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47 results on '"Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM)"'

1. Solving 2D linear isotropic elastodynamics by means of scalar potentials: a new challenge for finite elements

Author

Patrick Joly, Jerónimo Rodríguez, Sébastien Imperiale, Jorge Martínez, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela [Spain] (USC ), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Saclay, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Universidade de Santiago de Compostela, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine ( M3DISIM ), Laboratoire de mécanique des solides ( LMS ), École polytechnique ( X ) -MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique ( CNRS ) -École polytechnique ( X ) -MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique ( CNRS ) -Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Propagation des Ondes : Étude Mathématique et Simulation ( POEMS ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -Centre National de la Recherche Scientifique ( CNRS ), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
Numerical Analysis, Applied Mathematics, Isotropy, Scalar (mathematics), Mathematical analysis, General Engineering, 010103 numerical & computational mathematics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, Finite element method, Theoretical Computer Science, 010101 applied mathematics, Computational Mathematics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Computational Theory and Mathematics, Displacement field, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Time domain, 0101 mathematics, Scalar field, Helmholtz decomposition, Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; In this work we present a method for the computation of numerical solutions of 2D homogeneous isotropic elastodynamics equations by solving scalar wave equations. These equations act on the potentials of a Helmholtz decomposition of the displacement field and are decoupled inside the propagation domain. We detail how these equations are coupled at the boundary depending on the nature of the boundary condition satisfied by the displacement field. After presenting the case of rigid boundary conditions, that presents no specific difficulty, we tackle the challenging case of free surface boundary conditions that presents severe stability issues if a straightforward approach is used. We introduce an adequate functional framework as well as a time domain mixed formulation to circumvent these issues. Numerical results confirm the stability of the proposed approach.

Published
2018
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2. Modeling acto-myosin interaction: beyond the Huxley–Hill framework

Author

Chaintron, Louis-Pierre, Caruel, Matthieu, Kimmig, François, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Muscle contraction, Thermodynamics, Jump-diffusion process, [SPI.MECA.BIOM]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Biomechanics [physics.med-ph], Molecular motors, Poisson random measures, Applied mathematics, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Applied probability
Abstract

Contractile force in muscle tissue is produced by the interaction of myosin molecular motors that bind and pull on specific sites located on surrounding actin filaments. The classical framework set by the landmark works of A.F. Huxley and T.L. Hill to model this active system is build on the central assumption that thermal fluctuations of a given myosin motor are sufficiently small so that it cannot interact with more than one binding site at any time. In this paper we present the physiological and mathematical limitations of this approach to motivate a new formulation that circumvent them without resorting to the more complex multi-site model paradigm. The acto-myosin system is now described as a Markov process combining Langevin driftdiffusion and Poisson jumps dynamics. We show that the corresponding system of Stochastic Differential Equation is well-posed and derive its Partial Differential Equation analog in order to obtain the thermodynamic balance laws. We finally show that by applying standard elimination procedures, a modified version of the original Huxley-Hill framework can be obtained as a reduced version of our model. Theoretical results are supported by numerical simulations where the model outputs are compared to benchmark experimental data.

Published
2023

3. Reduced left ventricular dynamics modeling based on a cylindrical assumption

Author

Martin Genet, Jérôme Diaz, Dominique Chapelle, Philippe Moireau, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Computational Theory and Mathematics, Applied Mathematics, Modeling and Simulation, Computational mechanics, Continuum mechanics on manifold, Reduced-order modeling, Biomedical Engineering, [SPI.MECA.BIOM]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Biomechanics [physics.med-ph], Cardiac modeling, Molecular Biology, Software
Abstract

Biomechanical modeling and simulation is expected to play a significant role in the development of the next generation tools in many fields of medicine. However, full-fledged finite element models of complex organs such as the heart can be computationally very expensive, thus limiting their practical usability. Therefore, reduced models are much valuable to be used, e.g., for pre-calibration of full-fledged models, fast predictions, real-time applications, etc.. In this work, focused on the left ventricle, we develop a reduced model by defining reduced geometry & kinematics while keeping general motion and behavior laws, allowing to derive a reduced model where all variables & parameters have a strong physical meaning. More specifically, we propose a reduced ventricular model based on cylindrical geometry & kinematics, which allows to describe the myofiber orientation through the ventricular wall and to represent contraction patterns such as ventricular twist, two important features of ventricular mechanics. Our model is based on the original cylindrical model of [Guccione, McCulloch, & Waldman 1991; Guccione, Waldman, & McCulloch 1993], albeit with multiple differences: we propose a fully dynamical formulation, integrated into an open-loop lumped circulation model, and based on a material behavior that incorporates a fine description of contraction mechanisms; moreover, the issue of the cylinder closure has been completely reformulated; our numerical approach is novel as well, with consistent spatial (finite element) and time discretizations. Finally, we analyse the sensitivity of the model response to various numerical and physical parameters, and study its physiological response.

Published
2023
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4. Varying thin filament activation in the framework of the Huxley'57 model

Author

François Kimmig, Matthieu Caruel, Dominique Chapelle, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Modélisation et Simulation Multi-Echelle (MSME), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel

Subjects
Sarcomeres, Applied Mathematics, Biomedical Engineering, [SPI.MECA.BIOM]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Biomechanics [physics.med-ph], Cardiac modeling, Actins, Actin Cytoskeleton, Computational Theory and Mathematics, Thick and thin filament activation, Modeling and Simulation, Huxley'57 model, Calcium, Mathematical modeling, Molecular Biology, Software, Muscle Contraction
Abstract

International audience; Muscle contraction is triggered by the activation of the actin sites of the thin filament by calcium ions. It results that the thin filament activation level varies over time. Moreover, this activation process is also used as a regulation mechanism of the developed force. Our objective is to build a model of varying actin site activation level within the classical Huxley'57 two-state framework. This new model is obtained as an enhancement of a previously proposed formulation of the varying thick filament activation within the same framework [1]. We assume that the state of an actin site depends on whether it is activated and whether it forms a cross-bridge with the associated myosin head, which results in four possible states. The transitions between the actin site states are controlled by the global actin sites activation level and the dynamics of these transitions is coupled with the attachment-detachment process. A preliminary calibration of the model with experimental twitch contraction data obtained at varying sarcomere lengths is performed.

Published
2022
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5. An efficient numerical method for time domain electromagnetic wave propagation in co-axial cables

Author

Akram Beni-Hamad, Geoffrey Beck, Sébastien Imperiale, Patrick Joly, Beck, Geoffrey, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Paris (ENS-PSL), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Computational Mathematics, Numerical Analysis, Edge elements, Hybrid numerical method, Applied Mathematics, Coaxial cables, Telegrapher's models, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Maxwell's equation, Numerical simulation, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this work, we construct an efficient numerical method to solve 3D Maxwell’s equations in coaxial cables. Our strategy is based upon a hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objectives is to validate numerically generalized telegrapher’s models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article.

Published
2022
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6. Coupling reduced-order blood flow and cardiac models through energy-consistent strategies: modeling and discretization

Author

Sébastien Imperiale, François Kimmig, Federica Caforio, Philippe Moireau, Jessica Manganotti, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Karl-Franzens-Universität Graz, École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Institute of Mathematics and Scientific Computing, University of Graz

Subjects
Work (thermodynamics), Reduced-order models, Heart models, Discretization, Quantitative Biology::Tissues and Organs, 0206 medical engineering, 02 engineering and technology, 030204 cardiovascular system & hematology, Energy-preserving time-scheme, Reduced order, Systems engineering, Cardiovascular modelling, 03 medical and health sciences, Arterial segment, TA168, 0302 clinical medicine, Computational Science and Engineering, Engineering (miscellaneous), Physics, Coupling, Applied Mathematics, Mechanics of engineering. Applied mechanics, [SPI.MECA.BIOM]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Biomechanics [physics.med-ph], Mechanics, Blood flow, TA349-359, Dicrotic notch, 020601 biomedical engineering, Computer Science Applications, Shallow-water models, Ageing, Modeling and Simulation, Energy (signal processing), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this work we provide a novel energy-consistent formulation for the classical 1D formulation of blood flow in an arterial segment. The resulting reformulation is shown to be suitable for the coupling with a lumped (0D) model of the heart that incorporates a reduced formulation of the actin-myosin interaction. The coupling being consistent with energy balances, we provide a complete heart-circulation model compatible with thermodynamics hence stable numerically and informative physiologically. These latter two properties are verified by numerical experiments.

Published
2021
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7. Hierarchical modeling of length-dependent force generation in cardiac muscles and associated thermodynamically-consistent numerical schemes

Author

Dominique Chapelle, Philippe Moireau, François Kimmig, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Computer science, Computational Mechanics, Ocean Engineering, Context (language use), multi-scale modeling, Sarcomere, biomechanics, 03 medical and health sciences, Myosin head, 0302 clinical medicine, numerical methods, 030304 developmental biology, 0303 health sciences, Frank–Starling law of the heart, Hierarchical modeling, Applied Mathematics, Mechanical Engineering, Numerical analysis, [SPI.MECA.BIOM]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Biomechanics [physics.med-ph], Cardiac modeling, Multiscale modeling, multiscale modeling, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computational Mathematics, Computational Theory and Mathematics, Coupling (computer programming), Frank-Starling mechanism, sarcomere, Biological system, 030217 neurology & neurosurgery, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In the context of cardiac muscle modeling, the availability of the myosin heads in the sarcomeres varies over the heart cycle contributing to the Frank-Starling mechanism at the organ level. In this paper, we propose a new approach that allows to extend the Huxley'57 muscle contraction model equations to incorporate this variation. This extension is built in a thermodynamically consistent manner, and we also propose adapted numerical methods that satisfy thermodynamical balances at the discrete level. Moreover, this whole approach-both for the model and the numerics-is devised within a hierarchical strategy enabling the coupling of the microscopic sarcomere-level equations with the macroscopic tissue-level description. As an important illustration, coupling our model with a previously proposed simplified heart model, we demonstrate the ability of the modeling and numerical framework to capture the essential features of the Frank-Starling mechanism.

