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.
We consider a continuous-time optimization method based on a dynamical system, where a massive particle starting at rest moves in the conservative force field generated by the objective function, ...without any kind of friction. We formulate a restart criterion based on the mean dissipation of the kinetic energy, and we prove a global convergence result for strongly-convex functions. Using the Symplectic Euler discretization scheme, we obtain an iterative optimization algorithm. We have considered a discrete mean dissipation restart scheme, but we have also introduced a new restart procedure based on ensuring at each iteration a decrease of the objective function greater than the one achieved by a step of the classical gradient method. For the discrete conservative algorithm, this last restart criterion is capable of guaranteeing a qualitative convergence result. We apply the same restart scheme to the Nesterov Accelerated Gradient (NAG-C), and we use this restarted NAG-C as benchmark in the numerical experiments. In the smooth convex problems considered, our method shows a faster convergence rate than the restarted NAG-C. We propose an extension of our discrete conservative algorithm to composite optimization: in the numerical tests involving non-strongly convex functions with
ℓ
1
-regularization, it has better performances than the well known efficient Fast Iterative Shrinkage-Thresholding Algorithm, accelerated with an adaptive restart scheme.
Subgradient methods are the natural extension to the non-smooth case of the classical gradient descent for regular convex optimization problems. However, in general, they are characterized by slow ...convergence rates, and they require decreasing step-sizes to converge. In this paper we propose a subgradient method with constant step-size for composite convex objectives with
ℓ
1
-regularization. If the smooth term is strongly convex, we can establish a linear convergence result for the function values. This fact relies on an accurate choice of the element of the subdifferential used for the update, and on proper actions adopted when non-differentiability regions are crossed. Then, we propose an accelerated version of the algorithm, based on conservative inertial dynamics and on an adaptive restart strategy, that is guaranteed to achieve a linear convergence rate in the strongly convex case. Finally, we test the performances of our algorithms on some strongly and non-strongly convex examples.
•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.
We develop a parallel solver for the cardiac electro-mechanical coupling. The electric model consists of two non-linear parabolic partial differential equations (PDEs), the so-called Bidomain model, ...which describes the spread of the electric impulse in the heart muscle. The two PDEs are coupled with a non-linear elastic model, where the myocardium is considered as a nearly-incompressible transversely isotropic hyperelastic material. The discretization of the whole electro-mechanical model is performed by Q1 finite elements in space and a semi-implicit finite difference scheme in time. This approximation strategy yields at each time step the solution of a large scale ill-conditioned linear system deriving from the discretization of the Bidomain model and a non-linear system deriving from the discretization of the finite elasticity model. The parallel solver developed consists of solving the linear system with the Conjugate Gradient method, preconditioned by a Multilevel Schwarz preconditioner, and the non-linear system with a Newton–Krylov-Algebraic Multigrid solver. Three-dimensional parallel numerical tests on a Linux cluster show that the parallel solver proposed is scalable and robust with respect to the domain deformations induced by the cardiac contraction.
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.
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.
1 Dipartimento di Matematica, Università degli Studi di Pavia, Pavia, Italy; 2 Dipartimento di Matematica, Università degli Studi di Milano, Milan, Italy; and 3 Cardiovascular Research and Training ...Institute, University of Utah, Salt Lake City, Utah
Submitted 6 June 2007
; accepted in final form 16 August 2007
Unipolar electrograms (EGs) and hybrid (or unorthodox or unipolar) monophasic action potentials (HMAPs) are currently the only proposed extracellular electrical recording techniques for obtaining cardiac recovery maps with high spatial resolution in exposed and isolated hearts. Estimates of the repolarization times from the HMAP downstroke phase have been the subject of recent controversies. The goal of this paper is to computationally address the controversies concerning the HMAP information content, in particular the reliability of estimating the repolarization time from the HMAP downstroke phase. Three-dimensional numerical simulations were performed by using the anisotropic bidomain model with a region of short action potential durations. EGs, transmembrane action potentials (TAPs), and HMAPs elicited by an epicardial stimulation close or away from a permanently depolarized site were computed. The repolarization time was computed as the moment of EG fastest upstroke (RT eg ) during the T wave, of HMAP fastest downstroke (RT HMAP ), and of TAP fastest downstroke (RT tap ). The latter was taken as the gold standard for repolarization time. We also compared the times (RT90 HMAP , RT90 tap ) when the HMAP and TAP first reach 90% of their resting value during the downstroke. For all explored sites, the HMAP downstroke closely followed the TAP downstroke, which is the expression of local repolarization activity. Results show that HMAP and TAP markers are highly correlated, and both markers RT HMAP and RT eg (RT90 HMAP ) are reliable estimates of the TAP reference marker RT tap (RT90 tap ). Therefore, the downstroke phase of the HMAP contains valuable information for assessing repolarization times.
unipolar electrograms; monophasic action potential; bidomain model; heterogeneity; action potential duration
Address for reprint requests and other correspondence: P. Colli Franzone, Dipartimento di Matematica, Universita' degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy (e-mail: colli{at}imati.cnr.it )
► We compute anodal and cathodal strength–interval curves by 3D bidomain simulations. ► The results show detailed excitation patterns elicited by break and make responses. ► We identify novel ...excitation mechanisms at the break/make transition of the S–I curves. ► The model includes intramural fiber rotation and orthotropic anisotropy. ► The LRd model is augmented with the electroporation, outward, and funny currents.
The assessment and understanding of cardiac excitation mechanisms is very important for the development and improvement of implantable cardiac devices, pacing protocols, and arrhythmia treatments. Previous bidomain simulation studies have investigated cathodal and anodal make/break mechanisms of cardiac excitation and strength–interval (S–I) curves in two-dimensional sheets or cylindrical domains, that by symmetry reduce to the two-dimensional case. In this work, cathodal and anodal S–I curves are studied by means of detailed bidomain simulations which include: (i) three-dimensional cardiac slabs; (ii) transmural fiber rotation; (iii) unequal orthotropic anisotropy of the conducting media; (iv) incorporation of funny and electroporation currents in the ventricular membrane model. The predicted shape of cathodal and anodal S–I curves exhibit the same features of the S–I curves observed experimentally and the break/make transition coincides with the final descending phase of the S–I curves. Away from the break/make transition, only the break or make excitation mechanism is observed independently of the stimulus strength, whereas within an interval at the break/make transition, new paradoxical excitation behaviors are observed that depend on the stimulus strength.
•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.
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