1,721,071 research outputs found
An oscillation-free fully staggered algorithm for velocity-dependent active models of cardiac mechanics
Full text linkIn this paper we address an unresolved problem in the numerical modeling of cardiac electromechanics, that is the onset of numerical oscillations due to the dependence of force generation models on the fibers shortening velocity. A way to avoid numerical oscillations is to use monolithic schemes for the solution of the coupled problem of active-passive mechanics. However, staggered strategies, which foresee the sequential solution of the models of force generation and of tissue mechanics, are preferable, due to their reduced computational cost and low implementation effort. In this paper we propose a cure for this issue, by introducing, with respect to the standard staggered scheme, a numerically consistent stabilization term. This term is derived in virtue of the identification of the cause of instability in the mismatch between macroscopic and microscopic strains, inconsistently expressed in Lagrangian and Eulerian coordinates, respectively. By considering a model problem of active mechanics we prove that the proposed scheme is unconditionally absolutely stable (i.e. it is stable for any time step size), yet within a fully staggered framework. As such, the new scheme removes the non-physical oscillations, as we prove by applying it to three force generation models, namely the Niederer-Hunter-Smith model, the model by Land and coworkers, and the mean-field force generation model that we have recently proposed. (C) 2020 The Author(s). Published by Elsevier B.V.CMC
Accelerating the convergence to a limit cycle in 3D cardiac electromechanical simulations through a data-driven 0D emulator
Full text linkThe results of numerical simulations of cardiac electromechanics are typically characterized by a long transient before reaching a periodic solution known as limit cycle. This yields a serious computational overhead, as the only clinically relevant output is associated with such limit cycle. To accelerate the convergence to the limit cycle, we propose a strategy based on a surrogate model, wherein the computationally demanding 3D components are replaced by a 0D emulator, built through an automated data-driven algorithm on the basis of pressurevolume transients of as few as three heartbeats simulated with the 3D model. The 0D emulator, consisting of a time-dependent pressure-volume relationship, can provide the 3D model with an initial guess, such that in just two heartbeats a solution is reached that is as close to the limit cycle as the one obtained after more than 20 heartbeats with the 3D model. The 0D emulator is also recommended in many-query settings (e.g. when performing sensitivity analysis, parameter estimation and uncertainty quantification), that call for the repeated solution of the model for different values of the parameters. Indeed, the construction of the emulator does not have to be repeated when the parameters of the circulation model it is coupled with vary. Finally, should the parameters of the 3D electromechanical model vary as well, we propose a parametric emulator, obtained by interpolation of emulators constructed for given values of the parameters. This paper is accompanied by a Python library implementing the proposed algorithm, open to integration with existing cardiac solvers.CMC
Combining data assimilation and machine learning to build data-driven models for unknown long time dynamics—Applications in cardiovascular modeling
Full text linkWe propose a method to discover differential equations describing the long-term dynamics of phenomena featuring a multiscale behavior in time, starting from measurements taken at the fast-scale. Our methodology is based on a synergetic combination of data assimilation (DA), used to estimate the parameters associated with the known fast-scale dynamics, and machine learning (ML), used to infer the laws underlying the slow-scale dynamics. Specifically, by exploiting the scale separation between the fast and the slow dynamics, we propose a decoupling of time scales that allows to drastically lower the computational burden. Then, we propose a ML algorithm that learns a parametric mathematical model from a collection of time series coming from the phenomenon to be modeled. Moreover, we study the interpretability of the data-driven models obtained within the black-box learning framework proposed in this paper. In particular, we show that every model can be rewritten in infinitely many different equivalent ways, thus making intrinsically ill-posed the problem of learning a parametric differential equation starting from time series. Hence, we propose a strategy that allows to select a unique representative model in each equivalence class, thus enhancing the interpretability of the results. We demonstrate the effectiveness and noise-robustness of the proposed methods through several test cases, in which we reconstruct several differential models starting from time series generated through the models themselves. Finally, we show the results obtained for a test case in the cardiovascular modeling context, which sheds light on a promising field of application of the proposed methods
Combining physics-based and data-driven models: advancing the frontiers of research with scientific machine learning
