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Fluid-Solid Coupling for the Simulation of Left Ventricular Mechanics
Abstract
In this thesis, an anatomically accurate finite element model of the left ventricle is presented for coupled fluid- solid simulation of blood flow and tissue mechanics during passive filling. Beginning from continuum theory, general conservation principles – common to both blood flow and tissue mechanics – are derived in the ALE frame. The deviation of these underlying principles for fluid and solid mechanics is then discussed, and their respective weak forms shown. A coupling technique is then devised which allows non-conformity of the fluid- solid problems by introducing an additional weak constraint. It is then proven, for a linear fluid-solid system, that this coupling strategy produces unique stable solutions which, in the discrete setting, converge optimally (under certain restrictions on the solution’s smoothness). The methods discussed are subsequently implemented into a parallel software framework, and extensively verified using numerical experiments. The culmination of these efforts enables the creation of a left ventricular heart model (which is shown to have bounded a priori energy estimates), incorporating known information on tissue architecture and material properties. This model is then used to conduct a comprehensive energy analysis of passive filling in the model left ventricle.
... In this paper, we consider a monolithic ALE FSI technique that is able to use non-conforming meshes at the interface, 5,[15][16][17] such that meshes can be designed based on the requirements of the physics of the coupled subsystems leading to improved accuracy and to decreased computational cost (by avoiding underrefinement and overrefinement, respectively). Coupling of subdomain equations is achieved via introduction of an additional coupling domain and enforcing interface constraints by means of a Lagrange multiplier variable. ...
... Coupling of subdomain equations is achieved via introduction of an additional coupling domain and enforcing interface constraints by means of a Lagrange multiplier variable. The method has been studied regarding stability and convergence 15,18 and successfully applied to various biomedical engineering problems, such as the simulation of whole-heart and left ventricular mechanics. [15][16][17]19,20 Previously, the method has been used for coupling the non-conservative ALE Navier-Stokes equations and the governing equations for quasi-static/transient finite elasticity. ...
... The method has been studied regarding stability and convergence 15,18 and successfully applied to various biomedical engineering problems, such as the simulation of whole-heart and left ventricular mechanics. [15][16][17]19,20 Previously, the method has been used for coupling the non-conservative ALE Navier-Stokes equations and the governing equations for quasi-static/transient finite elasticity. It has been extended recently to enable modeling of turbulent flow phenomena by a stabilized cG(1)cG(1) scheme 21 to extend the use of the method over a larger range of Reynolds numbers. ...
This paper details the validation of a non-conforming arbitrary Lagrangian-Eulerian fluid-structure interaction (FSI) technique using a recently developed experimental 3D FSI benchmark problem. Numerical experiments for steady and transient test cases of the benchmark were conducted employing an inf-sup stable and a general Galerkin scheme. The performance of both schemes is assessed. Spatial refinement with three mesh refinement levels and fluid domain truncation with two fluid domain lengths are studied as well as employing a sequence of increasing time step sizes for steady-state cases. How quickly an approximate steady-state or periodic steady-state is reached is investigated and quantified based on error norm computations. Comparison of numerical results with experimental phase-contrast magnetic resonance imaging data shows very good overall agreement including governing of flow patterns observed in the experiment.
... This limitation stems largely from difficulties involved in coupling non-conforming domains as well as needs to maintain computational tractability. In Nordsletten [16], a linear system fluid–solid system was developed which was coupled using an additional Lagrange multiplier on the fluid–solid interface. Through the proper construction of inf-sup stable spaces, the linear fluid–solid system was proven to admit unique solutions and optimal error estimates. ...
... This scalar field is then used as a Lagrange multiplier, constraining the displacement field, u, to satisfy both Equations (3a) and (3b). Although the form of the quasi-static solid model in (3) varies from the conservation equations (1) used to model the blood, these two forms are closely related and the quasi-static system can be derived as a special case of the previously introduced ALE form [16] (though r s and r f generally vary). The mechanical model requires characterization of the constitutive form defining the relationship between r s , discussed later in Section 2.2.3, and the myocardial displacement, and the field of actively developed tension. ...
