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

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

Integrated Heart - Coupling multiscale and multiphysics models for the simulation of the cardiac function

Mathematical modelling of the human heart and its function can expand our understanding of various cardiac diseases, which remain the most common cause of death in the developed world. Like other physiological systems, the heart can be understood as a complex multiscale system involving interacting phenomena at the molecular, cellular, tissue, and organ levels. This article addresses the numerical modelling of many aspects of heart function, including the interaction of the cardiac electrophysiology system with contractile muscle tissue, the sub-cellular activation-contraction mechanisms, as well as the hemodynamics inside the heart chambers. Resolution of each of these sub-systems requires separate mathematical analysis and specially developed numerical algorithms, which we review in detail. By using specific sub-systems as examples, we also look at systemic stability, and explain for example how physiological concepts such as microscopic force generation in cardiac muscle cells, translate to coupled systems of differential equations, and how their stability properties influence the choice of numerical coupling algorithms. Several numerical examples illustrate three fundamental challenges of developing multiphysics and multiscale numerical models for simulating heart function, namely: (i) the correct upscaling from single-cell models to the entire cardiac muscle, (ii) the proper coupling of electrophysiology and tissue mechanics to simulate electromechanical feedback, and (iii) the stable simulation of ventricular hemodynamics during rapid valve opening and closure

Impact of uncertainties in cardiac mechanics simulations

Modeling the mechanics of the heart have led to considerable insights, but it still representes a complex and demanding computational problem, especially in a strongly coupled electromechanical setting. Passive cardiac tissue is commonly modeled as a hyperelastic, near-incompressible and orthotropic material, which are properties very challenging for the numerical solution of the model. In particular, near-incompressibility is known to cause numerical issues. In this work, some improvements were done in a cardiac mechanics simulator in order to be more efficient in the treatment of these numerical issues. With the improved solver for cardiac mechanics, it was possible to run problems with higher computational cost, such as sensitivity and uncertainty quantification analyses. This type of analysis has been a topic of scientific interest to assess the possibility of translating patient-specific simulations to clinical applications. However, personalized simulations are still challenging problems, because of the wide biological variability among patients, the uncertainties in experimental measurements and in the geometric representation of the heart. Due to these uncertainties in model inputs, it is difficult to define a reliable model that can be translated to clinical applications. Recent studies have focused on quantifying uncertainties for cardiac models in order to investigate how they can influence simulation results and, consequently, how we can make the models more reliable. Then, the present work also quantifies how uncertainties in the geometry can impact in quantities of interest from cardiac mechanics. The polynomial chaos approach was used to quantify uncertainties in geometries of the left ventricle during cardiac mechanics simulations. Initially, we performed some studies using simplified geometries during ventricular filling phase simulations and, after, we quantify uncertainties in more realistic geometries during the full cardiac cycle.A modelagem da mecânica cardíaca tem levado a descobertas interessantes, porém este continua sendo um problema complexo e de alta demanda computacional, especialmente em modelos eletromecânicos fortemente acoplados. O tecido cardíaco é geralmente considerado como um material hiperelástico, quase incompressível e ortotrópico, fatores que dificultam a solução numérica do modelo. Neste trabalho, melhorias foram realizadas em um simulador da mecânica cardíaca para tratar tais problemas numéricos de forma mais eficiente. Com este simulador mais eficiente foi possível tratar problemas que demandam de um maior esfoço computacional, como as análises de sensibilidade e quantificação de incertezas, onde várias simulações precisam ser realizadas. Este tipo de análise tem sido tópico de interesse científico para avaliar a possibilidade de usar simulações personalizadas por paciente em aplicações clínicas. Porém, estas simulações ainda são problemas desafiadores, por causa da grande variabilidade biológica entre pacientes e das incertezas em medidas experimentais e em representações geométricas do coração. Devido a estas incertezas em entradas do modelo, é difícil definir um modelo confiável que possa ser usado em aplicações clínicas. Estudos recentes têm se voltado à investigação de como estas incertezas podem influenciar no resultado de simulações e, consequentemente, descobrir como tornar os modelos mais confiáveis. Então, o presente trabalho quantifica incertezas nas geometrias usadas nas simulações para investigar como quantidades de interesse da mecânica cardíaca podem ser afetadas. A abordagem do polinômio caos é utilizada para a quantificação de incertezas em geometrias do ventrículo esquerdo submetidas a simulações da mecânica cardíaca. Inicialmente, as análises foram realizadas usando geometrias simplificadas em simulações da fase de preenchimento ventricular e, posteriormente, análises de quantificação de incertezas em geometrias mais realísticas submetidas a simulações do ciclo cardíaco completo são realizadas.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superio