Published
2021
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8. Sequential data assimilation for mechanical systems with complex image data: application to tagged-MRI in cardiac mechanics

Author

Philippe Moireau, Dominique Chapelle, Alexandre Imperiale, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Département Imagerie et Simulation pour le Contrôle (DISC), Laboratoire d'Intégration des Systèmes et des Technologies (LIST (CEA)), Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Laboratoire d'Intégration des Systèmes et des Technologies (LIST)

Subjects
Computer science, 0206 medical engineering, Image processing, 02 engineering and technology, Data type, Synthetic data, 030218 nuclear medicine & medical imaging, lcsh:TA168, 03 medical and health sciences, 0302 clinical medicine, Data assimilation, Medical imaging, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Engineering (miscellaneous), Flexibility (engineering), Measure (data warehouse), Tagged-MRI, business.industry, Applied Mathematics, Pattern recognition, Cardiac modeling, 020601 biomedical engineering, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, lcsh:Systems engineering, Modeling and Simulation, Patient-specific model, Artificial intelligence, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Focus (optics), business, lcsh:Mechanics of engineering. Applied mechanics, lcsh:TA349-359, Estimation
Abstract

Tagged Magnetic Resonance images (tagged-MRI) are generally considered to be the gold standard of medical imaging in cardiology. By imaging spatially-modulated magnetizations of the deforming tissue, indeed, this modality enables an assessment of intra-myocardial deformations over the heart cycle. The objective of the present work is to incorporate the most valuable information contained in tagged-MRI in a data assimilation framework, in order to perform joint state-parameter estimation for a complete biomechanical model of the heart. This type of estimation is the second major step, after initial anatomical personalization, for obtaining a genuinely patient-specific model that integrates the individual characteristics of the patient, an essential prerequisite for benefitting from the model predictive capabilities. Here, we focus our attention on proposing adequate means of quantitatively comparing the cardiac model with various types of data that can be extracted from tagged-MRI after an initial image processing step, namely, 3D displacements fields, deforming tag planes or grids, or apparent 2D displacements. This quantitative comparison—called discrepancy measure—is then used to feed a sequential data assimilation procedure. In the state estimation stage of this procedure, we also propose a new algorithm based on the prediction–correction paradigm, which provides increased flexibility and effectiveness in the solution process. The complete estimation chain is eventually assessed with synthetic data, produced by running a realistic model simulation representing an infarcted heart characterized by increased stiffness and reduced contractility in a given region of the myocardium. From this simulation we extract the 3D displacements, tag planes and grids, and apparent 2D displacements, and we assess the estimation with each corresponding discrepancy measure. We demonstrate that—via regional estimation of the above parameters—the data assimilation procedure allows to quantitatively estimate the biophysical parameters with good accuracy, thus simultaneously providing the location of the infarct and characterizing its seriousness. This shows great potential for combining a biomechanical heart model with tagged-MRI in order to extract valuable new indices in clinical diagnosis.

Published
2021
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9. Combining data assimilation and machine learning to build data-driven models for unknown long time dynamics—Applications in cardiovascular modeling

Author

Francesco Regazzoni, Philippe Moireau, Dominique Chapelle, Modeling and Scientific Computing [Milano] (MOX), Politecnico di Milano [Milan] (POLIMI), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
cardiovascular modeling, Computer science, Differential equation, data‐driven modeling, 0206 medical engineering, Biomedical Engineering, Context (language use), 02 engineering and technology, 030204 cardiovascular system & hematology, Machine learning, computer.software_genre, Field (computer science), Data-driven, 03 medical and health sciences, 0302 clinical medicine, multiscale problems, Molecular Biology, data assimilation, Interpretability, Parametric statistics, Research Article ‐ Fundamental, Artificial neural network, business.industry, Applied Mathematics, Models, Theoretical, 020601 biomedical engineering, Test case, machine learning, Computational Theory and Mathematics, data-driven modeling, Modeling and Simulation, Artificial intelligence, business, computer, artificial neural networks, Algorithms, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Software
Abstract

We 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., We 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, used to estimate the parameters associated with the known fast‐scale dynamics, and machine learning, used to infer the laws underlying the slow‐scale dynamics. 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.

Published
2021
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10. A flexible framework for sequential estimation of model parameters in computational hemodynamics

Author

Christopher J. Arthurs, Tobias Schaeffter, C. Alberto Figueroa, Philippe Moireau, Nan Xiao, Imaging Sciences and Biomedical Engineering Division [London], Guy's and St Thomas' Hospital [London]-King‘s College London, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Technical University of Berlin / Technische Universität Berlin (TU), Physikalisch-Technische Bundesanstalt [Berlin] (PTB), Depts. of Surgery and Biomedical Engineering, University of Michigan, École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Technical University Berlin

Subjects
Computer science, Calibration (statistics), 0206 medical engineering, Computational hemodynamics, 02 engineering and technology, Patient specific modeling, Synthetic data, lcsh:TA168, Stiffness, 03 medical and health sciences, 0302 clinical medicine, Data assimilation, Parameter estimation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Engineering (miscellaneous), Sequential estimation, Boundary conditions, Estimation theory, Applied Mathematics, Kalman filter, 020601 biomedical engineering, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, Flow (mathematics), lcsh:Systems engineering, Modeling and Simulation, Netlist, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], lcsh:Mechanics of engineering. Applied mechanics, lcsh:TA349-359, Algorithm, Kalman filtering, 030217 neurology & neurosurgery, Research Article
Abstract

A major challenge in constructing three dimensional patient specific hemodynamic models is the calibration of model parameters to match patient data on flow, pressure, wall motion, etc. acquired in the clinic. Current workflows are manual and time-consuming. This work presents a flexible computational framework for model parameter estimation in cardiovascular flows that relies on the following fundamental contributions. (i) A Reduced-Order Unscented Kalman Filter (ROUKF) model for data assimilation for wall material and simple lumped parameter network (LPN) boundary condition model parameters. (ii) A constrained least squares augmentation (ROUKF-CLS) for more complex LPNs. (iii) A “Netlist” implementation, supporting easy filtering of parameters in such complex LPNs. The ROUKF algorithm is demonstrated using non-invasive patient-specific data on anatomy, flow and pressure from a healthy volunteer. The ROUKF-CLS algorithm is demonstrated using synthetic data on a coronary LPN. The methods described in this paper have been implemented as part of the CRIMSON hemodynamics software package.

Published
2020
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11. Construction and convergence analysis of conservative second order local time discretisation for linear wave equations

Author

Juliette Chabassier, Sébastien Imperiale, Advanced 3D Numerical Modeling in Geophysics (Magique 3D), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Université Paris-Saclay, Modélisation et simulation de la propagation des ondes fondées sur des mesures expérimentales pour caractériser des milieux géophysiques et héliophysiques et concevoir des objets complexes (MAKUTU), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Polytechnique de Bordeaux (Bordeaux INP), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Discretization, 010103 numerical & computational mathematics, Time step, Mathematics Subject Classification.35L05, 65M12, 65M22, 65M60, 01 natural sciences, Applied mathematics, 0101 mathematics, [MATH]Mathematics [math], Mathematics, Numerical Analysis, Applied Mathematics, Domain decomposition methods, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], Wave equation, 010101 applied mathematics, Wave equations, Computational Mathematics, Local implicit scheme, Test case, Modeling and Simulation, Local time, Norm (mathematics), Time discretisation, Local time stepping, Converge analysis, Analysis
Abstract

International audience; In this work we present and analyse a time discretisation strategy for linear wave equations t hat aims at using locally in space the most adapted time discretisation among a family of implicit or explicit centered second order schemes. The proposed family of schemes is adapted to domain decomposition methods such as the mortar element method. They correspond in that case to local implicit schemes and to local time stepping. We show that, if some regularity properties of the solution are satisfied and if the time step verifies a stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence. Finally, we provide 1D and 2D numerical illustrations that confirm the obtained theoretical results and we compare our approach on 1D test cases to other existing local time stepping strategies for wave equations.

Published
2020
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12. Construction and analysis of fourth order, energy consistent, family of explicit time discretizations for dissipative linear wave equations

Author

Julien Diaz, Juliette Chabassier, Sébastien Imperiale, Advanced 3D Numerical Modeling in Geophysics (Magique 3D), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Université Paris-Saclay, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Numerical Analysis, Helmholtz equation, Applied Mathematics, Dissipative linear wave equation, Inverse, 010103 numerical & computational mathematics, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], Space (mathematics), Wave equation, 01 natural sciences, Stability (probability), Energy analysis, 010101 applied mathematics, Computational Mathematics, Matrix (mathematics), Modeling and Simulation, Convergence (routing), Dissipative system, Applied mathematics, High order time discretisation, 0101 mathematics, 1991 Mathematics Subject Classification. 00A71, 35L05, 85A20, 33C55, 65M60, Analysis, Mathematics
Abstract

International audience; This paper deals with the construction of a family of fourth order, energy consistent, explicit time discretizations for dissipative linear wave equations. The schemes are obtained by replacing the inversion of a matrix, that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations, by explicit approximations of its inverse. The stability of the schemes are studied using an energy analysis and a convergence analysis is carried out. Numerical results in 1D illustrate the space/time convergence properties of the schemes and their efficiency is compared to more classical time discretizations.

