BPX preconditioners for the Bidomain model of electrocardiology

The aim of this work is to develop a BPX preconditioner for the Bidomain model of electrocardiology. This model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction\u2013diffusion partial differential equation (PDE) and an elliptic linear PDE, modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations, the so-called membrane model, describing the ionic currents through the cellular membrane. The discretization of the coupled system by finite elements in space and semi-implicit finite differences in time yields at each time step the solution of an ill-conditioned linear system. The goal of the present study is to construct, analyze and numerically test a BPX preconditioner for the linear system arising from the discretization of the Bidomain model. Optimal convergence rate estimates are established and verified by two- and three-dimensional numerical tests on both structured and unstructured meshes. Moreover, in a full heartbeat simulation on a three-dimensional wedge of ventricular tissue, the BPX preconditioner is about 35% faster in terms of CPU times than ILU(0) and an Algebraic Multigrid preconditioner

A numerical study of scalable cardiac electro-mechanical solvers on HPC architectures

We introduce and study some scalable domain decomposition preconditioners for cardiac electro-mechanical 3D simulations on parallel HPC (High Performance Computing) architectures. The electro-mechanical model of the cardiac tissue is composed of four coupled sub-models: (1) the static finite elasticity equations for the transversely isotropic deformation of the cardiac tissue; (2) the active tension model describing the dynamics of the intracellular calcium, cross-bridge binding and myofilament tension; (3) the anisotropic Bidomain model describing the evolution of the intra- and extra-cellular potentials in the deforming cardiac tissue; and (4) the ionic membrane model describing the dynamics of ionic currents, gating variables, ionic concentrations and stretch-activated channels. This strongly coupled electro-mechanical model is discretized in time with a splitting semi-implicit technique and in space with isoparametric finite elements. The resulting scalable parallel solver is based on Multilevel Additive Schwarz preconditioners for the solution of the Bidomain system and on BDDC preconditioned Newton-Krylov solvers for the non-linear finite elasticity system. The results of several 3D parallel simulations show the scalability of both linear and non-linear solvers and their application to the study of both physiological excitation-contraction cardiac dynamics and re-entrant waves in the presence of different mechano-electrical feedbacks

Efficient Numerical Methods for Heart Simulation

The heart is one the most important organs in the human body and many other live creatures. The electrical activity in the heart controls the heart function, and many heart diseases are linked to the abnormalities in the electrical activity in the heart. Mathematical equations and computer simulation can be used to model the electrical activity in the heart. The heart models are challenging to solve because of the complexity of the models and the huge size of the problems. Several cell models have been proposed to model the electrical activity in a single heart cell. These models must be coupled with a heart model to model the electrical activity in the entire heart. The bidomain model is a popular model to simulate the propagation of electricity in myocardial tissue. It is a continuum-based model consisting of non-linear ordinary differential equations (ODEs) describing the electrical activity at the cellular scale and a system of partial differential equations (PDEs) describing propagation of electricity at the tissue scale. Because of this multi-scale, ODE/PDE structure of the model, splitting methods that treat the ODEs and PDEs in separate steps are natural candidates as numerical methods. First, we need to solve the problem at the cellular scale using ODE solvers. One of the most popular methods to solve the ODEs is known as the Rush-Larsen (RL) method. Its popularity stems from its improved stability over integrators such as the forward Euler (FE) method along with its easy implementation. The RL method partitions the ODEs into two sets: one for the gating variables, which are treated by an exponential integrator, and another for the remaining equations, which are treated by the FE method. The success of the RL method can be understood in terms of its relatively good stability when treating the gating variables. However, this feature would not be expected to be of benefit on cell models for which the stiffness is not captured by the gating equations. We demonstrate that this is indeed the case on a number of stiff cell models. We further propose a new partitioned method based on the combination of a first-order generalization of the RL method with the FE method. This new method leads to simulations of stiff cell models that are often one or two orders of magnitude faster than the original RL method. After solving the ODEs, we need to use bidomain solvers to solve the bidomain model. Two well-known, first-order time-integration methods for solving the bidomain model are the semi-implicit method and the Godunov operator-splitting method. Both methods decouple the numerical procedure at the cellular scale from that at the tissue scale but in slightly different ways. The methods are analyzed in terms of their accuracy, and their relative performance is compared on one-, two-, and three-dimensional test cases. As suggested by the analysis, the test cases show that the Godunov method is significantly faster than the semi-implicit method for the same level of accuracy, specifically, between 5 and 15 times in the cases presented. Second-order bidomain solvers can generally be expected to be more effective than first-order bidomain solvers under normal accuracy requirements. However, the simplest and the most commonly applied second-order method for the PDE step, the Crank-Nicolson (CN) method, may generate unphysical oscillations. We investigate the performance of a two-stage, L-stable singly diagonally implicit Runge-Kutta method for solving the PDEs of the bidomain model and present a stability analysis. Numerical experiments show that the enhanced stability property of this method leads to more physically realistic numerical simulations compared to both the CN and Backward Euler (BE) methods

