Identification of weakly coupled multiphysics problems. Application to the inverse problem of electrocardiography

This work addresses the inverse problem of electrocardiography from a new perspective, by combining electrical and mechanical measurements. Our strategy relies on the defini-tion of a model of the electromechanical contraction which is registered on ECG data but also on measured mechanical displacements of the heart tissue typically extracted from medical images. In this respect, we establish in this work the convergence of a sequential estimator which combines for such coupled problems various state of the art sequential data assimilation methods in a unified consistent and efficient framework. Indeed we ag-gregate a Luenberger observer for the mechanical state and a Reduced Order Unscented Kalman Filter applied on the parameters to be identified and a POD projection of the electrical state. Then using synthetic data we show the benefits of our approach for the estimation of the electrical state of the ventricles along the heart beat compared with more classical strategies which only consider an electrophysiological model with ECG measurements. Our numerical results actually show that the mechanical measurements improve the identifiability of the electrical problem allowing to reconstruct the electrical state of the coupled system more precisely. Therefore, this work is intended to be a first proof of concept, with theoretical justifications and numerical investigations, of the ad-vantage of using available multi-modal observations for the estimation and identification of an electromechanical model of the heart

Numerical simulation of electrocardiograms for full cardiac cycles in healthy and pathological conditions

This work is dedicated to the simulation of full cycles of the electrical activity of the heart and the corresponding body surface potential. The model is based on a realistic torso and heart anatomy, including ventricles and atria. One of the specificities of our approach is to model the atria as a surface, which is the kind of data typically provided by medical imaging for thin volumes. The bidomain equations are considered in their usual formulation in the ventricles, and in a surface formulation on the atria. Two ionic models are used: the Courtemanche-Ramirez-Nattel model on the atria, and the "Minimal model for human Ventricular action potentials" (MV) by Bueno-Orovio, Cherry and Fenton in the ventricles. The heart is weakly coupled to the torso by a Robin boundary condition based on a resistor- capacitor transmission condition. Various ECGs are simulated in healthy and pathological conditions (left and right bundle branch blocks, Bachmann's bundle block, Wolff-Parkinson-White syndrome). To assess the numerical ECGs, we use several qualitative and quantitative criteria found in the medical literature. Our simulator can also be used to generate the signals measured by a vest of electrodes. This capability is illustrated at the end of the article

Using parametric model order reduction for inverse analysis of large nonlinear cardiac simulations

Predictive high-fidelity finite element simulations of human cardiac mechanics commonly require a large number of structural degrees of freedom. Additionally, these models are often coupled with lumped-parameter models of hemodynamics. High computational demands, however, slow down model calibration and therefore limit the use of cardiac simulations in clinical practice. As cardiac models rely on several patient-specific parameters, just one solution corresponding to one specific parameter set does not at all meet clinical demands. Moreover, while solving the nonlinear problem, 90% of the computation time is spent solving linear systems of equations. We propose to reduce the structural dimension of a monolithically coupled structure-Windkessel system by projection onto a lower-dimensional subspace. We obtain a good approximation of the displacement field as well as of key scalar cardiac outputs even with very few reduced degrees of freedom, while achieving considerable speedups. For subspace generation, we use proper orthogonal decomposition of displacement snapshots. Following a brief comparison of subspace interpolation methods, we demonstrate how projection-based model order reduction can be easily integrated into a gradient-based optimization. We demonstrate the performance of our method in a real-world multivariate inverse analysis scenario. Using the presented projection-based model order reduction approach can significantly speed up model personalization and could be used for many-query tasks in a clinical setting

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.Postprint (published version

On the identification of multiple space dependent ionic parameters in cardiac electrophysiology modelling

In this paper, we consider the inverse problem of space dependent multiple ionic parameters identification in cardiac electrophysiology modelling from a set of observations. We use the monodomain system known as a state-of-the-art model in cardiac electrophysiology and we consider a general Hodgkin-Huxley formalism to describe the ionic exchanges at the microscopic level. This formalism covers many physiological transmembrane potential models including those in cardiac electrophysiology. Our main result is the proof of the uniqueness and a Lipschitz stability estimate of ion channels conductance parameters based on some observations on an arbitrary subdomain. The key idea is a Carleman estimate for a parabolic operator with multiple coefficients and an ordinary differential equation system

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