Fluid dynamics of an idealized left ventricle: the extended Nitsche's method for the treatment of heart valves as mixed time varying boundary conditions

In this work, we study the blood flow dynamics in idealized left ventricles (LV) of the human heart modelled by the Navier-Stokes equations with mixed time varying boundary conditions. The latter are introduced for simulating the functioning of the aortic and mitral valves. On the basis of the extended Nitsche's method firstly presented in [Juntunen and Stenberg, Mathematics of Computation, 2009], we propose a formulation allowing an efficient and straightforward numerical treatment of the opening and closing phases of the heart valves that are associated with different kind of boundary conditions, namely, natural and essential, switching during each heartbeat. Moreover, our formulation already includes terms preventing the numerical instabilities associated to backflow divergence, that is, nonphysical reinflow at the valves. We present and discuss numerical results for the LV obtained by means of isogeometric analysis for the spatial approximation with the aim of both analysing the formulation and showing the effectiveness of the approach. In particular, we show that the formulation allows to reproduce meaningful results even in idealized LV. Copyright (C) 2017 John Wiley & Sons, Ltd.

MATHICSE Technical Report: Fluid dynamics of an idealized left ventricle: the extended Nitsche’s method for the treatment of heart valves as mixed time varying boundary conditions

Alfio Quarteroni, Anna Tagliabue

In this work, we study the blood flow dynamics in idealized left ventricles (LV)of the human heart modelled by the Navier-Stokes equations with mixed time varying boundary conditions (BCs). The latter are introduced for simulating the functioning of the aortic and mitral valves. Based on the extended Nitsche’s method firstly presented in [Juntunen and Stenberg, Mathematicsof Computation, 2009], we propose a formulation allowing an efficient and straight forward numerical treatment of the opening and closing phases of the heart valves which are associated to different kind of BCs, namely natural and essential. Moreover, our formulation includes terms preventing the numerical instabilities associated to back flow divergence, i.e. nonphysical reinflow at the valves. We present and discuss numerical results for the LV obtained by means of Isogeometric Analysis for the spatial approximation with the aim of both analysing the formulation and showing the effectiveness of the approach. In particular, we show that the formulation allows to reproduce meaningful results even in idealized LV.

MATHICSE2015

Modeling the fluid dynamics of the heart: from blood flows in idealized left ventricles to patient-specific aortic valve simulations

Simone Deparis, Elena Faggiano, Marco Fedele, Davide Forti, Alfio Quarteroni, Anna Tagliabue

The computational fluid dynamics of the heart represents a challenging task both in terms of mathematical and numerical modeling; this is mainly due to the pulsatile nature of the blood flow, to its complex interaction with the valves, and, more in general, to the reciprocal action of the components responsible for the heart functioning. Even if one focuses on the study of the left ventricle, the blood flow patterns result to be significantly dependent on the mechanical contraction and relaxation of the muscle, on the conformation of the chamber, and on the interaction with the valves, which define a complex fluid-structure interaction problem. In this respect, also the blood flow in the aorta, and hence in the downstream circulation, is strongly affected by the aortic valve, whose behavior should be suitably mathematically and numerically modeled. In this work, we firstly focus on the study of the fluid dynamics inside the left ventricle in idealized configurations, for which we propose a mathematical model based on the Navier-Stokes equations endowed with mixed, time-dependent boundary conditions, which allow a simplified treatment of the aortic and mitral valves’ behavior. In this idealized setting, we perform numerical simulations which highlight the role and influence of modeling the valves to study and characterize the blood flow patterns inside the ventricle, as well as other parameters of clinical relevance. In addition, we consider a reduced, patient-specific fluid-structure-interaction model for the simulation of the blood flow through the aortic valve. Specifically, we propose an efficient coupled model which represents the valve dynamics by means of a zero-dimensional (0D) equation with the opening angle as primitive variable, while the blood flow by means of the full 3D Navier-Stokes equations. In this coupled model, the valve’s leaflets, which are reconstructed from MRI data of the open and closed configurations for a specific patient, influence the Navier-Stokes equations by means of resistive immersed surfaces, whose position depends on the opening angle of the valve. Moreover, the dynamics of the valve described by the 0D model is dependent on the velocity and pressure variables, specifically on the pressure jump and the flow rate through the valve itself. We per form patient-specific numerical simulations of the aortic valve based on this reduced 3D-0D model, for which we highlight its ability to correctly capture the fluid dynamics indicators expected for the patient.

2015

Dynamical Low Rank approximation of PDEs with random parameters

Eleonora Musharbash

In this work, we focus on the Dynamical Low Rank (DLR) approximation of PDEs equations with random parameters. This can be interpreted as a reduced basis method, where the approximate solution is expanded in separable form over a set of few deterministic basis functions at each time, with the peculiarity that both the deterministic modes and the stochastic coefficients are computed on the fly and are free to adapt in time so as best describe the structure of the random solution. Our first goal is to generalize and reformulate in a variational setting the Dynamically Orthogonal (DO) method, proposed by Sapsis and Lermusiaux (2009) for the approximation of fluid dynamic problems with random initial conditions. The DO method is reinterpreted as a Galerkin projection of the governing equations onto the tangent space along the approximate trajectory to the manifold M_S , given by the collection of all functions which can be expressed as a sum of S linearly independent deterministic modes combined with S linearly independent stochastic modes. Depending on the parametrization of the tangent space, one obtains a set of nonlinear differential equations, suitable for numerical integration, for both the coefficients and the basis functions of the approximate solution. By formalizing the DLR variational principle for parabolic PDEs with random parameters we establish a precise link with similar techniques developed in different contexts such as the Multi-Configuration Time-Dependent Hartree method in quantum dynamics and the Dynamical Low-Rank approximation in the finite dimensional setting. By the use of curvature estimates for the approximation manifold M_S , we derive a theoretical bound for the approximation error of the S-terms DO solution by the corresponding S-terms best approximation at each time instant. The bound is applicable for full rank DLR approximate solutions on the largest time interval in which the best S-terms approximation is continuously differentiable in time. Secondly, we focus on parabolic equations, especially incompressible Navier-Stokes equations, with random Dirichlet boundary conditions and we propose a DLR technique which allows for the strong imposition of such boundary conditions. We show that the DLR variational principle can be set in the constrained manifold of all S rank random fields with a prescribed value on the boundary, expressed in low-rank format, with rank M smaller than S. We characterize the tangent space to the constrained manifold by means of the Dual Dynamically Orthogonal formulation, in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions, consistent with the original problem. The same formulation is also used to conveniently include the incompressibility constraint when dealing with incompressible Navier-Stokes equations with random parameters. Finally, we extend the DLR approach for the approximation of wave equations with random parameters. We propose the Symplectic DO method, according to which the governing equation is rewritten in Hamiltonian form and the approximate solution is sought in the low dimensional manifold of all complex-valued random fields with fixed rank. Recast in the real setting, the approximate solution is expanded over a set of a few dynamical symplectic deterministic modes and satisfies the symplectic projection of the Hamiltonian system into the tangent space of the approximation manifold along the trajectory.