... This approach provides a mathematically stable method for coupling fluid and solid bodies of different discretizations. The introduction of the coupling constraints via an additional Lagrange multiplier has been proven to be well posed for linear coupled problems and demonstrated numerically in a number of test examples [17, 16]. Notably, this approach also ensures conservation of mechanical energy for the entire coupled system due to the weak Lagrange multiplier formulation—a necessary requirement for energetic analysis (see Section 3). ...
Understanding the underlying feedback mechanisms of fluid/solid coupling and the role it plays in heart function is crucial for characterizing normal heart function and its behavior in disease. To improve this understanding, an anatomically accurate computational model of fluid–solid mechanics in the left ventricle is presented which assesses both the passive diastolic and active systolic phases of the heart. Integrating multiple data which characterize the hemodynamical and tissue mechanical properties of the heart, a numerical approach was applied which allows non-conformity in an optimal finite element scheme (J. Comp. Phys., submitted; Fluid–solid coupling for the simulation of left ventricular mechanics, University of Oxford, 2009). This approach is applied to look specifically at left ventricular fluid/solid coupling, allowing quantitative assessment of blood flow through the left ventricle, pressure distributions, activation and the loss of mechanical energy due to viscous dissipation. Copyright © 2010 John Wiley & Sons, Ltd.
... To write the ALE weak form, consider the velocity and pressure, (v, p), on the reference domain, K 1 . The state variable spaces are defined as [64,65,54], ...
... In Problem 2, a solid mechanical system was introduced. Here, we follow the common approach of considering the hyperelastic body in the Lagrangian reference/coordinate frame [13,54]. Similar to the ALE case, reference and physical geometries (denoted K 2 and X 2 , respectively) are defined. ...
... While the solid map is given by the displacement field u h , the fluid map is arbitrary so long as it adheres to the movement on the coupled boundary. The selection of this map satisfies the weak Laplacian problem [58,54], ...
In this study, a Lagrange multiplier technique is developed to solve problems of coupled mechanics and is applied to the case of a Newtonian fluid coupled to a quasi-static hyperelastic solid. Based on theoretical developments in [57], an additional Lagrange multiplier is used to weakly impose displacement/velocity continuity as well as equal, but opposite, force. Through this approach, both mesh conformity and kinematic variable interpolation may be selected independently within each mechanical body, allowing for the selection of grid size and interpolation most appropriate for the underlying physics. In addition, the transfer of mechanical energy in the coupled system is proven to be conserved. The fidelity of the technique for coupled fluid–solid mechanics is demonstrated through a series of numerical experiments which examine the construction of the Lagrange multiplier space, stability of the scheme, and show optimal convergence rates. The benefits of non-conformity in multi-physics problems is also highlighted. Finally, the method is applied to a simplified elliptical model of the cardiac left ventricle.
... • the monolithic schemes [20,29,45,55,64,72,80,81,93,115], in which the nonlinear algebraic system arising from the discretization of the FSI problem is solved as a whole by means of Newton or inexact-Newton schemes. ...
... The FE-PPE approach described above is applied in this chapter to estimate relative pressures on a series of test cases with increasing complexity. The problems presented are solved using CHeart, a multi-physics finite elements solver designed by Nordsletten [140,141] and developed by the CHeart team at King's College London 3 . CHeart simulates the mechanical physics, fluid dynamics and electrophysiology of the human heart by combining fast, scaling codes in FORTRAN 90 to MPI parallel computing and advanced GPU technologies to enable computation of expensive problems using arbitrarily defined element types and interpolation orders. ...
The presence of obstructions to blood flow in the human cardiovascular system can lead to an inability to efficiently and effectively perfuse downstream tissues and increase the work demand of the heart. In this scenario, the pressure drop through a vascular obstruction is a biomarker adopted in clinical guidelines for the management of several disease conditions, such as aortic coarctation and valvular stenosis. While extensively used clinically, current methods and tools to assess the pressure drop suffer from either the associated risks of invasive catheterization procedures, or potential inaccuracies from non-invasive pressure estimations due to simplified formulations or inter-observer variability.