Efficient Computational Methods for Strongly Coupled Cardiac Electromechanics

Strongly coupled cardiac electromechanical models can further our understanding of the relative importance of feedback mechanisms in the heart, but computational challenges currently remain a major obstacle, which limit their widespread use. To address this issue, we present a set of efficient computational methods including an efficient adaptive cell model integration scheme and a solution method for the monodomain equations that maintains high conduction velocity for time steps greater than 0.1 ms. We also present a novel method for increasing the efficiency of simulating electromechanical coupling, ..

Efficient computational methods for strongly coupled cardiac electromechanics

Strongly coupled cardiac electromechanical models can further our understanding of the relative importance of feedback mechanisms in the heart, but computational challenges currently remain a major obstacle, which limit their widespread use. To address this issue, we present a set of efficient computational methods including an efficient adaptive cell model integration scheme and a solution method for the monodomain equations that maintains high conduction velocity for time steps greater than 0.1 ms. We also present a novel method for increasing the efficiency of simulating electromechanical coupling, ..

Electromechanical large scale computational models of the ventricular myocardium

Els models computacionals del cor són una eina important que pot donar als investigadors biomèdics una font addicional d’informació per entendre el funcionament del miocardi. Els models numèrics poden ajudar a interpretar dades experimentals i proporcionar informació complementària sobre mecanismes cardíacs que no poden ser determinats amb precisió mitjançant dispositius clínics clàssics. En aquesta tesi, s’apliquen tècniques de computació a gran escala per construir una eina computacional capaç d’executar-se en paral•lel en milers de processadors, permetent simulacions d’alta fidelitat en malles fines. Per simular el bombeig del cor, s’utilitza un esquema d’acoblament explícit entre les equacions electrofisiològiques en tres dimensions i la formulació en mecànica de sòlids. Per trobar la solució numèrica, s’utilitza el mètode d’elements finits. A més, s’implementen tècniques en assimilació de dades per a l’estimació efectiva dels paràmetres electrofisiològics i mecànics rellevants que apareixen a les equacions, la qual cosa ´es un pas crucial cap a un model cardíac sensible a cada pacient. El codi computacional s’aplica per simular problemes físics reals. S’estudia la propagació electromecànica en una geometria de conill, on es prova la sensibilitat del model a les variacions d’entrada. En particular, l’eina de càlcul s’utilitza per avaluar la influència del camp de fibres cardíaques en la contracció del teixit. Per desenvolupar una simulació cardíaca útil per a fins clínics, el model requereix la integració i combinació de la mecànica computacional i les tècniques de processament d’imatge més recents. El model resultant pot ser la base d’estudis teòrics sobre mecanismes de patologies, oferint als investigadors i cardiòlegs pistes addicionals per comprendre el funcionament del cor. Pot ajudar a la planificació de cirurgia i modelització, com és la predicció dels efectes de compostos farmacològics en el ritme cardíac o l’estudi de l’efecte de medicaments. Aquest projecte només és possible en un equip multidisciplinar, on grups especialitzats uneixen les seves forces en les respectives disciplines: cardiòlegs, investigadors imatge, bioenginyers i científics de la computació. El present model computacional del cor és un pas més cap a la creació d’un laboratori cardíac virtual.A cardiac computational model is a relevant tool that can give biomedical researchers an additional source of information to understand how the heart works. Numerical models can help to interpret experimental data and provide information about cardiac mechanisms that can not be determined accurately by classical clinical devices. In this thesis, High Performance Computing (HPC) techniques are used to build a cardiac computational tool, which is capable of running in parallel in thousands of processors, bioengineers and computational scientists. The present cardiac computational model is one further step towards the creation of a virtual lab, allowing high fidelity simulations on fine meshes. To simulate the pumping heart, an explicit coupling scheme between the three-dimensional electrophysiological equations and the solid mechanics formulation is used, solving the governing equations with finite element methods. Also, data assimilation techniques are implemented for the effective estimation of some relevant electrophysiological parameters, which is a crucial step towards the patient-sensitive cardiac model. The data assimilation techniques are assessed on synthetic data generated by the model. Finally, the computational code is applied to simulate real physical problems. The electromechanical propagation in a rabbit geometry is studied to test the sensitivity of the framework to input variations. Particularly, the computational tool is used to evaluate the influence of the fiber field in the contraction of the tissue. To develop a cardiac simulation useful for clinical purposes, the integrative model requires combining computational mechanics and image processing techniques via data assimilation methods. Coupled with the most advanced image processing analysis, the framework can be the base of theoretical studies into the mechanisms of cardiac pathologies. It can help surgery planning and cardiac modeling, such as the prediction of the impact of pharmacological compounds on the heart’s rhythm or to improve the knowledge of drug study, giving medical researchers additional hints to understand the heart. This realization is only possible in a multidisciplinary team, where specialized groups join forces in their respective disciplines: cardiologists, image researchers, bioengineers and computational scientists. The present cardiac computational model is one further step towards the creation of a virtual la

