Numerical approximation of cardiac electro-fluid-mechanical models:coupling strategies for large-scale simulation

The mathematical modeling of the heart involves several challenges, which are intrinsically related to the complexity of its function. A satisfactory cardiac model must be able to describe a wide range of different processes, such as the evolution of the transmembrane potential in the myocardium, the deformation caused by the muscles contraction, and the dynamics of the blood inside the heart chambers. In this work, we focus on the coupling of the electrophysiology, the active and the passive mechanics, and the fluid dynamics of the blood in the left ventricle (LV) of the human heart. The models describing the previously mentioned processes are called âsingle core modelsâ, and can be regarded as the building blocks of an âintegrated modelâ. In this thesis, we first review the isolated single core mathematical models for the description of the LV function, and discuss their space and time discretizations with particular emphasis on the coupling conditions. We consider both implicit and semi-implicit schemes for the time discretization. The fully discretized single core problems thus obtained are then combined to define integrated electromechanics and electrofluidmechanics problems. We then focus on the numerical coupling strategy for the electromechanics solver in the framework of the active strain formulation. First, we propose a monolithic strategy where the discretized core models are solved simultaneously; then, several novel segregated strategies, where the discretized core models are solved sequentially, are proposed and systematically compared with each other. The segregated strategies are obtained by exploiting a Godunov splitting scheme, which introduces a first order error on the solution. We show that, while the monolithic approach is more accurate and more stable for relatively large timesteps, segregated approaches allow to solve the integrated problem much more efficiently in terms of computational resources. Moreover, with segregated approaches, it is possible to use different timesteps for the different core models in a staggered fashion, thus further improving the computational efficiency of the schemes. The monolithic and the segregated strategies for the electromechanics are used to solve a benchmark problem with idealized geometry: the results are then compared in terms of accuracy and efficiency. We numerically confirm that the segregated strategies are accurate at least of order one. In light of the results obtained, we employ the proposed strategies to simulate the electromechanics of a subject-specific LV for a full heartbeat. We simulate both healthy and pathological scenarios: in the latter case, we account for an ischemic necrosis of the tissue and analyze several clinical indicators such as pressure-volume loops and the end systolic pressure-volume relationship. Finally, we use the proposed strategies to simulate the electrofluidmechanics of a realistic LV during the systolic phase of the heartbeat. When defining the integrated cardiac models, we establish a preprocess pipeline aimed at preparing geometries and data for both idealized and subject-specific simulations. The pipeline is succesfully used for the setting up of large scale simulations in a high performance computing framework, where the (strong and weak) scalability of the proposed coupling strategies is assessed

Effect of fibre orientation and bulk modulus on the electromechanical modelling of human ventricles

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

A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle

In this paper, we propose a monolithic algorithm for the numerical solution of the electromechanics model of the left ventricle in the human heart. Our coupled model integrates the monodomain equation with the Bueno-Orovio minimal model for electrophysiology and the Holzapfel-Ogden constitutive law for the passive mechanics of the myocardium; a distinguishing feature of our electromechanics model is the use of the active strain formulation for muscle contraction, which we exploit - for the first time in this context - by means of a transmurally variable active strain formulation. We use the Finite Element method for space discretization and Backward Differentiation Formulas for time discretization, which we consider for both implicit and semi-implicit schemes. We compare and discuss the two schemes in terms of computational efficiency as the semi-implicit scheme poses significant restrictions on the timestep size due to stability considerations, while the implicit scheme yields instead a nonlinear problem, which we solve by means of the Newton method. Emphasis is laid on preconditioning strategy of the linear solver, which we perform by factorizing a block Gauss-Seidel preconditioner in combination with combination with parallel preconditioners for each of the single core models composing the integrated electromechanics model. We carry out several numerical simulations in the High Performance Computing framework for both idealized and patient-specific left ventricle geometries and meshes, which we obtain by segmenting medical MRI images. We produce personalized pressure-volume loops by means of the computational procedure, which we use to synthetically interpret and analyze the outputs of the electromechanics model

Effect of fibre orientation and bulk modulus on the electromechanical modelling of human ventricles

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 bi-ventricle 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