GEMS: A Fully Integrated PETSc-Based Solver for Coupled Cardiac Electromechanics and Bidomain Simulations

Cardiac contraction is coordinated by a wave of electrical excitation which propagates through the heart. Combined modeling of electrical and mechanical function of the heart provides the most comprehensive description of cardiac function and is one of the latest trends in cardiac research. The effective numerical modeling of cardiac electromechanics remains a challenge, due to the stiffness of the electrical equations and the global coupling in the mechanical problem. Here we present a short review of the inherent assumptions made when deriving the electromechanical equations, including a general representation for deformation-dependent conduction tensors obeying orthotropic symmetry, and then present an implicit-explicit time-stepping approach that is tailored to solving the cardiac mono- or bidomain equations coupled to electromechanics of the cardiac wall. Our approach allows to find numerical solutions of the electromechanics equations using stable and higher order time integration. Our methods are implemented in a monolithic finite element code GEMS (Ghent Electromechanics Solver) using the PETSc library that is inherently parallelized for use on high-performance computing infrastructure. We tested GEMS on standard benchmark computations and discuss further development of our software

together with details of a mailing list and links to documentation and tutorials

Abstract Chaste -Cancer, Heart And Soft Tissue Environment -is an open source C++ library for the computational simulation of mathematical models developed for physiology and biology. Code development has been driven by two initial applications: cardiac electrophysiology and cancer development. A large number of cardiac electrophysiology studies have been enabled and performed, including high-performance computational investigations of defibrillation on realistic human cardiac geometries. New models for the initiation and growth of tumours have been developed. In particular, cellbased simulations have provided novel insight into the role of stem cells in the colorectal crypt. Chaste is constantly evolving and is now being applied to a far wider range of problems. The code provides modules for handling common scientific computing components, such as meshes and solvers for ordinary and partial differential equations (ODEs/PDEs). Re-use of these components avoids the need for researchers to 're-invent the wheel' with each new project, accelerating the rate of progress in new applications. Chaste is developed using industrially-derived techniques, in particular testdriven development, to ensure code quality, re-use and reliability. In this article we provide examples that illustrate the types of problems Chaste can be used to solve, which can be run on a desktop computer. We highlight some scientific studies that have used or are using Chaste, and the insights they have provided. The source code, both for specific releases and the development version, is available to download under an open source Berkeley Software Distribution (BSD) licence a