The primary aim of this thesis is to further develop non-invasive methods to estimate pressure drops, increasing current robustness and accuracy. Using the comprehensive spatio-temporal hemodynamic information provided by Four Dimensional Phase-Contrast Magnetic Resonance Imaging, an existing finite element formulation to compute pressure differences is evaluated, illustrating its sensitivity to data when estimating viscous flows and exploring potential approaches to address this. A novel formulation using the work-energy principle is then introduced and validated on in silico test cases, demonstrating an increased accuracy and robustness to noise and to the image segmentation process. Finally, the proposed method is applied for the assessment of the aortic valve function of a cohort of patients with various degree of stenosis, revealing a fundamental bias in the Bernoulli formulation taken in Doppler-based estimation.
... The problems under consideration were implemented in CHeart -a multi-physics software tool based on [66][67][68]55] and expanded by the CHeart team at KCL. All problems were solved on a Dell OPTIPLEX 990, quad-core (Intel Ò Core™ i7-2600 CPU @ 3.40 GHz), on an 2.1 GHz AMD Opteron™ Interlagos 32 processor and on an SGI with 640 2.67 GHz processors (Intel Ò Xeon Ò CPU E7-8837). ...
The Lagrange Multiplier (LM) and penalty methods are commonly used to enforce incompressibility and compressibility in models of cardiac mechanics. In this paper we show how both formulations may be equivalently thought of as a weakly penalized system derived from the statically condensed Perturbed Lagrangian formulation, which may be directly discretized maintaining the simplicity of penalty formulations with the convergence characteristics of LM techniques. A modified Shamanskii–Newton–Raphson scheme is introduced to enhance the nonlinear convergence of the weakly penalized system and, exploiting its equivalence, modifications are developed for the penalty form. Focusing on accuracy, we proceed to study the convergence behavior of these approaches using different interpolation schemes for both a simple test problem and more complex models of cardiac mechanics. Our results illustrate the well-known influence of locking phenomena on the penalty approach (particularly for lower order schemes) and its effect on accuracy for whole-cycle mechanics. Additionally, we verify that direct discretization of the weakly penalized form produces similar convergence behavior to mixed formulations while avoiding the use of an additional variable. Combining a simple structure which allows the solution of computationally challenging problems with good convergence characteristics, the weakly penalized form provides an accurate and efficient alternative to incompressibility and compressibility in cardiac mechanics.
An unresolved issue in patient–specific models of cardiac mechanics is the choice of an appropriate constitutive law, able to accurately capture the passive behavior of the myocardium, while still having uniquely identifiable parameters tunable from available clinical data. In this pa- per, we aim to facilitate this choice by examining the practical identifiability and model fidelity of constitutive laws often used in cardiac mechanics. Our analysis focuses on the use of novel 3D tagged MRI, providing detailed displacement information in three-dimensions. The practical identifiability of each law is examined by generating synthetic 3D tags from in silico simulations, allowing mapping of the objective function landscape over parame- ter space and comparison of minimizing parameter values with original ground truth values. Model fi- delity was tested by comparing these laws with the more complex transversely isotropic Guccione law, by characterising their passive end-diastolic pressure volume relation behavior, as well as by considering the in vivo case of a healthy volun- teer. These results show that a reduced form of the Holzapfel–Ogden law provides the best balance be- tween identifiability and model fidelity across the tests considered.
Two closely-related fictitious domain methods for solving problems involving multiple interfaces are introduced. Like other fictitious domain methods, the proposed methods simplify the task of finite element mesh generation and provide access to solvers that can take advantage of uniform structured grids. The proposed methods do not involve the Lagrange multipliers, which makes them quite different from existing fictitious domain methods. This difference leads to an advantageous form of the inf–sup condition, and allows one to avoid time-consuming integration over curvilinear surfaces. In principle, the proposed methods have the same rate of convergence as existing fictitious domain methods. Nevertheless it is shown that, at the cost of introducing additional unknowns, one can improve the quality of the solution near the interfaces. The methods are presented using a two-dimensional model problem formulated in the context of linearized theory of elasticity. The model problem is sufficient for presenting method details and mathematical foundations. Although the model problem is formulated in two dimensions and involves only one interface, there are no apparent conceptual difficulties to extending the methods to three dimensions and multiple interfaces. Further, it is possible to extend the methods to nonlinear problems involving multiple interfaces.