Full text linkScientific Machine Learning (SciML) is a recently emerged research field which combines physics-based and data-driven models for the numerical approximation of differential problems. Physics-based models rely on the physical understanding of the problem at hand, subsequent mathematical formulation, and numerical approximation. Data-driven models instead aim to extract relations between input and output data without arguing any causality principle underlining the available data distribution. In recent years, data-driven models have been rapidly developed and popularised. Such a diffusion has been triggered by a huge availability of data (the so-called big data), an increasingly cheap computing power, and the development of powerful Machine Learning (ML) algorithms. SciML leverages the physical awareness of physics-based models and, at the same time, the efficiency of data-driven algorithms. With SciML, we can inject physics and mathematical knowledge into ML algorithms. Yet, we can rely on data-driven algorithms' capability to discover complex and nonlinear patterns from data and improve the descriptive capacity of physics-based models. After recalling the mathematical foundations of digital modelling and ML algorithms, and presenting the most popular ML architectures, we discuss the great potential of a broad variety of SciML strategies in solving complex problems governed by Partial Differential Equations (PDEs). Finally, we illustrate the successful application of SciML to the simulation of the human cardiac function, a field of significant socio-economic importance that poses numerous challenges on both the mathematical and computational fronts. The corresponding mathematical model is a complex system of nonlinear ordinary and PDEs describing the electromechanics, valve dynamics, blood circulation, perfusion in the coronary tree, and torso potential. Despite the robustness and accuracy of physics-based models, certain aspects, such as unveiling constitutive laws for cardiac cells and myocardial material properties, as well as devising efficient reduced-order models to dominate the extraordinary computational complexity, have been successfully tackled by leveraging data-driven models
Reconstructing relaxed configurations in elastic bodies: Mathematical formulations and numerical methods for cardiac modeling
Full text linkModeling the behavior of biological tissues and organs often necessitates the knowledge of their shape in the absence of external loads. However, when their geometry is acquired in-vivo through imaging techniques, bodies are typically subject to mechanical deformation due to the presence of external forces, and the load-free configuration needs to be reconstructed. This paper addresses this crucial and frequently overlooked topic, known as the inverse elasticity problem (IEP), by delving into both theoretical and numerical aspects, with a particular focus on cardiac mechanics. In this work, we extend Shield's seminal work to determine the structure of the IEP with arbitrary material inhomogeneities and in the presence of both body and active forces. These aspects are fundamental in computational cardiology, and we show that they may break the variational structure of the inverse problem. In addition, we show that the inverse problem might have no solution even in the presence of constant Neumann boundary conditions and a polyconvex strain energy functional. We then present the results of extensive numerical tests to validate our theoretical framework, and to characterize the computational challenges associated with a direct numerical approximation of the IEP. Specifically, we show that this framework outperforms existing approaches both in terms of robustness and optimality, such as Sellier's iterative procedure, even when the latter is improved with acceleration techniques. A notable discovery is that multigrid preconditioners are, in contrast to standard elasticity, not efficient, where a one-level additive Schwarz and generalized Dryja–Smith–Widlund provide a much more reliable alternative. Finally, we successfully address the IEP for a full-heart geometry, demonstrating that the IEP formulation can compute the stress-free configuration in real-life scenarios where Sellier's algorithm proves inadequate
Machine learning for fast and reliable solution of time-dependent differential equations
Full text linkWe propose a data-driven Model Order Reduction (MOR) technique, based on Artificial Neural Networks (ANNs), applicable to dynamical systems arising from Ordinary Differential Equations (ODEs) or time-dependent Partial Differential Equations (PDEs). Unlike model-based approaches, the proposed approach is non-intrusive since it just requires a collection of input-output pairs generated through the high-fidelity (HF) ODE or PDE model. We formulate our model reduction problem as a maximum-likelihood problem, in which we look for the model that minimizes, in a class of candidate models, the error on the available input-output pairs. Specifically, we represent candidate models by means of ANNs, which we train to learn the dynamics of the HF model from the training input-output data. We prove that ANN models are able to approximate every time-dependent model described by ODEs with any desired level of accuracy. We test the proposed technique on different problems, including the model reduction of two large-scale models. Two of the HF systems of ODEs here considered stem from the spatial discretization of a parabolic and an hyperbolic PDE respectively, which sheds light on a promising field of application of the proposed technique