Fluid-electro-mechanical model of the human heart for supercomputers

The heart is a complex system. From the transmembrane cell activity to the spatial organization in helicoidal fibers, it includes several spatial and temporal scales. The heart muscle is surrounded by two main tissues that modulate how it deforms: the pericardium and the blood. The former constrains the epicardial surface and the latter exerts a force in the endocardium. The main function of this peculiar muscle is to pump blood to the pulmonary and systemic circulations. In this way, solid dynamics of the heart is as important as the induced fluid dynamics. Despite the work done in computational research of multiphysics heart modelling, there is no reference of a tightly-coupled scheme that includes electrophysiology, solid and fluid mechanics in a whole human heart. In this work, we propose, develop and test a fluid-electro-mechanical model of the human heart. To start, the heartbeat phenomenon is disassembled in the different composing problems. The first building block is the electrical activity of the myocytes, that induces the mechanical deformation of the myocardium. The contraction of the muscle reduces the intracavitary space, that pushes out the contained blood. At the same time, the inertia, pressure and viscous stresses in this fluid exerts a force on the solid wall. In this way, we can understand the heart as a fluid-electro-mechanical problem. All the models are implemented in Alya, the Barcelona Supercomputing Center simulation software. A multi-code approach is used, splitting the problem in a solid and a fluid domain. In the former, electrophysiology coupled with solid mechanics are solved. In the later, fluid dynamics in an arbitrary Lagrangian-Eulerian domain are computed. The equations are spatially discretized using the finite element method and temporally discretized using finite differences. Facilitated by the multi-code approach, a novel high performance quasi-Newton method is developed to deal with the intrinsic issues of fluid-structure interaction problems in iomechanics. All the schemes are optimized to run in massively parallel computers. A wide range of experiments are shown to validate, test and tune the numerical model. The different hypothesis proposed — as the critical effect of the atrium or the presence of pericardium — are also tested in these experiments. Finally, a normal heartbeat is simulated and deeply analyzed. This healthy computational heart is first diseased with a left bundle branch block. After this, its function is restored simulating a cardiac resynchronization therapy. Then, a third grade atrioventricular block is simulated in the healthy heart. In this case, the pathologic model is treated with a minimally invasive leadless intracardiac pacemaker. This requires to include the device in the geometrical description of the problem, solve the structural problem with the tissue, and the fluid-structure interaction problem with the blood. As final experiment, we test the parallel performance of the coupled solver. In the cases mentioned above, the results are qualitatively compared against experimental measurements, when possible. Finally, a first glance in a coupled fluid-electro-mechanical cardiovascular system is shown. This model is build adding a one dimensional model of the arterial network created by the Laboratório Nacional de Computação Científica in Petropolis, Brasil. Despite the artificial geometries used, the outflow curves are comparable with physiological observations. The model presented in this thesis is a step towards the virtual human heart. In a near future computational models like the presented in this thesis will change how pathologies are understood and