Cardiac electrophysiology and mechanoelectric feedback : modeling and simulation

Cardiac arrhythmia such as atrial and ventricular fibrillation are characterized by rapid and irregular electrical activity, which may lead to asynchronous contraction and a reduced pump function. Besides experimental and clinical studies, computer simulations are frequently applied to obtain insight in the onset and perpetuation of cardiac arrhythmia. In existing models, the excitable tissue is often modeled as a continuous two-phase medium, representing the intracellular and interstitial domains, respectively. A possible drawback of continuous models is the lack of flexibility when modeling discontinuities in the cardiac tissue. We introduce a discrete bidomain model in which the cardiac tissue is subdivided in segments, each representing a small number of cardiac cells. Active membrane behavior as well as intracellular coupling and interstitial currents are described by this model. Compared with the well-known continuous bidomain equations, our Cellular Bidomain Model is better aimed at modeling the structure of cardiac tissue, in particular anisotropy, myofibers, fibrosis, and gap junction remodeling. An important aspect of our model is the strong coupling between cardiac electrophysiology and cardiomechanics. Mechanical behavior of a single segment is modeled by a contractile element, a series elastic element, and a parallel elastic element. Active force generated by the sarcomeres is represented by the contractile element together with the series elastic element. The parallel elastic element describes mechanical behavior when the segment is not electrically stimulated. Contractile force is related to the intracellular calcium concentration, the sarcomere length, and the velocity of sarcomere shortening. By incorporating the influence of mechanical deformation on electrophysiology, mechanoelectric feedback can be studied. In our model, we consider the immediate influence of stretch on the action potential by modeling a stretch-activated current. Furthermore, we consider the adap- tation of ionic membrane currents triggered by changes in mechanical load. The strong coupling between cardiac electrophysiology and cardiac mechanics is a unique property of our model, which is reflected by its application to obtain more insight in the cause and consequences of mechanical feedback on cardiac electrophysiology. In this thesis, we apply the Cellular Bidomain Model in five different simulation studies to cardiac electrophysiology and mechanoelectric feedback. In the first study, the effect of field stimulation on virtual electrode polarization is studied in uniform, decoupled, and nonuniform cardiac tissue. Field stimulation applied on nonuniform tissue results in more virtual electrodes compared with uniform tissue. Spiral waves can be terminated in decoupled tissue, but not in uniform, homogeneous tissue. By gradually increasing local differences in intracellular conductivities, the amount and spread of virtual electrodes increases and spiral waves can be terminated. We conclude that the clinical success of defibrillation may be explained by intracellular decoupling and spatial heterogeneity present in normal and in pathological cardiac tissue. In the second study, the role of the hyperpolarization-activated inward current If is investigated on impulse propagation in normal and in pathological tissue. The effect of diffuse fibrosis and gap junction remodeling is simulated by reducing cellular coupling nonuniformly. As expected, the conduction velocity decreases when cellular coupling is reduced. In the presence of If, the conduction velocity increases both in normal and in pathological tissue. In our simulations, ectopic activity is present in regions with high expression of If and is facilitated by cellular uncoupling. We also found that an increased If may facilitate propagation of the action potential. Hence, If may prevent conduction slowing and block. Overexpression of If may lead to ectopic activity, especially when cellular coupling is reduced under pathological conditions. In the third study, the influence of the stretch-activated current Isac is investigated on impulse propagation in cardiac fibers composed of segments that are electrically and mechanically coupled. Simulations of homogeneous and inhomogeneous cardiac fibers have been performed to quantify the relation between conduction velocity and Isac under stretch. Conduction slowing and block are related to the amount of stretch and are enhanced by contraction of early-activated segments. Our observations are in agreement with experimental results and explain the large differences in intra-atrial conduction, as well as the increased inducibility of atrial fibrillation in acutely dilated atria. In the fourth study, we investigate the hypothesis that electrical remodeling is triggered by changes in mechanical work. Stroke work is determined for each segment by simulating the cardiac cycle. Electrical remodeling is simulated by adapting the L-type Ca2+ current ICa,L such that a homogeneous distribution of stroke work is obtained. With electrical remodeling, a more homogeneous shortening of the fiber is obtained, while heterogeneity in APD increases and the repolarization wave reverses. These results are in agreement with experimentally observed distributions of strain and APD and indicate that electrical remodeling leads to more homogeneous shortening during ejection. In the fifth study, we investigate the effect of stretch on the vulnerability to AF. The human atria are represented by a triangular mesh obtained from MRI data. To model acute dilatation, overall stretch is applied to the atria. In the presence of Isac, the membrane potential depolarizes, which causes inactivation of the sodium channels and results in conduction slowing or block. Inducibility of AF increases under stretch, which is explained by an increased dispersion in refractory period, conduction slowing, and local conduction block. Our observations explain the large differences in intra-atrial conduction measured in experiments and provide insight in the vulnerability to AF in dilated atria. In conclusion, our model is well-suited to describe cardiac electrophysiology and mechanoelectric feedback. For future applications, the model may be improved by taking into account new insights from cellular physiology, a more accurate geometry, and hemodynamics

