40 research outputs found
IGS: an IsoGeometric approach for Smoothing on surfaces
We propose an Isogeometric approach for smoothing on surfaces, namely
estimating a function starting from noisy and discrete measurements. More
precisely, we aim at estimating functions lying on a surface represented by
NURBS, which are geometrical representations commonly used in industrial
applications. The estimation is based on the minimization of a penalized
least-square functional. The latter is equivalent to solve a 4th-order Partial
Differential Equation (PDE). In this context, we use Isogeometric Analysis
(IGA) for the numerical approximation of such surface PDE, leading to an
IsoGeometric Smoothing (IGS) method for fitting data spatially distributed on a
surface. Indeed, IGA facilitates encapsulating the exact geometrical
representation of the surface in the analysis and also allows the use of at
least globally continuous NURBS basis functions for which the 4th-order
PDE can be solved using the standard Galerkin method. We show the performance
of the proposed IGS method by means of numerical simulations and we apply it to
the estimation of the pressure coefficient, and associated aerodynamic force on
a winglet of the SOAR space shuttle
Effect of fibre orientation and bulk modulus on the electromechanical modelling of human ventricles
This work concerns the mathematical and numerical modeling of the heart. The aim is to enhance the understanding of the cardiac function in both physiological and pathological conditions. Along this road, a challenge arises from the multi-scale and multi-physics nature of the mathematical problem at hand. In this paper, we propose an electromechanical model that, in bi-ventricle geometries, combines the monodomain equation, the Bueno-Orovio minimal ionic model, and the Holzapfel-Ogden strain energy function for the passive myocardial tissue modelling together with the active strain approach combined with a model for the transmurally heterogeneous thickening of the myocardium. Since the distribution of the electric signal is dependent on the fibres orientation of the ventricles, we use a Laplace-Dirichlet Rule-Based algorithm to determine the myocardial fibres and sheets configuration in the whole bi-ventricle. In this paper, we study the influence of different fibre directions and incompressibility constraint and penalization on the compressibility of the material (bulk modulus) on the pressure-volume relation simulating a full heart beat. The coupled electromechanical problem is addressed by means of a fully segregated scheme. The numerical discretization is based on the Finite Element Method for the spatial discretization and on Backward Differentiation Formulas for the time discretization. The arising non-linear algebraic system coming from application of the implicit scheme is solved through the Newton method. Numerical simulations are carried out in a patient-specific biventricle geometry to highlight the most relevant results of both electrophysiology and mechanics and to compare them with physiological data and measurements. We show how various fibre configurations and bulk modulus modify relevant clinical quantities such as stroke volume, ejection fraction and ventricle contractility
Deep learning-based reduced order models in cardiac electrophysiology
Predicting the electrical behavior of the heart, from the cellular scale to
the tissue level, relies on the formulation and numerical approximation of
coupled nonlinear dynamical systems. These systems describe the cardiac action
potential, that is the polarization/depolarization cycle occurring at every
heart beat that models the time evolution of the electrical potential across
the cell membrane, as well as a set of ionic variables. Multiple solutions of
these systems, corresponding to different model inputs, are required to
evaluate outputs of clinical interest, such as activation maps and action
potential duration. More importantly, these models feature coherent structures
that propagate over time, such as wavefronts. These systems can hardly be
reduced to lower dimensional problems by conventional reduced order models
(ROMs) such as, e.g., the reduced basis (RB) method. This is primarily due to
the low regularity of the solution manifold (with respect to the problem
parameters) as well as to the nonlinear nature of the input-output maps that we
intend to reconstruct numerically. To overcome this difficulty, in this paper
we propose a new, nonlinear approach which exploits deep learning (DL)
algorithms to obtain accurate and efficient ROMs, whose dimensionality matches
the number of system parameters. Our DL approach combines deep feedforward
neural networks (NNs) and convolutional autoencoders (AEs). We show that the
proposed DL-ROM framework can efficiently provide solutions to parametrized
electrophysiology problems, thus enabling multi-scenario analysis in
pathological cases. We investigate three challenging test cases in cardiac
electrophysiology and prove that DL-ROM outperforms classical projection-based
ROMs.Comment: 28 page
Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model
Abstract We describe the behavior of a deformable porous material by means of a poro-hyperelastic model that has been previously proposed in Chapelle and Moireau (2014) under general assumptions for mass and momentum balance and isothermal conditions for a two-component mixture of fluid and solid phases. In particular, we address here a linearized version of the model, based on the assumption of small displacements. We consider the mathematical analysis and the numerical approximation of the problem. More precisely, we carry out firstly the well-posedness analysis of the model. Then, we propose a numerical discretization scheme based on finite differences in time and finite elements for the spatial approximation; stability and numerical error estimates are proved. Particular attention is dedicated to the study of the saddle-point structure of the problem, that turns out to be interesting because velocities of the fluid phase and of the solid phase are combined into a single quasi-incompressibility constraint. Our analysis provides guidelines to select the componentwise polynomial degree of approximation of fluid velocity, solid displacement and pressure, to obtain a stable and robust discretization based on Taylor–Hood type finite element spaces. Interestingly, we show how this choice depends on the porosity of the mixture, i.e. the volume fraction of the fluid phase
Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations
Since myocardial fibers drive the electric signal propagation throughout the
myocardium, accurately modeling their arrangement is essential for simulating
heart electrophysiology (EP). Rule-Based-Methods (RBMs) represent a commonly
used strategy to include cardiac fibers in computational models. A particular
class of such methods is known as Laplace-Dirichlet-Rule-Based-Methods (LDRBMs)
since they rely on the solution of Laplace problems. In this work we provide a
unified framework, based on LDRBMs, for generating full heart muscle fibers.
