Overlapping Additive Schwarz (OAS) preconditioners are here constructed for isogeometric collocation discretizations of the system of linear elasticity in both two and three space dimensions. ...Isogeometric collocation methods are recent variants of isogeometric analysis based on the numerical approximation of the strong form of partial differential equations at appropriate collocation points. Numerical results in two and three dimensions show that two-level OAS preconditioners are scalable in the number of subdomains N, quasi-optimal with respect to the mesh size h and optimal with respect to the spline polynomial degree p. Moreover, two-level OAS preconditioners are more robust than one-level OAS and non-preconditioned GMRES solvers when the material tends to the incompressible limit, as well as in the presence of strong deformation of the NURBS geometry.
•Scar tissue border zone is a major determinant of the onset of simulated cardiac reentry•Reentrant cycles may develop within a sub-epicardial border zone wedging between unexcitable scars•Reentrant ...cycles may develop within a homogeneous sub-epicardial border zone covering a scar•Reentrant pathways follow epicardial fiber direction regardless of the endocardial stimulation site•Thin, rather than thick, sub-epicardial border zone facilitates the onset of reentry
Cardiac ventricular tachycardia (VT) is a life-threatening arrhythmia consisting of a well organized structure of reentrant electrical excitation pathways. Understanding the generation and maintenance of the reentrant mechanisms, which lead to the onset of VT induced by premature beats in presence of infarct scar, is one of the most important issues in current electrocardiology. We investigate, by means of numerical simulations, the role of infarct scar dimension, repolarization properties and anisotropic fiber structure of scar tissue border zone (BZ) in the genesis of VT. The simulations are based on the Bidomain model, a reaction-diffusion system of Partial Differential Equations, discretized by finite elements in space and implicit-explicit finite differences in time. The computational domain adopted is an idealized left ventricle affected by an infarct scar extending transmurally. We consider two different scenarios: i) the scar region extends along the entire transmural wall thickness, from endocardium to epicardium, with the exception of a BZ region shaped as a central sub-epicardial channel (CBZ); ii) the scar region extends transmurally along the ventricular wall, from endocardium to a sub-epicardial surface, and is surrounded by a BZ region (EBZ). In CBZ simulations, the results have shown that: i) the scar extent is a crucial element for the genesis of reentry; ii) the repolarization properties of the CBZ, in particular the reduction of IKs and IKr currents, play an important role in the genesis of reentrant VT. In EBZ simulations, since the possible reentrant pathway is not assigned a-priori, we investigate in depth where the entry and exit sites of the cycle of reentry are located and how the functional channel of reentry develops. The results have shown that: i) the interplay between the epicardial anisotropic fiber structure and the EBZ shape strongly affects the propensity that an endocardial premature stimulus generates a cycle of reentry; ii) reentrant pathways always develop along the epicardial fiber direction; iii) very thin EBZs rather than thick EBZs facilitate the onset of cycles of reentry; iv) the sustainability of cycles of reentry depends on the endocardial stimulation site and on the interplay between the epicardial breakthrough site, local fiber direction and BZ rim.
In this work, we investigate the influence of cardiac tissue deformation on re-entrant wave dynamics. We have developed a 3D strongly coupled electro-mechanical Bidomain model posed on an ideal ...monoventricular geometry, including fiber direction anisotropy and stretch-activated currents (SACs). The cardiac mechanical deformation influences the bioelectrical activity with two main mechanical feedback: (a) the geometric feedback (GEF) due to the presence of the deformation gradient in the diffusion coefficients and in a convective term depending on the deformation rate and (b) the mechano-electric feedback (MEF) due to SACs. Here, we investigate the relative contribution of these two factors with respect to scroll wave stability. We extend the previous works Keldermann et al., Am. J. Physiol. Heart Circ. Physiol. 299, H134-H143 (2010) and Hu et al., PLoS One 8(4), e60287 (2013) that were based on the Monodomain model and a simple non-selective linear SAC, while here we consider the full Bidomain model and both selective and non-selective components of SACs. Our simulation results show that the stability of cardiac scroll waves is influenced by MEF, which in case of low reversal potential of non-selective SACs might be responsible for the onset of ventricular fibrillation; GEF increases the scroll wave meandering but does not determine the scroll wave stability.