Published
2020
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13. A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer

Author

Philippe Moireau, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Saclay, École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
time-discrete non-linear systems, Control and Optimization, Observer (quantum physics), Discretization, optimal filtering, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Estimator, Context (language use), 010103 numerical & computational mathematics, Filter (signal processing), Kalman filter, 01 natural sciences, 010101 applied mathematics, Dynamic programming, Computational Mathematics, Discrete time and continuous time, Control and Systems Engineering, deterministic observer, Applied mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Hamilton-Jacobi-Bellman, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; In this work, we seek exact formulations of the optimal estimator and filter for a non-linear framework, as the Kalman filter is for a linear framework. The solution is well established with the Mortensen filter in a continuous-time setting, but we seek here its counterpart in a discrete-time context. We demonstrate that it is possible to pursue at the discrete-time level an exact dynamic programming strategy and we find an optimal estimator combining a prediction step using the model and a correction step using the data. This optimal estimator reduces to the discrete-time Kalman estimator when the operators are in fact linear. Furthermore, the strategy that consists of discretizing the least square criterion and then finding the exact estimator at the discrete level allows to determine a new time-scheme for the Mortensen filter which is proven to be consistent and unconditionally stable, with also a consistent and stable discretization of the underlying Hamilton-Jacobi-Bellman equation.

Published
2018
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14. Multiscale population dynamics in reproductive biology: singular perturbation reduction in deterministic and stochastic models

Author

Frédérique Clément, Romain Yvinec, Keltoum Chahour, Celine Bonnet, Marie Postel, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Etudes et Recherche en Mathématiques Appliquées (LERMA), Ecole Mohammadia d'Ingénieurs (EMI), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Analysis and Control of Unsteady Models for Engineering Sciences (ACUMES), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Dynamiques de populations multi-échelles pour des systèmes physiologiques (MUSCA), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Physiologie de la reproduction et des comportements [Nouzilly] (PRC), Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Mathématiques et Informatique Appliquées du Génome à l'Environnement [Jouy-En-Josas] (MaIAGE), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Physiologie de la reproduction et des comportements [Nouzilly] (PRC), Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut National de la Recherche Agronomique (INRA)-Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Institut Français du Cheval et de l'Equitation [Saumur]-Institut National de la Recherche Agronomique (INRA), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Institut Français du Cheval et de l'Equitation [Saumur] (IFCE)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Français du Cheval et de l'Equitation [Saumur] (IFCE)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Mathématiques et Informatique Appliquées du Génome à l'Environnement [Jouy-En-Josas] (MaIAGE), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Institut Français du Cheval et de l'Equitation [Saumur] (IFCE)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Mathématiques et Informatique Appliquées du Génome à l'Environnement [Jouy-En-Josas] (MaIAGE), and Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)

Subjects
0301 basic medicine, Singular perturbation, Stochastic modelling, Differential equation, Quantitative Biology::Tissues and Organs, [SDV]Life Sciences [q-bio], Population, 030209 endocrinology & metabolism, Quantitative Biology::Cell Behavior, 03 medical and health sciences, 0302 clinical medicine, Convergence (routing), QA1-939, FOS: Mathematics, Applied mathematics, Limit (mathematics), [MATH]Mathematics [math], Quantitative Biology - Populations and Evolution, education, Tissues and Organs (q-bio.TO), Mathematics, T57-57.97, education.field_of_study, Applied mathematics. Quantitative methods, Probability (math.PR), Ode, Populations and Evolution (q-bio.PE), Quantitative Biology - Tissues and Organs, Nonlinear system, 030104 developmental biology, FOS: Biological sciences, Mathematics - Probability
Abstract

In this study, we describe different modeling approaches for ovarian follicle population dynamics, based on either ordinary (ODE), partial (PDE) or stochastic (SDE) differential equations, and accounting for interactions between follicles. We put a special focus on representing the population-level feedback exerted by growing ovarian follicles onto the activation of quiescent follicles. We take advantage of the timescale difference existing between the growth and activation processes to apply model reduction techniques in the framework of singular perturbations. We first study the linear versions of the models to derive theoretical results on the convergence to the limit models. In the nonlinear cases, we provide detailed numerical evidence of convergence to the limit behavior. We reproduce the main semi-quantitative features characterizing the ovarian follicle pool, namely a bimodal distribution of the whole population, and a slope break in the decay of the quiescent pool with aging., Comment: preprint version of a proceeding of CEMRACS 2018: http://smai.emath.fr/cemracs/cemracs18/

Published
2020
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15. Signed-distance function based non-rigid registration of image series with varying image intensity

Author

Tomáš Oberhuber, Kateřina Škardová, Radomir Chabiniok, Jaroslav Tintěra, Department of Mathematics [Prague] (FNSPE), Czech Technical University in Prague (CTU), Institute for Clinical and Experimental Medicine (IKEM), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), School of Biomedical Engineering & Imaging Sciences [London], King‘s College London-Guy's and St Thomas' Hospital [London], École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Guy's and St Thomas' Hospital [London]-King‘s College London, École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
Image Series, business.industry, Computer science, [SDV.IB.IMA]Life Sciences [q-bio]/Bioengineering/Imaging, Applied Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Optical flow, Signed distance function, Image segmentation, Mutual information, 030204 cardiovascular system & hematology, Object (computer science), Real image, 03 medical and health sciences, 0302 clinical medicine, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], 030220 oncology & carcinogenesis, Discrete Mathematics and Combinatorics, Computer vision, Segmentation, Artificial intelligence, business, Analysis
Abstract

International audience; In this paper we propose a method for locally adjusted optical flow-based registration of multimodal images, which uses the segmentation of the object of interest and its representation by the signed-distance function (OF dist method). We deal with non-rigid registration of the image series acquired by the Modiffied Look-Locker Inversion Recovery (MOLLI) magnetic resonance imaging sequence, which is used for a pixel-wise estimation of T 1 relaxation time. The spatial registration of the images within the series is necessary to compensate the patient's imperfect breath-holding. The evolution of intensities and a large variation of image contrast within the MOLLI image series, together with the myocardium of left ventricle (the object of interest) typically not being the most distinct object in the scene, makes the registration challenging. The paper describes all components of the proposed OF dist method and their implementation. The method is then compared to the performance of a standard mutual information maximization-based registration method, applied either to the original image (MIM) or to the signed-distance function (MIM dist). Several experiments with synthetic and real MOLLI images are carried out. On synthetic image with a single object, MIM performed the best, while OF dist and MIM dist provided better results on synthetic images with more than one object and on real images. When applied to signed-distance function of two objects of interest, MIM dist provided a larger registration error (but more homogeneously distributed) compared to OF dist. For the real MOLLI image series with left ventricle pre-segmented using a level-set method, the proposed OF dist registration performed the best, as is demonstrated visually and by measuring the increase of mutual information in the object of interest and its neighborhood.

Published
2020
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16. A multiscale mathematical model of cell dynamics during neurogenesis in the mouse cerebral cortex

Author

Marie Postel, Frédérique Clément, Alice Karam, Sylvie Schneider-Maunoury, Guillaume Pézeron, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Biologie du Développement [IBPS] (LBD), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de Biologie Paris Seine (IBPS), Institut National de la Santé et de la Recherche Médicale (INSERM)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Physiologie moléculaire et adaptation (PhyMA), Muséum national d'Histoire naturelle (MNHN)-Centre National de la Recherche Scientifique (CNRS), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Biologie du Développement [Paris] (LBD), Sorbonne Université (SU)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Laboratoire de Biologie du Développement (LBD), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Evolution des régulations endocriniennes (ERE), and École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France

Subjects
Time varying transfer rates, [SDV]Life Sciences [q-bio], Multiscale mathematical modeling, Biochemistry, Mice, 0302 clinical medicine, Neural Stem Cells, Structural Biology, Numerical simulations, Mouse mutant for Ftm/Rpgrip1l, lcsh:QH301-705.5, [SDV.BDD]Life Sciences [q-bio]/Development Biology, Cerebral Cortex, Neurons, 0303 health sciences, education.field_of_study, Applied Mathematics, Neurogenesis, Cell Cycle, Neural stem cell, Computer Science Applications, medicine.anatomical_structure, Cerebral cortex, lcsh:R858-859.7, Cell cycle indexes, Cell Division, Research Article, Population, [SDV.BC]Life Sciences [q-bio]/Cellular Biology, Biology, lcsh:Computer applications to medicine. Medical informatics, Models, Biological, 03 medical and health sciences, Development of the cerebral cortex, medicine, Animals, Humans, Progenitor cell, education, Molecular Biology, Mitosis, 030304 developmental biology, Progenitor, Cell dynamics, Proteins, Neural progenitors, Cytoskeletal Proteins, Disease Models, Animal, lcsh:Biology (General), Mutation, Neuron, Neuroscience, 030217 neurology & neurosurgery
Abstract

Background Neurogenesis in the murine cerebral cortex involves the coordinated divisions of two main types of progenitor cells, whose numbers, division modes and cell cycle durations set up the final neuronal output. To understand the respective roles of these factors in the neurogenesis process, we combine experimental in vivo studies with mathematical modeling and numerical simulations of the dynamics of neural progenitor cells. A special focus is put on the population of intermediate progenitors (IPs), a transit amplifying progenitor type critically involved in the size of the final neuron pool. Results A multiscale formalism describing IP dynamics allows one to track the progression of cells along the subsequent phases of the cell cycle, as well as the temporal evolution of the different cell numbers. Our model takes into account the dividing apical progenitors (AP) engaged into neurogenesis, both neurogenic and proliferative IPs, and the newborn neurons. The transfer rates from one population to another are subject to the mode of division (proliferative, or neurogenic) and may be time-varying. The model outputs are successfully fitted to experimental cell numbers from mouse embryos at different stages of cortical development, taking into account IPs and neurons, in order to adjust the numerical parameters. We provide additional information on cell kinetics, such as the mitotic and S phase indexes, and neurogenic fraction. Conclusions Applying the model to a mouse mutant for Ftm/Rpgrip1l, a gene involved in human ciliopathies with severe brain abnormalities, reveals a shortening of the neurogenic period associated with an increased influx of newborn IPs from apical progenitors at mid-neurogenesis. Our model can be used to study other mouse mutants with cortical neurogenesis defects and can be adapted to study the importance of progenitor dynamics in cortical evolution and human diseases.

