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Register a new userA cardiac electromechanics model coupled with a lumped parameters model for closed-loop blood circulation. Part II: numerical approximation
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Added by
Matteo Salvador
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In the framework of accurate and efficient segregated schemes for 3D cardiac electromechanics and 0D cardiovascular models, we propose here a novel numerical approach to address the coupled 3D-0D problem introduced in Part I of this two-part series of papers. We combine implicit-explicit schemes to solve the different cardiac models in a multiphysics setting. We properly separate and manage the different time and space scales related to cardiac electromechanics and blood circulation. We employ a flexible and scalable intergrid transfer operator that enables to interpolate Finite Element functions among different meshes and, possibly, among different Finite Element spaces. We propose a numerical method to couple the 3D electromechanical model and the 0D circulation model in a numerically stable manner within a fully segregated fashion. No adaptations are required through the different phases of the heartbeat. We also propose a robust algorithm to reconstruct the stress-free reference configuration. Due to the computational cost associated with the numerical solution of this inverse problem, the reference configuration recovery algorithm comes along with a novel projection technique to precisely recover the unloaded geometry from a coarser representation of the computational domain. We show the convergence property of our numerical schemes by performing an accuracy study through grid refinement. To prove the biophysical accuracy of our computational model, we also address different scenarios of clinical interest in our numerical simulations by varying preload, afterload and contractility. Indeed, we simulate physiologically relevant behaviors and we reproduce meaningful results in the context of cardiac function.
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We propose an integrated electromechanical model of the human heart, with focus on the left ventricle, wherein biophysically detailed models describe the different physical phenomena concurring to the cardiac function. We model the subcellular generation of active force by means of an Artificial Neural Network, which is trained by a suitable Machine Learning algorithm from a collection of pre-computed numerical simulations of a biophysically detailed, yet computational demanding, high-fidelity model. To provide physiologically meaningful results, we couple the 3D electromechanical model with a closed-loop 0D (lumped parameters) model describing the blood circulation in the whole cardiovascular network. We prove that the 3D-0D coupling of the two models is compliant with the principle of energy conservation, which is achieved in virtue of energy-consistent boundary conditions that account for the interaction among cardiac chambers within the computational domain, pericardium and surrounding tissue. We thus derive an overall balance of mechanical energy for the 3D-0D model. This provides a quantitative insight into the energy utilization, dissipation and transfer among the different compartments of the cardiovascular network and during different stages of the heartbeat. In virtue of this new model and the energy balance, we propose a new validation tool of heart energy usage against relationships used in the daily clinical practice. Finally, we provide a mathematical formulation of an inverse problem aimed at recovering the reference configuration of one or multiple cardiac chambers, starting from the stressed configuration acquired from medical imaging. This is fundamental to correctly initialize electromechanical simulations. Numerical methods and simulations of the 3D-0D model will be detailed in Part II.
Two crucial factors for accurate numerical simulations of cardiac electromechanics, which are also essential to reproduce the synchronous activity of the heart, are: i) accounting for the interaction between the heart and the circulatory system that determines pressures and volumes loads in the heart chambers; ii) reconstructing the muscular fiber architecture that drives the electrophysiology signal and the myocardium contraction. In this work, we present a 3D biventricular electromechanical model coupled with a 0D closed-loop model of the whole cardiovascular system that addresses the two former crucial factors. With this aim, we introduce a boundary condition for the mechanical problem that accounts for the neglected part of the domain located on top of the biventricular basal plane and that is consistent with the principles of momentum and energy conservation. We also discuss in detail the coupling conditions that stand behind the 3D and the 0D models. We perform electromechanical simulations in physiological conditions using the 3D-0D model and we show that our results match the experimental data of relevant mechanical biomarkers available in literature. Furthermore, we investigate different arrangements in cross-fibers active contraction. We prove that an active tension along the sheet direction counteracts the myofiber contraction, while the one along the sheet-normal direction enhances the cardiac work. Finally, several myofiber architectures are analysed. We show that a different fiber field in the septal area and in the transmural wall effect the pumping functionality of the left ventricle.
In this paper, we propose a direct Eulerian generalized Riemann problem (GRP) scheme for a blood flow model in arteries. It is an extension of the Eulerian GRP scheme, which is developed by Ben-Artzi, et. al. in J. Comput. Phys., 218(2006). By using the Riemann invariants, we diagonalize the blood flow system into a weakly coupled system, which is used to resolve rarefaction wave. We also use Rankine-Hugoniot condition to resolve the local GRP formulation. We pay special attention to the acoustic case as well as the sonic case. The extension to the two dimensional case is carefully obtained by using the dimensional splitting technique. We test that the derived GRP scheme is second order accuracy.
A new two-dimensional model for blood flows in arteries with arbitrary cross sections is derived. The model consists of a system of balance laws for conservation of mass and balance of momentum in the axial and angular directions. The equations are derived by applying asymptotic analysis to the incompressible Navier-Stokes equations in narrow, large vessels and integrating in the radial direction in each cross section. The main properties of the system are discussed and a positivity-preserving well-balanced central-upwind scheme is presented. The merits of the scheme will be tested in a variety of scenarios. In particular, numerical results of simulations using an idealized aorta model are shown. We analyze the time evolution of the blood flow under different initial conditions such as perturbations to steady states consisting of a bulging in the vessels wall. We consider different situations given by distinct variations in the vessels elasticity.
Performing analysis, optimization and control using simulations of many-particle systems is computationally demanding when no macroscopic model for the dynamics of the variables of interest is available. In case observations on the macroscopic scale can only be produced via legacy simulator code or live experiments, finding a model for these macroscopic variables is challenging. In this paper, we employ time-lagged embedding theory to construct macroscopic numerical models from output data of a black box, such as a simulator or live experiments. Since the state space variables of the constructed, coarse model are dynamically closed and observable by an observation function, we call these variables closed observables. The approach is an online-offline procedure, as model construction from observation data is performed offline and the new model can then be used in an online phase, independent of the original. We illustrate the theoretical findings with numerical models constructed from time series of a two-dimensional ordinary differential equation system, and from the density evolution of a transport-diffusion system. Applicability is demonstrated in a real-world example, where passengers leave a train and the macroscopic model for the density flow onto the platform is constructed with our approach. If only the macroscopic variables are of interest, simulation runtimes with the numerical model are three orders of magnitude lower compared to simulations with the original fine scale model. We conclude with a brief discussion of possibilities of numerical model construction in systematic upscaling, network optimization and uncertainty quantification.
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