•The presence of intrinsic transmural cellular APD heterogeneities is not fully masked by electrotonic current flow or by the presence of the mechanical deformation.•Despite the presence of ...transmural APD heterogeneities, the recovery process follows the activation sequence and there is no significant transmural repolarization gradient.•With or without transmural APD heterogeneities, epicardial electrograms always display the same wave shape and discordance between the polarity of QRS complex and T-wave.•The main effects of the mechanical deformation are an increase of the dispersion of repolarization time and APD, when computed over the total cardiac domain and over the endo- and epicardial surfaces, while there is a slight decrease along the transmural direction.
The aim of this work is to investigate, by means of numerical simulations, the influence of myocardial deformation due to muscle contraction and relaxation on the cardiac repolarization process in presence of transmural intrinsic action potential duration (APD) heterogeneities. The three-dimensional electromechanical model considered consists of the following four coupled components: the quasi-static transversely isotropic finite elasticity equations for the deformation of the cardiac tissue; the active tension model for the intracellular calcium dynamics and cross-bridge binding; the anisotropic Bidomain model for the electrical current flow through the deforming cardiac tissue; the membrane model of ventricular myocytes, including stretch-activated channels. The numerical simulations are based on our finite element parallel solver, which employs Multilevel Additive Schwarz preconditioners for the solution of the discretized Bidomain equations and Newton-Krylov methods for the solution of the discretized non-linear finite elasticity equations. Our findings show that: (i) the presence of intrinsic transmural cellular APD heterogeneities is not fully masked by electrotonic current flow or by the presence of the mechanical deformation; (ii) despite the presence of transmural APD heterogeneities, the recovery process follows the activation sequence and there is no significant transmural repolarization gradient; (iii) with or without transmural APD heterogeneities, epicardial electrograms always display the same wave shape and discordance between the polarity of QRS complex and T-wave; (iv) the main effects of the mechanical deformation are an increase of the dispersion of repolarization time and APD, when computed over the total cardiac domain and over the endo- and epicardial surfaces, while there is a slight decrease along the transmural direction.
A balancing domain decomposition by constraints (BDDC) preconditioner with a novel scaling, introduced by Dohrmann for problems with more than one variable coefficient and here denoted as deluxe ...scaling, is extended to isogeometric analysis of scalar elliptic problems. This new scaling turns out to be more powerful than the standard $\rho$- and stiffness scalings considered in a previous isogeometric BDDC study. Our $h$-analysis shows that the condition number of the resulting deluxe BDDC preconditioner is scalable with a quasi-optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces. Extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings, in particular for large isogeometric polynomial degree and high regularity. PUBLICATION ABSTRACT
Isogeometric Schwarz preconditioners are constructed and analyzed for both compressible elasticity in primal formulation and almost incompressible elasticity in mixed formulation. These ...preconditioners require the solution of local elasticity problems on overlapping subdomains forming a decomposition of the problem domain and the solution of a coarse elasticity problem associated with the subdomain coarse mesh. An h-analysis of the preconditioner for the primal formulation of compressible elasticity yields an optimal convergence rate bound that is scalable in the number of subdomains and is linear in the ratio between subdomain and overlap sizes. Extensive numerical experiments in 2D and 3D confirm this theoretical bound and show that an analogous bound holds for the mixed formulation of almost incompressible elasticity. The numerical tests also show the good preconditioner performance with respect to the polynomial degree p and regularity k of the isogeometric basis functions, as well as with respect to the presence of discontinuous elastic coefficients in composite materials and to domain deformation.
The aim of this work is to construct and analyze a FETI-DP type domain decomposition preconditioner for isogeometric discretizations of the Stokes and mixed linear elasticity systems. This method ...extends to the isogeometric analysis context the preconditioner previously proposed by Tu and Li (2015) for finite element discretizations of the Stokes system. The resulting isogeometric FETI-DP algorithm is proven to be scalable in the number of subdomains and has a quasi-optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes. Extensive two-dimensional numerical experiments validate the theory, investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity, and show its robustness with respect to domain deformation, material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces.