Published
2019
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17. Conservative and entropy controlled remap for multi-material ALE simulations with space-staggered schemes

Author

Alexandra Claisse, Patrick Le Tallec, Alexis Marbœuf, Centre d'Études de Limeil-Valenton (CEA-DAM), Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Physics and Astronomy (miscellaneous), Computer science, media_common.quotation_subject, Second law of thermodynamics, Monotonic function, 010103 numerical & computational mathematics, [SPI.MECA.MSMECA]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Materials and structures in mechanics [physics.class-ph], Energy conservation, 01 natural sciences, symbols.namesake, Multi-material ALE schemes, Staggered remap, Applied mathematics, 0101 mathematics, Total energy, media_common, Numerical Analysis, Applied Mathematics, Multi material, Staggered schemes, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, symbols, Hydrodynamics, Momentum conservation, Special care, Lagrangian, Entropy control
Abstract

International audience; The remapping strategy is crucial in any Arbitrary Lagrangian-Eulerian (ALE) algorithm based on a Lagrange-plus-remap paradigm. This step is particularly challenging for space-staggered schemes since inconsistencies may appear between cell centered and node centered fields after remap if no special care is taken [1], [2], [3]. We propose here a space-staggered remapping strategy focusing on conservation properties and entropy control. The proposed algorithm conserves mass, total energy and respects the Second Law of Thermodynamics (for robustness) up to round-off errors. This is achieved at a low computational cost by introducing a consistent, explicit and local post processing of the linear momentum after remap. This new method is then analyzed showing that the strict momentum conservation is sacrificed. It is now conserved to the scheme's order, such as entropy. Other classical properties such that the “DeBar consistency” [4], the continuity with the Lagrangian step and the monotonicity are also discussed. This work is developed in the context of the intersection-based (or overlay-based) remap. Therefore, the rezoned mesh does not have to be close to the Lagrangian one and, even if it is not considered here, our study can be easily extended to rezoning strategies which modify the mesh connectivity.

Published
2019
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18. Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains

Author

Sébastien Imperiale, Philippe Moireau, Antoine Tonnoir, Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI), Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Control and Optimization, Observer (quantum physics), Reconstruction algorithm, 010103 numerical & computational mathematics, 01 natural sciences, Domain (software engineering), 010101 applied mathematics, Computational Mathematics, Wave equation in unbounded domain, Control and Systems Engineering, Robustness (computer science), Bounded function, back and forth observer, Convergence (routing), Initial value problem, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, state estimation, 0101 mathematics, data assimilation, Mathematics
Abstract

International audience; We are interested in reconstructing the initial condition of a wave equation in an unbounded domain configuration from measurements available in time on a subdomain. To solve this problem, we adopt an iterative strategy of reconstruction based on observers and time reversal adjoint formulations. We prove the convergence of our reconstruction algorithm with perfect measurements and its robustness to noise. Moreover, we develop a complete strategy to practically solve this problem on a bounded domain using artificial transparent boundary conditions to account for the exterior domain. Our work then demonstrates that the consistency error introduced by the use of approximate transparent boundary conditions is compensated by the stabilization properties obtained from the use of the available measurements, hence allowing to still be able to reconstruct the unknown initial condition.; Nous nous intéressons au problème de recontruction de la donnée initiale pour uné equation d'ondes posée en domaine non bornéà l'aide de mesures en temps sur un sous domaine. Pour résoudre ce problème, nous utilisons une approche itérative de reconstruction basée sur une formulation type observateurs et retournement temporel. On prouve la convergence de l'algorithme de reconstruction dans le cas de mesures non bruitées et sa stabilité par rapport à des mesures bruitées. De plus, nous développons une méthodologie complète pour résoudre numériquement le problème en domaine borné en utilisant des conditions transparentes (absorbantes) sur les frontières artificielles bornant le domaine. Nous montrons en particulier que l'erreur de consistance introduite par les conditions absorbantes (non exactes) est compensée par les propriétés de stabilisation obtenues grâce aux mesures, ce qui permet d'être toujours en mesure de reconstituer la donnée initiale.

Published
2019
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19. Analysis and calibration of a linear model for structured cell populations with unidirectional motion : Application to the morphogenesis of ovarian follicles

Author

Frédérique Clément, Frédérique Robin, Romain Yvinec, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Physiologie de la reproduction et des comportements [Nouzilly] (PRC), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Institut Français du Cheval et de l'Equitation [Saumur]-Institut National de la Recherche Agronomique (INRA), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de la Recherche Agronomique (INRA)-Institut Français du Cheval et de l'Equitation [Saumur] (IFCE)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Institut National de la Recherche Agronomique (INRA)-Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Inria Saclay - Ile de France, and Institut National de la Recherche Agronomique (INRA)-Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Calibration (statistics), [SDV]Life Sciences [q-bio], Context (language use), 01 natural sciences, Quantitative Biology::Cell Behavior, 010104 statistics & probability, 03 medical and health sciences, AMS subject classifications. 35L65, 60K15, 60J80, 92D25, Mathematics - Analysis of PDEs, Convergence (routing), Parameter calibration, Applied mathematics, 0101 mathematics, Quantitative Biology - Populations and Evolution, 030304 developmental biology, Mathematics, 0303 health sciences, Biological data, Structured cell populations, Applied Mathematics, Linear model, Division (mathematics), McKendrick-VonFoerster model, Stable distribution, Structural identifiability, Renewal equations, Identifiability, Multi-type age dependent branching processes, Mathematics - Probability
Abstract

International audience; We analyze a multi-type age dependent model for cell populations subject to unidirectional motion, in both a stochastic and deterministic framework. Cells are distributed into successive layers; they may divide and move irreversibly from one layer to the next. We adapt results on the large-time convergence of PDE systems and branching processes to our context, where the Perron-Frobenius or Krein-Rutman theorem cannot be applied. We derive explicit analytical formulas for the asymptotic cell number moments, and the stable age distribution. We illustrate these results numerically and we apply them to the study of the morphodynamics of ovarian follicles. We prove the structural parameter identifiability of our model in the case of age independent division rates. Using a set of experimental biological data, we estimate the model parameters to fit the changes in the cell numbers in each layer during the early stages of follicle development.

Published
2019
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20. A benchmark problem to evaluate implementational issues for three-dimensional flows of incompressible fluids subject to slip boundary conditions

Author

Helena Švihlová, Radomir Chabiniok, Karel Tůma, Jaroslav Hron, Josef Málek, Alena Jarolímová, Kumbakonam R. Rajagopal, Keshava Rajagopal, University of Texas Southwestern Medical Center [Dallas], Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Czech Technical University in Prague (CTU), Faculty of Mathematics and Physics [Praha/Prague], Charles University [Prague] (CU), Texas A&M University [College Station], University of Texas, Houston College of Medicine and Houston Heart, École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
Computer science, Numerical analysis, Boundary (topology), General Medicine, Mixed finite element method, Vorticity, Engineering (General). Civil engineering (General), 01 natural sciences, 010305 fluids & plasmas, Physics::Fluid Dynamics, 010101 applied mathematics, Incompressible flow, 0103 physical sciences, Benchmark (computing), Compressibility, Applied mathematics, Boundary value problem, [PHYS.MECA.BIOM]Physics [physics]/Mechanics [physics]/Biomechanics [physics.med-ph], TA1-2040, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We consider flows of an incompressible Navier-Stokes fluid in a tubular domain with Navier's slip boundary condition imposed on the impermeable wall. We focus on several implementational issues associated with this type of boundary conditions within the framework of the standard Taylor-Hood mixed finite element method and present the computational results for flows in a tubular domain of finite length with one inlet and one outlet. In particular, we present the details regarding variants of the Nitsche method concerning the incorporation of the impermeability condition on the wall. We also find that the manner in which the normal to the boundary is numerically implemented influences the nature of the computational results. As a benchmark, we set up steady flows in a tube of finite length and compare the computational results with the analytical solutions. Finally, we identify various quantities of interest, such as the dissipation, wall shear stress, vorticity, pressure drop, and provide their precise mathematical definitions. We document how well these quantities are computationally approximated in the case of the benchmark. Although the geometry of the benchmark is simple, the correct computational results require careful selection of numerical methods and surprisingly non-trivial computational resources. Our goal is to test, using the setting with a known analytical solution, a robust computational tool that would be suitable for computations on real complex geometries that have relevance to problems in engineering and medicine. The model parameters in our computations are chosen based on flows in large arteries.

Published
2021
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21. Numerical analysis for an energy-stable total discretization of a poromechanics model with inf-sup stability

Author

Philippe Moireau, Dominique Chapelle, Bruno Burtschell, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Saclay, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Spacetime, Discretization, energy estimates, Applied Mathematics, Numerical analysis, Poromechanics, poromechanics, 010103 numerical & computational mathematics, incompressibility, 01 natural sciences, Stability (probability), 010101 applied mathematics, inf-sup, Nonlinear system, [PHYS.MECA.SOLID]Physics [physics]/Mechanics [physics]/Solid mechanics [physics.class-ph], Compressibility, Applied mathematics, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, total discretization, Energy (signal processing), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; We consider a previously proposed general nonlinear poromechanical formulation, and we derive a linearized version of this model. For this linearized model, we obtain an existence result and we propose a complete discretization strategy - in time and space - with a special concern for issues associated with incompressible or nearly-incompressible behavior. We provide a detailed mathematical analysis of this strategy, the main result being an error estimate uniform with respect to the compressibility parameter. We then illustrate our approach with detailed simulation results and we numerically investigate the importance of the assumptions made in the analysis, including the fulfillment of specific inf-sup conditions.