Computational modelling of the human heart and multiscale simulation of its electrophysiological activity aimed at the treatment of cardiac arrhythmias related to ischaemia and Infarction

[ES] Las enfermedades cardiovasculares constituyen la principal causa de morbilidad y mortalidad a nivel mundial, causando en torno a 18 millones de muertes cada año. De entre ellas, la más común es la enfermedad isquémica cardíaca, habitualmente denominada como infarto de miocardio (IM). Tras superar un IM, un considerable número de pacientes desarrollan taquicardias ventriculares (TV) potencialmente mortales durante la fase crónica del IM, es decir, semanas, meses o incluso años después la fase aguda inicial. Este tipo concreto de TV normalmente se origina por una reentrada a través de canales de conducción (CC), filamentos de miocardio superviviente que atraviesan la cicatriz del infarto fibrosa y no conductora. Cuando los fármacos anti-arrítmicos resultan incapaces de evitar episodios recurrentes de TV, la ablación por radiofrecuencia (ARF), un procedimiento mínimamente invasivo realizado mediante cateterismo en el laboratorio de electrofisiología (EF), se usa habitualmente para interrumpir de manera permanente la propagación eléctrica a través de los CCs responsables de la TV. Sin embargo, además de ser invasivo, arriesgado y requerir mucho tiempo, en casos de TVs relacionadas con IM crónico, hasta un 50% de los pacientes continúa padeciendo episodios recurrentes de TV tras el procedimiento de ARF. Por tanto, existe la necesidad de desarrollar nuevas estrategias pre-procedimiento para mejorar la planificación de la ARF y, de ese modo, aumentar esta tasa de éxito relativamente baja. En primer lugar, realizamos una revisión exhaustiva de la literatura referente a los modelos cardiacos 3D existentes, con el fin de obtener un profundo conocimiento de sus principales características y los métodos usados en su construcción, con especial atención sobre los modelos orientados a simulación de EF cardíaca. Luego, usando datos clínicos de un paciente con historial de TV relacionada con infarto, diseñamos e implementamos una serie de estrategias y metodologías para (1) generar modelos computacionales 3D específicos de paciente de ventrículos infartados que puedan usarse para realizar simulaciones de EF cardíaca a nivel de órgano, incluyendo la cicatriz del infarto y la región circundante conocida como zona de borde (ZB); (2) construir modelos 3D de torso que permitan la obtención del ECG simulado; y (3) llevar a cabo estudios in-silico de EF personalizados y pre-procedimiento, tratando de replicar los verdaderos estudios de EF realizados en el laboratorio de EF antes de la ablación. La finalidad de estas metodologías es la de localizar los CCs en el modelo ventricular 3D para ayudar a definir los objetivos de ablación óptimos para el procedimiento de ARF. Por último, realizamos el estudio retrospectivo por simulación de un caso, en el que logramos inducir la TV reentrante relacionada con el infarto usando diferentes configuraciones de modelado para la ZB. Validamos nuestros resultados mediante la reproducción, con una precisión razonable, del ECG del paciente en TV, así como en ritmo sinusal a partir de los mapas de activación endocárdica obtenidos invasivamente mediante sistemas de mapeado electroanatómico en este último caso. Esto permitió encontrar la ubicación y analizar las características del CC responsable de la TV clínica. Cabe destacar que dicho estudio in-silico de EF podría haberse efectuado antes del procedimiento de ARF, puesto que nuestro planteamiento está completamente basado en datos clínicos no invasivos adquiridos antes de la intervención real. Estos resultados confirman la viabilidad de la realización de estudios in-silico de EF personalizados y pre-procedimiento de utilidad, así como el potencial del abordaje propuesto para llegar a ser en un futuro una herramienta de apoyo para la planificación de la ARF en casos de TVs reentrantes relacionadas con infarto. No obstante, la metodología propuesta