The prelims comprise: Properties of infinite Voronoi diagramsProperties of Poisson Voronoi diagramsUses of Poisson Voronoi diagramsSimulating Poisson Voronoi and Delaunay cellsProperties of Poisson Voronoi cellsStochastic processes induced by Poisson VoronoidiagramsSectional Voronoi diagramsAdditively weighted Poisson Voronoi diagrams: the Johnson-Mehl modelHigher order Poisson Voronoi diagramsPoisson Voronoi diagrams on the surface of a sphereProperties of Poisson Delaunay cellsOther random Voronoi diagrams
Apres avoir rappele quelques resultats sur les maillages de Voronoi et Delaunay, on prouve, sous certaines hypotheses, que toute triangulation d'un polygone peut etre modifiee par une suite de procedures de changement de diagonale afin d'obtenir une triangulation de Delaunay. On en deduit plusieurs caracterisations des triangulations de Delaunay
A mathematical theory is developed for a class of a-posteriors error estimates of finite element solutions. It is based on a general formulation of the finite element method in terms of certain bilinear forms on suitable Hilbert spaces. The main theorem gives an error estimate in terms of localized quantities which can be computed approximately. The estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same. The theoretical results also lead to a heuristic characterization of optimal meshes, which in turn suggests a strategy for adaptive mesh refinement. Some numerical examples show the approach to be very effective.
The driven-cavity problem, a renowned bench-mark problem of computational, incompressible fluid dynamics, is physically unrealistic insofar as the inherent boundary singularities (where the moving lid meets the stationary walls) imply the necessity of an infinite force to drive the flow: this follows from G.I. Taylor's analysis of the so-called scaper problem. Using a boundary integral equation (BIE) formulation employing a suitable Green's function, we investigate herein, in the Strokes approximation, the effect of introducing small “leaks” to replace the singularities, thus rendering the problem physically realizable, The BIE approach used here incorporates functional forms of both the asymptotic far-field and singular near-field solution behaviours, in order to improve the accuracy of the numerical solution. Surprisingly, we find that the introduction of the leaks effects notably the global flow field a distance of the order of 100 leak widths away from the leaks. However, we observe that, as the leak width tends to zero, there is exellent agreement between our results and Taylor's thus justifying the use of the seemingly unrealizble boundary conditions in the driven-cavity problem. We also discover that the far-field, asymptotic, closed-form solution mentioned above is a remarkably accurate representation of the flow even in the near-field. Several streamline plots, over a range of spatial scales, are presented.
The centerline velocity profiles obtained from the solution of the two- and three-dimensional representations of the lid driven cavity flow problem are compared for different Reynolds numbers. Two configurations were used in this study: a unit cavity and a cavity with an aspect ratio of 2. The Reynolds numbers ranged from 100 to 5000 for all of the configurations studied. A new method of extending the Jacobi collocation technique called spectral difference is developed in this paper together with a unique computational grid. In addition, an iterative method for solving the pressure problem is also developed. This new numerical method allowed the calculation of three-dimensional Navier-Stokes equations to be performed in computers with very modest computational capabilities such as workstations.
In this article we discuss a fictitious domain method for the numerical solutions of three-dimensional elliptic problems with Dirichlet boundary conditions and also of the Navier-Stokes equations modeling incompressible viscous flow. The methodology for the Navier-Stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection, imbedding and incompressibility. Due to the decoupling, fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry. The resulting methodology is applied to two-dimensional unsteady external incompressible viscous flow problems and three-dimensional Stokes problems.
The central problem in modelling the multi–dimensional mechanics of the heart is in identifying functional forms and parameter of the constitutive equations, which describe the material properties of the resting and active, normal and diseased myocardium.
The constitutive properties of myocardium are three dimensional, anisotropic, nonlinear and time dependent. Formulating usefu constitutive laws requires a combination of multi–axial tissue testing in vitro, microstructural modelling based on quantitative morphology, statistical parameter estimation, and validation with measurement from intact hearts. Recent models capture some important properties of healthy and diseased myocardium including: the nonlinea interactions between the responses to different loading patterns; the influence of the laminar myofibre sheet architecture the effects of transverse stresses developed by the myocytes; and the relationship between collagen fibre architecture an mechanical properties in healing scar tissue after myocardial infarction.