Published
2019
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22. Apprehending the effects of mechanical deformations in cardiac electrophysiology - An homogenization approach

Author

Jean-Frédéric Gerbeau, Annabelle Collin, Philippe Moireau, Dominique Chapelle, Sébastien Imperiale, Institut Polytechnique de Bordeaux (Bordeaux INP), Modélisation Mathématique pour l'Oncologie (MONC), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], UNICANCER-UNICANCER-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Inria Siège, Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Physics, Cell specific, 0303 health sciences, Cardiac electrophysiology, Applied Mathematics, Multiphysics, 0206 medical engineering, Microscopic level, [SPI.MECA.BIOM]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Biomechanics [physics.med-ph], 02 engineering and technology, 020601 biomedical engineering, Homogenization (chemistry), 03 medical and health sciences, Classical mechanics, [SDV.MHEP.CSC]Life Sciences [q-bio]/Human health and pathology/Cardiology and cardiovascular system, Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 030304 developmental biology
Abstract

International audience; We follow a formal homogenization approach to investigate the effects of mechanical deformations in electrophysiology models relying on a bidomain description of ionic motion at the microscopic level. To that purpose, we extend these microscopic equations to take into account the mechanical deformations, and proceed by recasting the problem in the framework of classical two-scale homogenization in periodic media, and identifying the equations satisfied by the first coefficients in the formal expansions. The homogenized equations reveal some interesting effects related to the microstructure - and associated with a specific cell problem to be solved to obtain the macroscopic conductivity tensors - in which mechanical deformations play a non-trivial role, i.e. do not simply lead to a standard bidomain problem posed in the deformed configuration. We then present detailed numerical illustrations of the homogenized model with coupled cardiac electrical-mechanical simulations - all the way to ECG simulations - albeit without taking into account the abundantly-investigated effect of mechanical deformations in ionic models, in order to focus here on other effects. And in fact our numerical results indicate that these other effects are numerically of a comparable order, and therefore cannot be disregarded.

Published
2019
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23. Thermodynamic properties of muscle contraction models and associated discrete-time principles

Author

Dominique Chapelle, Philippe Moireau, François Kimmig, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Saclay, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Large class, 74F25 ·74H15 · 65M12 ·35Q79 and 92C45, Discretization, 02 engineering and technology, Clausius–Duhem inequality, 01 natural sciences, lcsh:TA168, muscle contraction, 0203 mechanical engineering, medicine, Computational Science and Engineering, Statistical physics, 0101 mathematics, [PHYS.MECA.BIOM]Physics [physics]/Mechanics [physics]/Biomechanics [physics.med-ph], Engineering (miscellaneous), thermodynamically consistent time-discretization, Mathematics, Clausius-Duhem inequality, Continuum mechanics, Applied Mathematics, sliding filaments, Computer Science Applications, 010101 applied mathematics, 020303 mechanical engineering & transports, Discrete time and continuous time, lcsh:Systems engineering, Modeling and Simulation, medicine.symptom, lcsh:Mechanics of engineering. Applied mechanics, lcsh:TA349-359, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Muscle contraction
Abstract

International audience; Considering a large class of muscle contraction models accounting for actin-myosin interaction, we present a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances. Moreover, we propose a complete discretization strategy for which we are also able to obtain discrete versions of the thermodynamic balances and other properties. Our major objective is to show how the thermodynamics of such models can be tracked after discretization, including when they are coupled to a macroscopic muscle formulation in the realm of continuum mechanics. Our approach allows to carefully identify the sources of energy and entropy in the system, and to follow them up to the numerical applications.

Published
2019
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24. Delayed feedback control method for computing the cyclic steady states of evolution problems

Author

Patrick Le Tallec, Ustim Khristenko, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Work (thermodynamics), Computer science, Mechanical Engineering, Computational Mechanics, General Physics and Astronomy, 010103 numerical & computational mathematics, [SPI.MECA]Engineering Sciences [physics]/Mechanics [physics.med-ph], Space (mathematics), 01 natural sciences, Computer Science Applications, Term (time), Mechanics of Materials, Limit cycle, 0103 physical sciences, Convergence (routing), Applied mathematics, Initial value problem, Transient (computer programming), Limit (mathematics), 0101 mathematics, 010306 general physics, ComputingMilieux_MISCELLANEOUS
Abstract

This work is focused on fast techniques for computing the cyclic steady states of evolution problems in non-linear mechanics with space–time periodicity conditions. In industrial applications, in order to avoid the inversion of very large matrices, such a cyclic solution is usually computed as an asymptotic limit of the associated initial value problem with arbitrary initial data. However, when the relaxation time is high, convergence to the limit cycle can be very slow. In such cases nonetheless, one is not interested in the transient solution, but only in a fast access to the limit cycle. Thus, in this work we modify the problem, introducing the time-delayed feedback control, which is widely used for stabilization of unstable periodic orbits. In our framework it is applied to an initially stable system in order to accelerate its convergence to the limit cycle. Moreover, the control term, based on the space–time periodicity error, includes both shifts in time and in space. Our main result is the optimal form of the control term for a very general class of linear evolution problems, providing the fastest convergence to the cyclic solution, which has been further extended and studied in the non-linear case . Efficiency of the method increases with the problem’s relaxation time. The method has been tested using academic applications and compared to the non-controlled asymptotic convergence as well as to the Newton–Krylov shooting algorithm. Finally, the method has been implemented into the Michelin industrial code, applied to a full 3D rolling tyre model.

Published
2018
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25. Convergence of discrete-time Kalman filter estimate to continuous-time estimate for systems with unbounded observation

Author

Atte Aalto, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics and Systems Analysis [Helsinki], Aalto University, University of Luxembourg [Luxembourg], École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Temporal discretization, 0209 industrial biotechnology, Control and Optimization, Applied Mathematics, Infinite-dimensional systems, Boundary control systems, 02 engineering and technology, Sampled data, MSC2010: 93E11, 47D06, 93C05, 60G15, temporal discretization, sampled data, 020901 industrial engineering & automation, Control and Systems Engineering, Optimization and Control (math.OC), Signal Processing, FOS: Mathematics, Mathematics [G03] [Physical, chemical, mathematical & earth Sciences], Mathématiques [G03] [Physique, chimie, mathématiques & sciences de la terre], Kalman filter, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control
Abstract

In this article, we complement recent results on the convergence of the state estimate obtained by applying the discrete-time Kalman filter on a time-sampled continuous-time system. As the temporal discretization is refined, the estimate converges to the continuous-time estimate given by the Kalman--Bucy filter. We shall give bounds for the convergence rates for the variance of the discrepancy between these two estimates. The contribution of this article is to generalize the convergence results to systems with unbounded observation operators under different sets of assumptions, including systems with diagonaliz-able generators, systems with admissible observation operators, and systems with analytic semigroups. The proofs are based on applying the discrete-time Kalman filter on a dense, numerable subset on the time interval [0,T] and bounding the increments obtained. These bounds are obtained by studying the regularity of the underlying semigroup and the noise-free output., arXiv admin note: text overlap with arXiv:1408.1275

Published
2018
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26. Energy decay and stability of a perfectly matched layer for the wave equation

Author

Maryna Kachanovska, Sébastien Imperiale, Daniel H. Baffet, Marcus J. Grote, Department of Mathematics [Basel], University of Basel (Unibas), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Polytechnique de Paris (IP Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0056,LMH,LabEx Mathématique Hadamard(2011), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Numerical Analysis, Wave propagation, Applied Mathematics, Mathematical analysis, General Engineering, 010103 numerical & computational mathematics, Space (mathematics), Wave equation, 01 natural sciences, Stability (probability), Theoretical Computer Science, 010101 applied mathematics, Computational Mathematics, Perfectly matched layer, Computational Theory and Mathematics, Preprint, 0101 mathematics, [MATH]Mathematics [math], Constant (mathematics), Software, Energy (signal processing), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

In Grote and Sim (Efficient PML for the wave equation. Preprint, arXiv:1001.0319 [math:NA], 2010; in: Proceedings of the ninth international conference on numerical aspects of wave propagation (WAVES 2009, held in Pau, France, 2009), pp 370–371), a PML formulation was proposed for the wave equation in its standard second-order form. Here, energy decay and $$L^2$$ stability bounds in two and three space dimensions are rigorously proved both for continuous and discrete formulations with constant damping coefficients. Numerical results validate the theory.

Published
2018
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27. Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model

Author

Annabelle Collin, Sébastien Imperiale, Modélisation Mathématique pour l'Oncologie (MONC), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], UNICANCER-UNICANCER-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Université Paris-Saclay, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Nondimensionalization, Applied Mathematics, Quantitative Biology::Tissues and Organs, 010102 general mathematics, Mathematical analysis, Bidomain model, 01 natural sciences, Homogenization (chemistry), 010101 applied mathematics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, Symbolic convergence theory, 0101 mathematics, Mathematics
Abstract

International audience; The aim of this paper is to provide a complete mathematical analysis of the periodic homogenization procedure that leads to the macroscopic bidomain model in cardiac elec-trophysiology. We consider space-dependent and tensorial electric conductivities as well as space-dependent physiological and phenomenological non-linear ionic models. We provide the nondimensionalization of the bidomain equations and derive uniform estimates of the solutions. The homogenization procedure is done using 2-scale convergence theory which enables us to study the behavior of the non-linear ionic models in the homogenization process .

