Phenomenological and physiological approaches in the study of cardiac alternans: Theoretical background, mathematical modeling and numerical simulations

This project presents the theoretical background in electrophysiology that is used as a basis in the development of numerical methods for the simulation of heart’s electrical activity. We discuss the mathematical models currently used in cardiac electrophysiology research to simulate the propagation of an action potential wave, and then focus on the description of the ionic currents. This study allows us to anticipate the numerical model sensitivity with regard to the discretization, with the aim of recovering the physiological behaviors of the tissue (i.e. conduction velocity, electrical restitution, etc). Thereafter, we introduce the pseudo-ECG signal reconstruction method from the action potential map. This leads us to examine the approximation of such a signal using the finite element basis coming from the discretization of the monodomain equations. We discuss then a particular cardiac rhythm disorder: action potential alternans. In a second step, we perform an extended simulation study in order to validate the previous numerical methods. We carefully check the convergence of the conduction velocity with respect to the mesh size, and we bring out the advantage of phenomenological approach in terms of electrical restitutions and computational cost. We reproduce typical rhythm disorders and show that the pseudo-ECG signal is able to detect them. Finally, we discuss the influence of anisotropy in cardiac wave propagation and alternans development, characterizing the effect of fibers direction and pacing site in the resulting spatio-temporal distribution.

Part load flow in radial centrifugal pumps

Olivier Braun

Centrifugal pumps are required to sustain a stable operation of the system they support under all operating conditions. Minor modifications of the surfaces defining the pump's water passage can influence the tendency to unstable system operation significantly. The action of such modifications on the flow are yet not fully understood, leading to costly trial and error approaches in the solution of instability problems. The part-load flow in centrifugal pumps is inherently time-dependent due to the interaction of the rotating impeller with the stationary diffuser (Rotor-Stator Interaction, RSI). Furthermore, adverse pressure gradients in the pump diffuser may cause flow separation, potentially inducing symmetry-breaking non-uniformities, either spatially stationary or rotating and either steady or intermittent. Rotating stall, characterized by the presence of distinct cells of flow separation on the circumference, rotating at a fraction of the impeller revolution rate, has been observed in thermal and hydraulic turbomachines. Due to its complexity, the part-load flow in radial centrifugal pumps is a major challenge for numerical flow simulation methods. The present study investigates the part-load flow in radial centrifugal pumps and pump-turbines by experimental and numerical methods, the latter using a finite volume discretization of the Reynolds-averaged Navier-Stokes (RANS) equation. Physical phenomena of part load flow are evidenced based on three case studies, and the ability of numerical simulation methods to reproduce part-load flow in radial centrifugal pumps qualitatively and quantitatively is assessed. A numerical study of the flow in a high specific speed radial pump-turbine using steady approaches and the hypothesis of angular periodicity between neighboring blade channels evidences the relation of sudden flow topology changes with an increase of viscous losses, impacting on the energy-discharge characteristic, and thus increasing the risk of unstable operation. When the flow rate drops below a critical threshold, the straight through-flow with flow separation zones attached to the guide vanes changes to an asymmetrical flow. Energy is drawn off the mean flow and dissipated in a large vortex-like structure. Besides flow separation in some diffuser channels, time-dependent numerical simulations of the flow in a double suction pump evidence a flow rate imbalance between both impeller sides interacting with asymmetric flow separation in the diffuser. Viscous losses increase substantially as this imbalance occurs, the resulting segment of positive slope in the energy-discharge characteristic is found for a flow rate sensibly different from measurements. Different modes of rotating stall are identified by transient pressure measurements in a low-specific-speed pump-turbine, showing 3 to 5 zones of separated flow, rotating at 0.016 to 0.028 times impeller rotation rate, depending on discharge. For operating conditions where stall with 4 cells is most pronounced, velocity is measured by Laser-Doppler methods at locations of interest. The velocity field is reconstructed with respect to the passage of stall cells by definition of a stall phase obtained from simultaneous transient pressure measurements. Time-dependent numerical simulation predicting rotating stall with 4 cells shows velocity fields that are in reasonable agreement with the measured velocity fields, but occurring at a sensibly higher flow rate than found from experiments. In consideration of the quantitative shortcomings of the numerical simulation, a novel modelling approach is proposed: Replacing the costly 3-dimensional simulation of the major part of the impeller channels by a 1-dimensional model allows a significant economy in computational resources, allowing an improved modeling for the remainder of the domain at constant computational cost. The model is validated with the challenging cases of rotating stall and impeller side flow rate imbalance. The satisfying coherence of the results with the simulation including the entire impeller channels qualifies this approach for numerous turbomachinery applications. It could also provide improved, time-dependent boundary conditions for draft tube vortex rope simulations at reasonable computational cost. Parameter studies modifying deliberately some quantities of mean flow and turbulence at the modeled boundary surfaces can be implemented in the framework of the method.

EPFL2009

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

Toni Mikael Lassila, Alfio Quarteroni, Simone Rossi, Ricardo Esteban Ruiz Baier

Mathematical modeling 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 modeling 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. (C) 2016 Elsevier B.V. All rights reserved.

Elsevier Science Sa2017

MATHICSE Technical Report : Integrated Heart – Coupling multiscale and multiphysics models for the simulation of the cardiac function

Toni Mikael Lassila, Alfio Quarteroni, Simone Rossi, Ricardo Esteban Ruiz Baier

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 haemodynamics 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 haemodynamics during rapid valve opening and closure.

MATHICSE2016

Part load flow in radial centrifugal pumps

Olivier Braun

EPFL2009