segregated method for the electro-mechanical modelling of both human ventricles in which we exploited a transversal and isotropic model for the mechanical activation
Summary
The heart has the role of pumping deoxigenated blood to the lungs to get oxygen and, simultaneously, delivers blood rich of all sort of vital substances to tissues and organs through the arterial circulatory system [32]. The contraction of the muscle yields a rapid increase of pressure inside the ventricular cavities, which allows the heart valves to open and close in precise sequence and induces the periodic filling and ejection of blood from the ventricles and the atria [15, 33, 35, 36, 45,46,47] This physiological process underline an intrinsic multi-scale nature. We analyse the effect of fibre direction and bulk modulus on the pressure-volume relationship through electromechanical modelling for both the left and right ventricles [4].
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