Published
2018
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28. Coupled variational formulations of linear elasticity and the DPG methodology

Author

Brendan Keith, Federico Fuentes, Leszek Demkowicz, Patrick Le Tallec, Institute for Computational Engineering and Sciences [Austin] (ICES), University of Texas at Austin [Austin], Instituto Venezolano de Investigaciones Cientificas (IVIC), Université Paris-Saclay, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Inria Saclay - Ile de France, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Numerical Analysis, Work (thermodynamics), Physics and Astronomy (miscellaneous), Applied Mathematics, Linear elasticity, Mathematical analysis, Mode (statistics), 65N12, 65N30, 010103 numerical & computational mathematics, Numerical Analysis (math.NA), [SPI.MECA.MSMECA]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Materials and structures in mechanics [physics.class-ph], 01 natural sciences, Stability (probability), Finite element method, Domain (mathematical analysis), Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Convergence (routing), FOS: Mathematics, Mathematics - Numerical Analysis, 0101 mathematics, Variety (universal algebra), Mathematics
Abstract

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually well-posed family of broken variational formulations of the original PDE. It can be exploited to solve challenging problems in a variety of physical scenarios where stability or a particular mode of convergence is desired in a part of the domain. The linear elasticity equations are solved in this work, but the approach can be applied to other equations as well. The broken variational formulations, which are essentially extensions of more standard formulations, are characterized by the presence of mesh-dependent broken test spaces and interface trial variables at the boundaries of the elements of the mesh. This allows necessary information to be naturally transmitted between adjacent subdomains, resulting in coupled variational formulations which are then proved to be globally well-posed. They are solved numerically using the DPG methodology, which is especially crafted to produce stable discretizations of broken formulations. Finally, expected convergence rates are verified in two different and illustrative examples., Comment: 24 pages, 5 figures

Published
2017
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29. Effective and energy-preserving time discretization for a general nonlinear poromechanical formulation

Author

Dominique Chapelle, Bruno Burtschell, Philippe Moireau, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Saclay, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Inria Saclay - Ile de France

Subjects
Coupling, Mathematical optimization, Work (thermodynamics), Discretization, Mechanical Engineering, 010103 numerical & computational mathematics, Dissipation, 01 natural sciences, Computer Science Applications, 010101 applied mathematics, Nonlinear system, Modeling and Simulation, Component (UML), Applied mathematics, General Materials Science, 0101 mathematics, Energy (signal processing), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Civil and Structural Engineering, Free energy principle, Mathematics
Abstract

Proposed time discretization scheme for general two-phase poromechanical model.Discrete energy estimate with the total free energy of the mixture, in a general nonlinear framework.Numerical examples with representative test problems. We consider a general nonlinear poromechanical model, formulated based on fundamental thermodynamics principle, suitable for representing the coupling of rapid internal fluid flows with large deformations of the solid, and compatible with a wide class of constitutive behavior. The objective of the present work is to propose for this model a time discretization scheme of the partitioned type, to allow the use of existing time schemes and possibly separate solvers for each component of the model, i.e. for the fluid and the solid. To that purpose, we adapt and extend an earlier proposed approach devised for fluid-structure interaction in an Arbitrary Lagrangian-Eulerian framework. We then establish an energy estimate for the resulting time scheme, in a form that is consistent with the underlying energy principle in the poromechanical formulation, up to some numerical dissipation effects and some perturbations that we have carefully identified and assessed. In addition, we provide some numerical illustrations of our numerical strategy with test problems that present typical features of large strains and rapid fluid flows, and also a case of singular transition related to total drainage. An example of challenging application envisioned for this model and associated numerical coupling scheme concerns the perfusion of the heart.

Published
2017
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30. Output error minimizing back and forth nudging method for initial state recovery

Author

Atte Aalto, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
0209 industrial biotechnology, General Computer Science, Observer (quantum physics), Back and forth nudging, Hardware_PERFORMANCEANDRELIABILITY, 02 engineering and technology, Output error minimization, 01 natural sciences, Linear dynamical system, 020901 industrial engineering & automation, Operator (computer programming), Control theory, Convergence (routing), FOS: Mathematics, Hardware_INTEGRATEDCIRCUITS, Applied mathematics, Variational data assimilation, 0101 mathematics, Electrical and Electronic Engineering, Observers, Mathematics - Optimization and Control, Mathematics, Mechanical Engineering, Observable, Dissipation, 010101 applied mathematics, Optimization and Control (math.OC), Control and Systems Engineering, Dissipative system, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Constant (mathematics), State estimation
Abstract

We show that for linear dynamical systems with skew-adjoint generators, the initial state estimate given by the back and forth nudging method with colocated feedback, converges to the minimizer of the discrepancy between the measured and simulated outputs - given that the observer gains are chosen suitably and the system is exactly observable. If the system's generator A is essentially skew-adjoint and dissipative (with not too much dissipation), the colocated feedback has to be corrected by the operator e^{At}e^{A*t} in order to obtain such convergence. In some special cases, a feasible approximation for this operator can be found analytically. The case with wave equation with constant dissipation will be demonstrated., Comment: This is the preprint version of the article. The final, published version is available on the journal's website

Published
2016
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31. Delayed Feedback Control Method For Calculating Space-Time Periodic Solutions Of Viscoelastic Problems

Author

Patrick Le Tallec, Ustim Khristenko, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), M. Papadrakakis, V. Papadopoulos, G. Stefanou, V. Plevris, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
0209 industrial biotechnology, Space time, Spectrum (functional analysis), 02 engineering and technology, [SPI.MECA.MSMECA]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Materials and structures in mechanics [physics.class-ph], Optimal control, 01 natural sciences, Viscoelasticity, Term (time), 020901 industrial engineering & automation, Vis- coelasticity, Control theory, 0103 physical sciences, Convergence (routing), Applied mathematics, Initial value problem, Steady rolling, Limit (mathematics), Delayed feedback control, 010306 general physics, Space-time periodic solution, Mathematics
Abstract

We are interested in fast techniques for calculating a periodic solution to viscoelastic evolution problems with a space-time periodic condition of ”rolling” type. Such a solution is usually computed as an asymptotic limit of the initial value problem with arbitrary initial data. We want to invent a control method, accelerating the convergence. The main idea is to modify our problem by introducing a feedback control term, based on a periodicity error, in order to accelerate the convergence to the desired periodic solution of the problem. First, an abstract evolution problem has been studied. From the analytic solution of the modified (controlled) problem, an efficient control has been found, optimizing the spectrum of the problem. The proposed control term can be mechanically interpreted, and its efficiency increases with the relaxation time. In order to confirm numerically the theoretical results, a finite element simulation has been carried out on a full 2D model for a steady rolling of a viscoelastic tyre with periodic sculpture. It has demonstrated that the controlled solution converges indeed faster than the non-controlled one, and that the efficiency of the method increases with the problem’s relaxation time, that is when the memory of the underlying problem is large.

Published
2016

32. Estimation from moments measurements for amyloid depolymerisation

Author

Aurora Armiento, Human Rezaei, Marie Doumic, Philippe Moireau, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Modelling and Analysis for Medical and Biological Applications (MAMBA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Unité de recherche Virologie et Immunologie Moléculaires (VIM (UR 0892)), Institut National de la Recherche Agronomique (INRA), ERC 306321, European Project: 306321,EC:FP7:ERC,ERC-2012-StG_20111012,SKIPPERAD(2012), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Unité de recherche Virologie et Immunologie Moléculaires (VIM), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
0301 basic medicine, Models, Molecular, Protein Folding, [SDV]Life Sciences [q-bio], Second moment of area, Protein aggregation, 01 natural sciences, oligomer, Polymerization, chemistry.chemical_compound, Amyloid disease, Static light scattering, MESH: amyloid, state estimation, education.field_of_study, Applied Mathematics, Neurodegenerative Diseases, General Medicine, 010101 applied mathematics, Biochemistry, protein stability, Modeling and Simulation, inverse problem, Thioflavin, Protein folding, General Agricultural and Biological Sciences, Biological system, Algorithms, Statistics and Probability, Amyloid, Prions, Population, General Biochemistry, Genetics and Molecular Biology, prion, 03 medical and health sciences, Protein Aggregates, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Animals, Humans, Computer Simulation, 0101 mathematics, education, data assimilation, MESH: protein stability, General Immunology and Microbiology, Models, Theoretical, Moment (mathematics), Kinetics, 030104 developmental biology, chemistry, transport equation, Protein Multimerization
Abstract

International audience; Estimating reaction rates and size distributions of protein polymers is an important step for understanding the mechanisms of protein misfolding and aggregation, a key feature for amyloid diseases. This study aims at {setting} this framework problem when the experimental measurements consist in the time-dynamics of a moment of the population (\emph{i.e.} for instance the total polymerised mass, as in Thioflavine T measurements, or the second moment measured by Static Light Scattering). We propose a general methodology, and we solve the problem theoretically and numerically in the case of a depolymerising system. We then apply our method to experimental data of degrading oligomers, and conclude that smaller aggregates of ovPrP protein should be more stable than larger ones. This has an important biological implication, since it is commonly admitted that small oligomers constitute the most cytotoxic species during prion misfolding process.

Published
2015
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33. A Luenberger observer for reaction-diffusion models with front position data

Author

Dominique Chapelle, Annabelle Collin, Philippe Moireau, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Modélisation Mathématique pour l'Oncologie (MONC), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], UNICANCER-UNICANCER-Inria Bordeaux - Sud-Ouest, European Union’s Seventh Framework Programme for research, technological development and demonstration, under grant agreement #611823 (VP2HF Project), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux]

Subjects
Numerical Analysis, Physics and Astronomy (miscellaneous), Observer (quantum physics), Eikonal equation, Estimation theory, Applied Mathematics, Cardiac electrophysiology, Kalman filter, Reaction-diffusion model, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, Computational Mathematics, Nonlinear system, Image processing, Control theory, Front propagation, Modeling and Simulation, Data assimilation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], State observer, Asymptotic expansion, Alpha beta filter, Mathematics
Abstract

International audience; We propose a Luenberger observer for reaction-diffusion models with propagating front features, and for data associated with the location of the front over time. Such models are considered in various application fields, such as electrophysiology, wild-land fire propagation and tumor growth modeling. Drawing our inspiration from image processing methods, we start by proposing an observer for the eikonal-curvature equation that can be derived from the reaction-diffusion model by an asymptotic expansion. We then carry over this observer to the underlying reaction-diffusion equation by an "inverse asymptotic analysis", and we show that the associated correction in the dynamics has a stabilizing effect for the linearized estimation error. We also discuss the extension to joint state-parameter estimation by using the earlier-proposed ROUKF strategy. We then illustrate and assess our proposed observer method with test problems pertaining to electrophysiology modeling, including with a realistic model of cardiac atria. Our numerical trials show that state estimation is directly very effective with the proposed Luenberger observer, while specific strategies are needed to accurately perform parameter estimation – as is usual with Kalman filtering used in a nonlinear setting – and we demonstrate two such successful strategies.