treated, and the way biomedical devices are designed.El corazón es un sistema complejo. Desde la actividad celular hasta la organización espacial en fibras helicoidales, incluye gran cantidad de escalas espaciales y temporales. El corazón está rodeado principalmente por dos tejidos que modulan su deformación: el pericardio y la sangre. El primero restringe el movimiento del epicardio, mientras el segundo ejerce fuerza sobre el endocardio. La función principal de este músculo es bombear sangre a la circulación sistémica y a la pulmonar. Así, la deformación del miocardio es tan importante como la fluidodinámica inducida. Al día de hoy, solo se han propuesto modelos parciales del corazón. Ninguno de los modelos publicados resuelve electrofisiología, mecánica del sólido, y dinámica de fluidos en una geometría completa del corazón. En esta tesis, proponemos, desarrollamos y probamos un modelo fluido -electro -mecánico del corazón. Primero, el problema del latido cardíaco es descompuesto en los distintos subproblemas. El primer bloque componente es la actividad eléctrica de los miocitos, que inducen la deformación mecánica del miocardio. La contratación de este músculo, reduce el espacio intracavitario, que empuja la sangre contenida. Al mismo tiempo, la inercia, presión y fuerzas viscosas del fluido inducen una presión sobre la pared del sólido. De esta manera, podemos entender el latido cardíaco como un problema fluido-electro-mecánico. Los modelos son implementados en Alya, el software de simulación del Barcelona Supercomputing Center. Se utiliza un diseño multi-código, separando el problema según el dominio en sólido y fluido. En el primero, se resuelve electrofisiología acoplado con mecánica del sólido. En el segundo, fluido dinámica en un dominio arbitrario Lagrangiano-Euleriano. Las ecuaciones son discretizadas espacial y temporalmente utilizando elementos finitos y diferencias finitas respectivamente. Facilitado por el diseño multi-codigo, se desarrolló un novedoso método quasi-Newton de alta performance, pensado específicamente para lidiar con los problemas intrínsecos de interacción fluido-estructura en biomecánica. Todos los esquemas fueron optimizados para correr en ordenadores masivamente paralelos.Se presenta un amplio espectro de experimentos con el fin de validar, probar y ajustar el modelo numérico. Las diferentes hipótesis propuestas tales como el efecto producido por la presencia de las aurículas o el pericardio son también demostradas en estos experimentos. Finalmente un latido normal es simulado y sus resultados son analizados con profundidad. El corazón computacional sano es, primeramente enfermado de un bloqueo de rama izquierda. Posteriormente se restaura la función normal mediante la terapia de resincronización cardíaca. Luego se afecta al corazón de un bloqueo atrioventricular de tercer grado. Esta patología es tratada mediante la implantación de un marcapasos intracardíaco. Para esto, se requiere incluir el dispositivo en la descripción geométrica, resolver el problema estructural con el tejido y la interacción fluido-estructura con la sangre. Como experimento numérico final, se prueba el desempeño paralelo del modelo acoplado.Finalmente, se muestran resultados preliminares para un modelo fluido-electro-mecánico del sistema cardiovascular. Este modelo se construye agregando un modelo unidimensional del árbol arterial. A pesar de las geometrías artificiales usadas, la curva de flujo en la raíz aórtica es comparable con observaciones experimentales. El modelo presentado aquí representa un avance hacia el humano virtual. En un futuro, modelos similares, cambiarán la forma en la que se entienden y tratan las enfermedades y la forma en la que los dispositivos biomédicos son diseñados