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This manuscript is concerned with a novel, unified finite element approach to fully coupled cardiac electromechanics. The intrinsic coupling arises from both the excitation-induced contraction of cardiac cells and the deformation-induced generation of current due to the opening of ion channels. In contrast to the existing numerical approaches suggested in the literature, which devise staggered algorithms through distinct numerical methods for the respective electrical and mechanical problems, we propose a fully implicit, entirely finite element-based modular approach. To this end, the governing differential equations that are coupled through constitutive equations are recast into the corresponding weak forms through the conventional isoparametric Galerkin method. The resultant non-linear weighted residual terms are then consistently linearized. The system of coupled algebraic equations obtained through discretization is solved monolithically. The put-forward modular algorithmic setting leads to an unconditionally stable and geometrically flexible framework that lays a firm foundation for the extension of constitutive equations towards more complex ionic models of cardiac electrophysiology and the strain energy functions of cardiac mechanics. The performance of the proposed approach is demonstrated through three-dimensional illustrative initial boundary-value problems that include a coupled electromechanical analysis of a biventricular generic heart model.

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SUMMARYThis work deals with the computational modeling of passive myocardial tissue within the framework of mixed, non-linear finite element methods. We consider a recently proposed, convex, anisotropic hyperelastic model that accounts for the locally orthotropic micro-structure of cardiac muscle. A coordinate-free representation of anisotropy is incorporated through physically relevant invariants of the Cauchy-Green deformation tensors and structural tensors of the corresponding material symmetry group. This model, which has originally been designed for exactly incompressible deformations, is extended towards entirely three-dimensional inhomogeneous deformations by additively decoupling the strain energy function into volumetric and isochoric parts along with the multiplicative split of the deformation gradient. This decoupled constitutive structure is then embedded in a mixed finite element formulation through a threefield Hu-Washizu functional whose simultaneous variation with respect to the independent pressure, dilatation, and placement fields results in the associated Euler-Lagrange equations, thereby minimizing the potential energy. This weak form is then consistently linearized for uniform-pressure elements within the framework of an implicit finite element method. To demonstrate the performance of the proposed approach, we present a three-dimensional finite element analysis of a generic biventricular heart model, subjected to physiological ventricular pressure. The parameters employed in the numerical analysis are identified by solving an optimization problem based on six simple shear experiments on explanted cardiac tissue.

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SUMMARYThe key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element-based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global-local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell-specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh-Nagumo model for oscillatory pacemaker cells and the Aliev-Panfilov model for non-oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non-linear system of equations is solved with an incremental iterative Newton-Raphson solution procedure. Since this framework only introduces one single scalar-valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two-dimensional patient-specific cardiac excitation problem.

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We introduce a novel constitutive model for growing soft biological tissue and study its performance in two characteristic cases of mechanically-induced wall thickening of the heart. We adopt the concept of an incompatible growth configuration introducing the multiplicative decomposition of the deformation gradient into an elastic and a growth part. The key feature of the model is the definition of the evolution equation for the growth tensor which we motivate by pressure-overload induced sarcomerogenesis. In response to the deposition of sarcomere units on the molecular level, the individual heart muscle cells increase in diameter, and the wall of the heart becomes progressively thicker. We present the underlying constitutive equations and their algorithmic implementation within an implicit nonlinear finite element framework. To demonstrate the features of the proposed approach, we study two classical growth phenomena in the heart: left and right ventricular wall thickening in response to systemic and pulmonary hypertension.