Published
2015
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34. Identification of weakly coupled multiphysics problems. Application to the inverse problem of electrocardiography

Author

Jean-Frédéric Gerbeau, Philippe Moireau, Cesare Corrado, Modélisation et calculs pour l'électrophysiologie cardiaque (CARMEN), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-IHU-LIRYC, Université Bordeaux Segalen - Bordeaux 2-CHU Bordeaux [Bordeaux]-CHU Bordeaux [Bordeaux], Numerical simulation of biological flows (REO), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, IHU-LIRYC, Université Bordeaux Segalen - Bordeaux 2-CHU Bordeaux [Bordeaux]-Université Bordeaux Segalen - Bordeaux 2-CHU Bordeaux [Bordeaux]-Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest

Subjects
Identification, Physics and Astronomy (miscellaneous), Computer science, Multiphysics, Synthetic data, Data assimilation, Control theory, FOS: Mathematics, Mathematics - Numerical Analysis, State observer, Electromechanics, Numerical Analysis, Applied Mathematics, Estimator, Reduced-Order Unscented Kalman Filter, Numerical Analysis (math.NA), Kalman filter, Inverse problem, Proper orthogonal decomposition, Computer Science Applications, Electrocardiogram, Computational Mathematics, Modeling and Simulation, Identifiability, [SDV.IB]Life Sciences [q-bio]/Bioengineering, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; 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.

Published
2015
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35. Data assimilation of time under-sampled measurements using observers, the wave-like equation example

Author

Alexandre Imperiale, Nicolae Cîndea, Philippe Moireau, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Control and Optimization, Discretization, numerical analysis, Numerical analysis, Sampling (statistics), Luenberger observers, Wave equation, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, interpolation, ACM: G.: Mathematics of Computing, Term (time), Computational Mathematics, Data assimilation, Control and Systems Engineering, Control theory, Applied mathematics, time under-sampled measurements, Spurious relationship, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Interpolation, Mathematics
Abstract

International audience; We propose a sequential data assimilation scheme using Luenberger type observers when only some space restricted time under-sampled measurements are available. More precisely, we consider a wave-like equation for which we assume known the restriction of the solution to an open non-empty subset of the spatial domain and for some time samples (typically the sampling step in time is much larger than the time discretization step). To assimilate the available data, two strategies are proposed and analyzed. The first strategy consists in assimilating data only if they are available and the second one in assimilating interpolation of the available data at all the discretization times. In order to tackle the spurious high frequencies which appear when we discretize the wave equation, for both strategies, we introduce a numerical viscous term. In this case, we prove some error estimates between the exact solution and our observers. Numerical simulations illustrate the theoretical results in the case of the one dimensional wave equation.

Published
2015
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36. Displacement Reconstructions in Ultrasound Elastography

Author

Sébastien Imperiale, Guillaume Bal, Department of Applied Physics and Applied Mathematics [New York] (APAM), Columbia University [New York], Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
medicine.medical_specialty, Computer science, General Mathematics, Acoustics, Physics::Medical Physics, 01 natural sciences, Displacement (vector), 030218 nuclear medicine & medical imaging, 03 medical and health sciences, symbols.namesake, 0302 clinical medicine, medicine, Medical imaging, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Reconstruction procedure, medicine.diagnostic_test, business.industry, Applied Mathematics, Ultrasound, Inverse problem, 010101 applied mathematics, Interferometry, Fourier transform, symbols, Elastography, Radiology, business
Abstract

International audience; We consider the reconstruction of internal elastic displacements from ultrasound measurements , which finds applications in the medical imaging modality called elastography. By appropriate interferometry and windowed Fourier transforms of the ultrasound measurements, we propose a reconstruction procedure of the vectorial structure of spatially varying elastic displacements in biological tissues. This provides a modeling and generalization of scalar reconstruction procedures routinely used in elastography. The proposed algorithm is justified using a single scattering approximation and local asymptotic analysis. Its validity is assessed by numerical simulations.

Published
2015
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37. Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models

Author

Marina Vidrascu, Dominique Chapelle, Cyril Touzé, Unité de Mécanique (UME), École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Numerical simulation of biological flows (REO), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
Discretization, Computational Mechanics, Shell (structure), Ocean Engineering, Geometry, Bifurcation diagram, Quadratic equation, [PHYS.MECA.STRU]Physics [physics]/Mechanics [physics]/Structural mechanics [physics.class-ph], stiffness evaluation, Mathematics, Coupling, MITC elements, bifurcation diagram, reduced-order models, Applied Mathematics, Mechanical Engineering, Mathematical analysis, geometric nonlinearity, Finite element method, Computational Mathematics, Nonlinear system, Computational Theory and Mathematics, [SPI.MECA.STRU]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph], finite elements, Reduction (mathematics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We propose a direct method for computing modal coupling coefficients - due to geometrically nonlinear effects - for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the finite element formulation, with specificities pertaining to the shell elements considered, namely, here elements of the ''Mixed Interpolation of Tensorial Components'' family (MITC). Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing - with a continuation procedure - the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes (NNMs), and we find that the most important part of the dynamics can be predicted with a single oscillator equation.

Published
2014
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38. Mathematical modelling of multi conductor cables

Author

Geoffrey Beck, Sébastien Imperiale, Patrick Joly, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris

Subjects
Asymptotic analysis, Applied Mathematics, Mathematical analysis, Lossy compression, Electromagnetic radiation, Conductor, symbols.namesake, Maxwell's equations, Maxwell equations, Electromagnetic field solver, symbols, Discrete Mathematics and Combinatorics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Telegrapher models, Analysis, Mathematics
Abstract

International audience; This paper proposes a formal justification of simplified 1D models for the propagation of electromagnetic waves in thin non-homogeneous lossy conductor cables. Our approach consists in deriving these models from an asymptotic analysis of 3D Maxwell’s equations. In essence, we extend and complete previous results to the multi-wires case.

Published
2014
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39. Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

Author

Sébastien Imperiale, Guillaume Bal, Cédric Bellis, François Monard, Department of Applied Physics and Applied Mathematics [New York] (APAM), Columbia University [New York], Matériaux et Structures (M&S), Laboratoire de Mécanique et d'Acoustique [Marseille] (LMA ), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Seattle], University of Washington [Seattle], École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Quantitative parameter identification, Applied Mathematics, Mathematical analysis, Linear elasticity, Isotropy, Stability (probability), Regularization (mathematics), Computer Science Applications, Theoretical Computer Science, Mathematics - Analysis of PDEs, Signal Processing, Isotropic solid, FOS: Mathematics, Numerical differentiation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Direct integration of a beam, Uniqueness, Internal data, Elastography, Lamé system, Mathematical Physics, Analysis of PDEs (math.AP), Mathematics, Regularization method
Abstract

Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach., Comment: 23 pages, 7 figures

Published
2014
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40. Wrinkle development analysis in thin sail-like structures using MITC shell finite elements

Author

Daniele Trimarchi, Dominic Taunton, Stephen R. Turnock, D. Chapelle, Marina Vidrascu, Fluid Structure Interactions Research Group, University of Southampton, Numerical simulation of biological flows (REO), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Modeling, analysis and control in computational structural dynamics (MACS), Inria Saclay - Ile de France, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Engineering, Shell (structure), MITC Shells, 02 engineering and technology, Type (model theory), 01 natural sciences, Measure (mathematics), Stability (probability), 010305 fluids & plasmas, 0203 mechanical engineering, 0103 physical sciences, Convergence (routing), Wrinkling Index, business.industry, Applied Mathematics, General Engineering, Structural engineering, Sail Modeling, Computer Graphics and Computer-Aided Design, Wrinkling, Finite element method, Range (mathematics), 020303 mechanical engineering & transports, Test case, business, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We propose a method of modelling sail type structures which captures the wrinkling behaviour of such structures. The method is validated through experimental and analytical test cases, particularly in terms of wrinkling prediction. An enhanced wrinkling index is proposed as a valuable measure characterizing the global wrinkling development on the deformed structure. The method is based on a pseudo-dynamic finite element procedure involving non-linear MITC shell elements. The major advantage compared to membrane models generally used for this type of analysis is that no ad hoc wrinkling model is required to control the stability of the structure. We demonstrate our approach to analyse the behaviour of various structures with spherical and cylindrical shapes, characteristic of downwind sails over a rather wide range of shape and constitutive parameters. In all cases convergence is reached and the overall flying shape is most adequately represented, which shows that our approach is a most valuable alternative to standard techniques to provide deeper insight into the physical behaviour. Limitations appear only in some very special instances in which local wrinkling-related instabilities are extremely high and would require specific additional treatments, out of the scope of the present study.