Biophysically detailed mathematical models of multiscale cardiac active mechanics
Full text linkWe propose four novel mathematical models, describing the microscopic mechanisms of force generation in the cardiac muscle tissue, which are suitable for multiscale numerical simulations of cardiac electromechanics. Such models are based on a biophysically accurate representation of the regulatory and contractile proteins in the sarcomeres. Our models, unlike most of the sarcomere dynamics models that are available in the literature and that feature a comparable richness of detail, do not require the time-consuming Monte Carlo method for their numerical approximation. Conversely, the models that we propose only require the solution of a system of PDEs and/or ODEs (the most reduced of the four only involving 20 ODEs), thus entailing a significant computational efficiency. By focusing on the two models that feature the best trade-off between detail of description and identifiability of parameters, we propose a pipeline to calibrate such parameters starting from experimental measurements available in literature. Thanks to this pipeline, we calibrate these models for room-temperature rat and for body-temperature human cells. We show, by means of numerical simulations, that the proposed models correctly predict the main features of force generation, including the steady-state force-calcium and force-length relationships, the length-dependent prolongation of twitches and increase of peak force, the force-velocity relationship. Moreover, they correctly reproduce the Frank-Starling effect, when employed in multiscale 3D numerical simulation of cardiac electromechanics
Universal Solution Manifold Networks (USM-Nets): Non-Intrusive Mesh-Free Surrogate Models for Problems in Variable Domains
Full text linkWe introduce universal solution manifold network (USM-Net), a novel surrogate model, based on artificial neural networks (ANNs), which applies to differential problems whose solution depends on physical and geometrical parameters. We employ a mesh-less architecture, thus overcoming the limitations associated with image segmentation and mesh generation required by traditional discretization methods. Our method encodes geometrical variability through scalar landmarks, such as coordinates of points of interest. In biomedical applications, these landmarks can be inexpensively processed from clinical images. We present proof-of-concept results obtained with a data-driven loss function based on simulation data. Nonetheless, our framework is non-intrusive and modular, as we can modify the loss by considering additional constraints, thus leveraging available physical knowledge. Our approach also accommodates a universal coordinate system, which supports the USM-Net in learning the correspondence between points belonging to different geometries, boosting prediction accuracy on unobserved geometries. Finally, we present two numerical test cases in computational fluid dynamics involving variable Reynolds numbers as well as computational domains of variable shape. The results show that our method allows for inexpensive but accurate approximations of velocity and pressure, avoiding computationally expensive image segmentation, mesh generation, or re-training for every new instance of physical parameters and shape of the domain
Machine learning of multiscale active force generation models for the efficient simulation of cardiac electromechanics
Full text linkHigh fidelity (HF) mathematical models describing the generation of active force in the cardiac muscle tissue typically feature a large number of state variables to capture the intrinsically complex underlying subcellular mechanisms. With the aim of drastically reducing the computational burden associated with the numerical solution of these models, we propose a machine learning method that builds a reduced order model (ROM); this is obtained as the best-approximation of the HF model within a class of candidate differential equations based on Artificial Neural Networks (ANNs). Within a semiphysical (gray-box) approach, an ANN learns the dynamics of the HF model from input–output pairs generated by the HF model itself (i.e. non-intrusively), being additionally informed with some a priori knowledge about the HF model. The ANN-based ROM, with just two internal variables, can accurately reproduce the results of the HF model, that instead features more than 2000 variables, under several physiological and pathological working regimes of the cell. We then propose a multiscale 3D cardiac electromechanical model, wherein active force generation is described by means of the previously trained ANN. We achieve a very favorable balance between accuracy of the result (order of 10−3 for the main cardiac biomarkers) and computational efficiency (with a speedup of about one order of magnitude), still relying on a biophysically detailed description of the microscopic force generation phenomenon
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