requiere de notables mejoras y validación por medio de es[CA] Les malalties cardiovasculars constitueixen la principal causa de morbiditat i mortalitat a nivell mundial, causant entorn a 18 milions de morts cada any. De elles, la més comuna és la malaltia isquèmica cardíaca, habitualment denominada infart de miocardi (IM). Després de superar un IM, un considerable nombre de pacients desenvolupen taquicàrdies ventriculars (TV) potencialment mortals durant la fase crònica de l'IM, és a dir, setmanes, mesos i fins i tot anys després de la fase aguda inicial. Aquest tipus concret de TV normalment s'origina per una reentrada a través dels canals de conducció (CC), filaments de miocardi supervivent que travessen la cicatriu de l'infart fibrosa i no conductora. Quan els fàrmacs anti-arítmics resulten incapaços d'evitar episodis recurrents de TV, l'ablació per radiofreqüència (ARF), un procediment mínimament invasiu realitzat mitjançant cateterisme en el laboratori de electrofisiologia (EF), s'usa habitualment per a interrompre de manera permanent la propagació elèctrica a través dels CCs responsables de la TV. No obstant això, a més de ser invasiu, arriscat i requerir molt de temps, en casos de TVs relacionades amb IM crònic fins a un 50% dels pacients continua patint episodis recurrents de TV després del procediment d'ARF. Per tant, existeix la necessitat de desenvolupar noves estratègies pre-procediment per a millorar la planificació de l'ARF i, d'aquesta manera, augmentar la taxa d'èxit, que es relativament baixa. En primer lloc, realitzem una revisió exhaustiva de la literatura referent als models cardíacs 3D existents, amb la finalitat d'obtindre un profund coneixement de les seues principals característiques i els mètodes usats en la seua construcció, amb especial atenció sobre els models orientats a simulació de EF cardíaca. Posteriorment, usant dades clíniques d'un pacient amb historial de TV relacionada amb infart, dissenyem i implementem una sèrie d'estratègies i metodologies per a (1) generar models computacionals 3D específics de pacient de ventricles infartats capaços de realitzar simulacions de EF cardíaca a nivell d'òrgan, incloent la cicatriu de l'infart i la regió circumdant coneguda com a zona de vora (ZV); (2) construir models 3D de tors que permeten l'obtenció del ECG simulat; i (3) dur a terme estudis in-silico de EF personalitzats i pre-procediment, tractant de replicar els vertaders estudis de EF realitzats en el laboratori de EF abans de l'ablació. La finalitat d'aquestes metodologies és la de localitzar els CCs en el model ventricular 3D per a ajudar a definir els objectius d'ablació òptims per al procediment d'ARF. Finalment, a manera de prova de concepte, realitzem l'estudi retrospectiu per simulació d'un cas, en el qual aconseguim induir la TV reentrant relacionada amb l'infart usant diferents configuracions de modelatge per a la ZV. Validem els nostres resultats mitjançant la reproducció, amb una precisió raonable, del ECG del pacient en TV, així com en ritme sinusal a partir dels mapes d'activació endocardíac obtinguts invasivament mitjançant sistemes de mapatge electro-anatòmic en aquest últim cas. Això va permetre trobar la ubicació i analitzar les característiques del CC responsable de la TV clínica. Cal destacar que aquest estudi in-silico de EF podria haver-se efectuat abans del procediment d'ARF, ja que el nostre plantejament està completament basat en dades clíniques no invasius adquirits abans de la intervenció real. Aquests resultats confirmen la viabilitat de la realització d'estudis in-silico de EF personalitzats i pre-procediment d'utilitat, així com el potencial de l'abordatge proposat per a arribar a ser en un futur una eina de suport per a la planificació de l'ARF en casos de TVs reentrants relacionades amb infart. No obstant això, la metodologia proposada requereix de notables millores i validació per mitjà d'estudis de simulació amb grans cohorts de pacients.