Published
2013
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41. Data Assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description

Author

Anne-Céline Boulanger, Philippe Moireau, Jacques Sainte-Marie, Benoît Perthame, Numerical Analysis, Geophysics and Ecology (ANGE), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Modelling and Analysis for Medical and Biological Applications (MAMBA), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Conservation law, Applied Mathematics, General Mathematics, First-order partial differential equation, shallow water system, 35L40, 35L65, 65M08, 76D55, 93E11, Burgers' equation, Nonlinear system, hyperbolic conservation law, Data assimilation, Optimization and Control (math.OC), Convergence (routing), kinetic formulation, nudging, FOS: Mathematics, Applied mathematics, State observer, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Variable (mathematics), Mathematics
Abstract

International audience; Developing robust data assimilation methods for hyperbolic conservation laws is a challenging subject. Those PDEs indeed show no dissipation effects and the input of additional information in the model equations may introduce errors that propagate and create shocks. We propose a new approach based on the kinetic description of the conservation law. A kinetic equation is a first order partial differential equation in which the advection velocity is a free variable. In certain cases, it is possible to prove that the nonlinear conservation law is equivalent to a linear kinetic equation. Hence, data assimilation is carried out at the kinetic level, using a Luenberger observer also known as the nudging strategy in data assimilation. Assimilation then resumes to the handling of a BGK type equation. The advantage of this framework is that we deal with a single "linear" equation instead of a nonlinear system and it is easy to recover the macroscopic variables. The study is divided into several steps and essentially based on functional analysis techniques. First we prove the convergence of the model towards the data in case of complete observations in space and time. Second, we analyze the case of partial and noisy observations. To conclude, we validate our method with numerical results on Burgers equation and emphasize the advantages of this method with the more complex Saint-Venant system.

Published
2013
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42. A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems -- Analysis, assessments and applications to parameter estimation

Author

Jacques Sainte-Marie, Dominique Chapelle, Philippe Moireau, Asven Gariah, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Modeling, analysis and control in computational structural dynamics (MACS), Inria Paris-Rocquencourt, Numerical Analysis, Geophysics and Ecology (ANGE), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Centre d'Études Techniques Maritimes et Fluviales (CETMEF), Avant création Cerema, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Mathematical optimization, Proper Orthogonal Decomposition, FitzHugh-Nagumo equations, 010103 numerical & computational mathematics, 01 natural sciences, Applied mathematics, 0101 mathematics, Remainder, Galerkin method, Mathematics, Numerical Analysis, Sequential estimation, Estimation theory, Applied Mathematics, Cardiac modeling, Real image, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Parameter variations, Proper orthogonal decomposition, A priori and a posteriori, Estimation, Mathematics Subject Classification: 65M60, 35A35, 35B45, 93E10, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Interpolation
Abstract

We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder – in the construction procedure – and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the FitzHugh − Nagumo 1D model. Finally, we give detailed illustrations of the approach in two parameter estimation applications, the first in a variational estimation framework with the FitzHugh − Nagumo model, and the second with a beating heart mechanical model for which we employ a sequential estimation method to characterize model parameters using real image data in a clinical case.

Published
2013
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43. A surface-based electrophysiology model relying on asymptotic analysis and motivated by cardiac atria modeling

Author

Annabelle Collin, Dominique Chapelle, Jean-Frédéric Gerbeau, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Numerical simulation of biological flows (REO), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Surface (mathematics), 0303 health sciences, Asymptotic analysis, Mathematical optimization, Computational model, Field (physics), Computer science, Computational electrophysiology, Applied Mathematics, Thin domains, Cardiac modeling, 01 natural sciences, 010101 applied mathematics, 03 medical and health sciences, Modeling and Simulation, Convergence (routing), Statistical physics, Limit (mathematics), 0101 mathematics, AMS Subject Classification: 22E46, 53C35, 57S20, Anisotropy, Representation (mathematics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 030304 developmental biology
Abstract

International audience; Computational electrophysiology is a very active field with tremendous potential in medical applications, albeit leads to highly intensive simulations. We here propose a surface-based electrophysiology formulation, motivated by the modeling of thin structures such as cardiac atria, which greatly reduces the size of the computational models. Moreover, our model is specifically devised to retain the key features associated with the anisotropy in the diffusion effects induced by the fiber architecture, with rapid variations across the thickness which cannot be adequately represented by naive averaging strategies. Our proposed model relies on a detailed asymptotic analysis in which we identify a limit model and establish strong convergence results. We also provide detailed numerical assessments which confirm an excellent accuracy of the surface-based model -- compared with the reference 3D model -- including in the representation of a complex phenomenon, namely, spiral waves.

Published
2013
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44. Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section

Author

Patrick Joly, Sébastien Imperiale, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Département Imagerie et Simulation pour le Contrôle (DISC), Laboratoire d'Intégration des Systèmes et des Technologies (LIST), Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Laboratoire d'Intégration des Systèmes et des Technologies (LIST (CEA))

Subjects
Physics, Numerical Analysis, Coaxial cable, Wave propagation, Applied Mathematics, Mathematical analysis, Scalar (mathematics), Lossy compression, Electromagnetic radiation, law.invention, Computational Mathematics, symbols.namesake, Maxwell's equations, law, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Electric potential, Coaxial
Abstract

International audience; In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. The full 3D Maxwell equations, that described the propagation of current and elec- tric potential in such cables, are classically not tackled directly, but instead a 1D scalar model known as the telegraphist's model is used. We aim at justifying, by means of asymptotic analysis, a time-domain "homogenized" telegraphist's model. This model, which includes a non-local in time op- erator, is obtained via asymptotic analysis, for a lossy coaxial cable whose cross-section is not homogeneous.

Published
2013
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45. Mathematical and numerical modelling of piezoelectric sensors

Author

Patrick Joly, Sébastien Imperiale, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Numerical Analysis, Piezoelectric sensor, Applied Mathematics, Computation, Numerical analysis, Mathematical analysis, Finite difference, Piezoelectricity, 010103 numerical & computational mathematics, quasi-static approximation, 01 natural sciences, Finite element method, 010101 applied mathematics, Computational Mathematics, Quasistatic approximation, Modeling and Simulation, ultrasonic sensors, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Electric potential, 0101 mathematics, Analysis, Mathematics
Abstract

International audience; The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element / finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented.

Published
2012

46. Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section

Author

Patrick Joly, Sébastien Imperiale, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)

Subjects
Mathematical optimization, Applied Mathematics, Mechanical Engineering, 010102 general mathematics, Mathematical analysis, 3d model, 01 natural sciences, Electromagnetic radiation, 010101 applied mathematics, Cross section (physics), Telegraphist's equation, Non homogeneous, Coaxial cables, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Coaxial, Error estimates, Non-homogeneous cross-section, Mathematics
Abstract

This paper is the first contribution towards the rigorous justification of asymptotic 1D models for the time-domain simulation of the propagation of electromagnetic waves in coaxial cables. Our general objective is to derive error estimates between the “exact” solution of the full 3D model and the “approximate” solution of the 1D model known as the Telegraphist’s equation.

Published
2012
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47. Stochastic nonlinear model for somatic cell population dynamics during ovarian follicle activation

Author

Frédérique Clément, Frédérique Robin, Romain Yvinec, Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Dynamiques de populations multi-échelles pour des systèmes physiologiques (MUSCA), Mathématiques et Informatique Appliquées du Génome à l'Environnement [Jouy-En-Josas] (MaIAGE), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Physiologie de la reproduction et des comportements [Nouzilly] (PRC), Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Physiologie de la reproduction et des comportements [Nouzilly] (PRC), Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Institut Français du Cheval et de l'Equitation [Saumur]-Institut National de la Recherche Agronomique (INRA), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Inria Saclay - Ile de France, Institut Français du Cheval et de l'Equitation [Saumur] (IFCE)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Français du Cheval et de l'Equitation [Saumur] (IFCE)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Mathématiques et Informatique Appliquées du Génome à l'Environnement [Jouy-En-Josas] (MaIAGE), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Institut Français du Cheval et de l'Equitation [Saumur] (IFCE)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École polytechnique (X)-Inria Saclay - Ile de France, Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Mathématiques et Informatique Appliquées du Génome à l'Environnement [Jouy-En-Josas] (MaIAGE), and Institut Français du Cheval et de l'Equitation [Saumur]-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)

Subjects
Cell type, Somatic cell, Finite state projection, [SDV]Life Sciences [q-bio], Cell, Population, Stochastic coupling techniques, Maximum likelihood estimates, Biology, Models, Biological, 01 natural sciences, 010305 fluids & plasmas, 03 medical and health sciences, Ovarian Follicle, Precursor cell, 0103 physical sciences, medicine, Animals, Ovarian follicle, education, Embedded Markov chain, 60J85 · 60J28 · 92D25 · 62M05, Cell Proliferation, 030304 developmental biology, Likelihood Functions, Stochastic Processes, 0303 health sciences, education.field_of_study, Stochastic cell population model, Cell growth, Applied Mathematics, First passage time, Agricultural and Biological Sciences (miscellaneous), Cell biology, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], medicine.anatomical_structure, Nonlinear Dynamics, Modeling and Simulation, Quantitative Biology - Cell Behavior, Female, First-hitting-time model, Mathematics - Probability
Abstract

In mammals, female germ cells are sheltered within somatic structures called ovarian follicles, which remain in a quiescent state until they get activated, all along reproductive life. We investigate the sequence of somatic cell events occurring just after follicle activation, starting by the awakening of precursor somatic cells, and their transformation into proliferative cells. We introduce a nonlinear stochastic model accounting for the joint dynamics of the two cell types, and allowing us to investigate the potential impact of a feedback from proliferative cells onto precursor cells. To tackle the key issue of whether cell proliferation is concomitant or posterior to cell awakening, we assess both the time needed for all precursor cells to awake, and the corresponding increase in the total cell number with respect to the initial cell number. Using the probabilistic theory of first passage times, we design a numerical scheme based on a rigorous Finite State Projection and coupling techniques to compute the mean extinction time and the cell number at extinction time. We find that the feedback term clearly lowers the number of proliferative cells at the extinction time. We calibrate the model parameters using an exact likelihood approach. We carry out a comprehensive comparison between the initial model and a series of submodels, which helps to select the critical cell events taking place during activation, and suggests that awakening is prominent over proliferation., Comment: Accepted in Journal of Mathematical Biology

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