First, we review existing ventricular LDRBMs providing a communal mathematical
description and introducing also some modeling improvements with respect to the
existing literature. We then carry out a systematic comparison of LDRBMs based
on meaningful biomarkers produced by numerical EP simulations. Next we propose,
for the first time, a LDRBM to be used for generating atrial fibers. The new
method, tested both on idealized and realistic atrial models, can be applied to
any arbitrary geometries. Finally, we present numerical results obtained in a
realistic whole heart where fibers are included for all the four chambers using
the discussed LDRBMs
Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium
We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation
of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison
between basis functions of degree p ≥ 1 and globally C
k
-continuous, with k = 0 or p − 1, to find
the most accurate approximation of a propagating front with the minimal number of degrees of freedom.
We show that B-spline basis functions of degree p ≥ 1, which are C
p−1
-continuous capture accurately the
front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that,
for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally,
we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with
physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical
scheme retains its favorable approximation properties also in a more realistic setting
Reduced basis method and error estimation for parametrized optimal control problems with control constraints
We propose a Reduced Basis method for the solution of parametrized optimal control problems with control constraints for which we extend the method proposed in Dedè, L. (SIAM J. Sci. Comput. 32:997, 2010) for the unconstrained problem. The case of a linear-quadratic optimal control problem is considered with the primal equation represented by a linear parabolic partial differential equation. The standard offline-online decomposition of the Reduced Basis method is employed with the Finite Element approximation as the "truth" one for the offline step. An error estimate is derived and an heuristic indicator is proposed to evaluate the Reduced Basis error on the optimal control problem at the online step; also, the indicator is used at the offline step in a Greedy algorithm to build the Reduced Basis space. We solve numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems. © 2011 Springer Science+Business Media, LLC
Modeling the cardiac electromechanical function: A mathematical journey
In this paper we introduce the electromechanical mathematical model of the human heart. After deriving it from physical first principles, we discuss its mathematical properties and the way numerical methods can be set up to obtain numerical approximations of the (otherwise unachievable) mathematical solutions. The major challenges that we need to face-e.g., possible lack of initial and boundary data, the trade off between increasing the accuracy of the numerical model and its computational complexity-are addressed. Numerical tests here presented have a twofold aim: to show that numerical solutions match the expected theoretical rate of convergence, and that our model can provide a preliminary valuable tool to face problems of clinical relevance
Isogeometric Analysis of a Phase Field Model for Darcy Flows with Discontinuous Data
The authors consider a phase field model for Darcy flows with discontinuous data in porous media; specifically, they adopt the Hele-Shaw-Cahn-Hillard equations of [Lee, Lowengrub, Goodman, Physics of Fluids, 2002] to model flows in the Hele-Shaw cell through a phase field formulation which incorporates discontinuities of physical data, namely density and viscosity, across interfaces. For the spatial approximation of the problem, the authors use NURBS—based isogeometric analysis in the framework of the Galerkin method, a computational framework which is particularly advantageous for the solution of high order partial differential equations and phase field problems which exhibit sharp but smooth interfaces. In this paper, the authors verify through numerical tests the sharp interface limit of the phase field model which in fact leads to an internal discontinuity interface problem; finally, they show the efficiency of isogeometric analysis for the numerical approximation of the model by solving a benchmark problem, the so-called “rising bubble” problem