[EN] Cardiovascular diseases represent the main cause of morbidity and mortality worldwide, causing around 18 million deaths every year. Among these diseases, the most common one is the ischaemic heart disease, usually referred to as myocardial infarction (MI). After surviving to a MI, a considerable number of patients develop life-threatening ventricular tachycardias (VT) during the chronic stage of the MI, that is, weeks, months or even years after the initial acute phase. This particular type of VT is typically sustained by reentry through slow conducting channels (CC), which are filaments of surviving myocardium that cross the non-conducting fibrotic infarct scar. When anti-arrhythmic drugs are unable to prevent recurrent VT episodes, radiofrequency ablation (RFA), a minimally invasive procedure performed by catheterization in the electrophysiology (EP) laboratory, is commonly used to interrupt the electrical conduction through the CCs responsible for the VT permanently. However, besides being invasive, risky and time-consuming, in the cases of VTs related to chronic MI, up to 50% of patients continue suffering from recurrent VT episodes after the RFA procedure. Therefore, there exists a need to develop novel pre-procedural strategies to improve RFA planning and, thereby, increase this relatively low success rate. First, we conducted an exhaustive review of the literature associated with the existing 3D cardiac models in order to gain a deep knowledge about their main features and the methods used for their construction, with special focus on those models oriented to simulation of cardiac EP. Later, using a clinical dataset of a chronically infarcted patient with a history of infarct-related VT, we designed and implemented a number of strategies and methodologies to (1) build patient-specific 3D computational models of infarcted ventricles that can be used to perform simulations of cardiac EP at the organ level, including the infarct scar and the surrounding region known as border zone (BZ); (2) construct 3D torso models that enable to compute the simulated ECG; and (3) carry out pre-procedural personalized in-silico EP studies, trying to replicate the actual EP studies conducted in the EP laboratory prior to the ablation. The goal of these methodologies is to allow locating the CCs into the 3D ventricular model in order to help in defining the optimal ablation targets for the RFA procedure. Lastly, as a proof-of-concept, we performed a retrospective simulation case study, in which we were able to induce an infarct-related reentrant VT using different modelling configurations for the BZ. We validated our results by reproducing with a reasonable accuracy the patient's ECG during VT, as well as in sinus rhythm from the endocardial activation maps invasively recorded via electroanatomical mapping systems in this latter case. This allowed us to find the location and analyse the features of the CC responsible for the clinical VT. Importantly, such in-silico EP study might have been conducted prior to the RFA procedure, since our approach is completely based on non-invasive clinical data acquired before the real intervention. These results confirm the feasibility of performing useful pre-procedural personalized in-silico EP studies, as well as the potential of the proposed approach to become a helpful tool for RFA planning in cases of infarct-related reentrant VTs in the future. Nevertheless, the developed methodology requires further improvements and validation by means of simulation studies including large cohorts of patients.During the carrying out of this doctoral thesis, the author Alejandro Daniel López Pérez was financially supported by the Ministerio de Economía, Industria y Competitividad of Spain through the program Ayudas para contratos predoctorales para la formación de doctores, with the grant number BES-2013-064089.López Pérez, AD. (2019). Computational modelling of the human heart and multiscale simulation of its electrophysiological activity aimed at the treatment of cardiac arrhythmias related to ischaemia and Infarction [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/124973TESI