•We construct a block FETI-DP preconditioner for isogeometric analysis of Stokes and mixed linear elasticity systems.•We prove that the resulting algorithm is scalable in the number of subdomains and has a quasi-optimal convergence rate bound.•Extensive two-dimensional numerical experiments validate the theoretical estimates and show the robustness of the method with respect to domain deformation, material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces.
Up to one-third of patients undergoing cardiac resynchronization therapy (CRT) are nonresponders. Multipoint bicathodic and cathodic-anodal left ventricle (LV) stimulations could overcome this ...clinical challenge, but their effectiveness remains controversial. Here we evaluate the performance of such stimulations through both in vivo and in silico experiments, the latter based on computer electromechanical modeling. Seven patients, all candidates for CRT, received a quadripolar LV lead. Four stimulations were tested: right ventricular (RVS); conventional single point biventricular (S-BS); multipoint biventricular bicathodic (CC-BS) and multipoint biventricular cathodic-anodal (CA-BS). The following parameters were processed: QRS duration; maximal time derivative of arterial pressure (dPdtmax); systolic arterial pressure (Psys); and stroke volume (SV). Echocardiographic data of each patient were then obtained to create an LV geometric model. Numerical simulations were based on a strongly coupled Bidomain electromechanical coupling model.
Considering the in vivo parameters, when comparing S-BS to RVS, there was no significant decrease in SV (from 45 ± 11 to 44 ± 20 ml) and 6% and 4% increases of dPdtmax and Psys, respectively. Focusing on in silico parameters, with respect to RVS, S-BS exhibited a significant increase of SV, dPdtmax and Psys. Neither the in vivo nor in silico results showed any significant hemodynamic and electrical difference among S-BS, CC-BS and CA-BS configurations.
These results show that CC-BS and CA-BS yield a comparable CRT performance, but they do not always yield improvement in terms of hemodynamic parameters with respect to S-BS. The computational results confirmed the in vivo observations, thus providing theoretical support to the clinical experiments.
•We study the effectiveness of biventricular multipoint bicathodic (CC-BS) and cathodic-anodal (CA-BS) pacing.•We develop computer models of left ventricular (LV) electro-mechanical activity based on echocardiographic data of patients.•We compare in vivo electrical and hemodynamic results obtained during CRT implants to in silico computer models of LV pacing.•CC-BS and CA-BS are comparable in terms of hemodynamic outputs.•CC-BS and CA-BS do not improve hemodynamic outputs with respect to conventional single point biventricular pacing.
The aim of this work is to design and study a Balancing Domain Decomposition by Constraints (BDDC) solver for the nonlinear elasticity system modeling the mechanical deformation of cardiac tissue. ...The contraction–relaxation process in the myocardium is induced by the generation and spread of the bioelectrical excitation throughout the tissue and it is mathematically described by the coupling of cardiac electro-mechanical models consisting of systems of partial and ordinary differential equations. In this study, the discretization of the electro-mechanical models is performed by Q1 finite elements in space and semi-implicit finite difference schemes in time, leading to the solution of a large-scale linear system for the bioelectrical potentials and a nonlinear system for the mechanical deformation at each time step of the simulation. The parallel mechanical solver proposed in this paper consists in solving the nonlinear system with a Newton–Krylov-BDDC method, based on the parallel solution of local mechanical problems and a coarse problem for the so-called primal unknowns. Three-dimensional parallel numerical tests on different machines show that the proposed parallel solver is scalable in the number of subdomains, quasi-optimal in the ratio of subdomain to mesh sizes, and robust with respect to tissue anisotropy.
Parallel numerical simulations of excitation and recovery in three-dimensional myocardial domains are presented. The simulations are based on the anisotropic Bidomain and Monodomain models, including ...intramural fiber rotation and orthotropic or axisymmetric anisotropy of the intra- and extra-cellular conductivity tensors. The Bidomain model consist of a system of two reaction–diffusion equations, while the Monodomain model consists of one reaction–diffusion equation. Both models are coupled with the phase I Luo–Rudy membrane model describing the ionic currents. Simulations of excitation and repolarization sequences on myocardial slabs of different sizes show how the distribution of the action potential durations (APD) is influenced by both the anisotropic electrical conduction and the fiber rotation. This influence occurs in spite of the homogeneous intrinsic properties of the cell membrane. The APD dispersion patterns are closely correlated to the anisotropic curvature of the excitation wavefront.
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