arXiv subjects

Boyce E. Griffith

Publications and source records attributed to Boyce E. Griffith.

44 records · Page 3Linked to original sources

arXiv2017

Image-based immersed boundary model of the aortic root

Each year, approximately 300,000 heart valve repair or replacement procedures are performed worldwide, including approximately 70,000 aortic valve replacement surgeries in the United States alone. This paper describes progress in constructing anatomically and physiologically realistic immersed boundary (IB) models of the dynamics of the aortic root and ascending aorta. This work builds on earlier IB models of fluid-structure interaction (FSI) in the aortic root, which previously achieved realistic hemodynamics over multiple cardiac cycles, but which also were limited to simplified aortic geometries and idealized descriptions of the biomechanics of the aortic valve cusps. By contrast, the model described herein uses an anatomical geometry reconstructed from patient-specific computed tomography angiography (CTA) data, and employs a description of the elasticity of the aortic valve leaflets based on a fiber-reinforced constitutive model fit to experimental tensile test data. Numerical tests show that the model is able to resolve the leaflet biomechanics in diastole and early systole at practical grid spacings. The model is also used to examine differences in the mechanics and fluid dynamics yielded by fresh valve leaflets and glutaraldehyde-fixed leaflets similar to those used in bioprosthetic heart valves. Although there are large differences in the leaflet deformations during diastole, the differences in the open configurations of the valve models are relatively small, and nearly identical hemodynamics are obtained in all cases considered.

physics.med-ph↗

arXiv2017

Geometric multigrid for an implicit-time immersed boundary method

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the structure and Eulerian variables to describe the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. These tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100--1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50--200 times more efficient than the explicit method.

math.NA↗

arXiv2017

Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions

The immersed boundary (IB) method is a mathematical and numerical framework for problems of fluid-structure interaction, treating the particular case in which an elastic structure is immersed in a viscous incompressible fluid. The IB approach to such problems is to describe the elasticity of the immersed structure in Lagrangian form, and to describe the momentum, viscosity, and incompressibility of the coupled fluid-structure system in Eulerian form. Interaction between Lagrangian and Eulerian variables is mediated by integral equations with Dirac delta function kernels. The IB method provides a unified formulation for fluid-structure interaction models involving both thin elastic boundaries and also thick viscoelastic bodies. In this work, we describe the application of an adaptive, staggered-grid version of the IB method to the three-dimensional simulation of the fluid dynamics of the aortic heart valve. Our model describes the thin leaflets of the aortic valve as immersed elastic boundaries, and describes the wall of the aortic root as a thick, semi-rigid elastic structure. A physiological left-ventricular pressure waveform is used to drive flow through the model valve, and dynamic pressure loading conditions are provided by a reduced (zero-dimensional) circulation model that has been fit to clinical data. We use this model and method to simulate aortic valve dynamics over multiple cardiac cycles. The model is shown to approach rapidly a periodic steady state in which physiological cardiac output is obtained at physiological pressures. These realistic flow rates are not specified in the model, however. Instead, they emerge from the fluid-structure interaction simulation.

cs.CE↗

arXiv2017

Hybrid finite difference/finite element immersed boundary method

The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach employs a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach.

math.NA↗

arXiv2016

Scalable smoothing strategies for a geometric multigrid method for the immersed boundary equations

The immersed boundary (IB) method is a widely used approach to simulating fluid-structure interaction (FSI). Although explicit versions of the IB method can suffer from severe time step size restrictions, these methods remain popular because of their simplicity and generality. In prior work (Guy et al., Adv Comput Math, 2015), some of us developed a geometric multigrid preconditioner for a stable semi-implicit IB method under Stokes flow conditions; however, this solver methodology used a Vanka-type smoother that presented limited opportunities for parallelization. This work extends this Stokes-IB solver methodology by developing smoothing techniques that are suitable for parallel implementation. Specifically, we demonstrate that an additive version of the Vanka smoother can yield an effective multigrid preconditioner for the Stokes-IB equations, and we introduce an efficient Schur complement-based smoother that is also shown to be effective for the Stokes-IB equations. We investigate the performance of these solvers for a broad range of material stiffnesses, both for Stokes flows and flows at nonzero Reynolds numbers, and for thick and thin structural models. We show here that linear solver performance degrades with increasing Reynolds number and material stiffness, especially for thin interface cases. Nonetheless, the proposed approaches promise to yield effective solution algorithms, especially at lower Reynolds numbers and at modest-to-high elastic stiffnesses.

math.NA↗

arXiv2016

A continuum mechanics-based musculo-mechanical model for esophageal transport

In this work, we extend our previous esophageal transport model using an immersed boundary (IB) method with discrete fiber-based structures, to one using a continuum mechanics-based model that is approximated based on finite elements (IB-FE). To deal with the leakage of flow when the Lagrangian mesh becomes coarser than the fluid mesh, we employ adaptive interaction quadrature points for Lagrangian-Eulerian interaction equations based on a previous work. In particular, we introduce a new anisotropic adaptive interaction quadrature rule. The new rule permits us to vary the interaction quadrature points not only at each time-step and element but also at different orientations per element. For the material model, we extend our previous fiber-based model to a continuum-based model. We first study a case in which a three-dimensional short tube is dilated. Results match very well with those from the implicit FE method. We remark that in our IB-FE case, the three-dimensional tube undergoes a very large deformation and the Lagrangian mesh-size becomes about 6 times of Eulerian mesh-size. To validate the method in handling fiber-matrix material models, we perform a second study on dilating a long fiber-reinforced tube. Errors are small when we compare numerical solutions with analytical solutions. The technique is then applied to the problem of esophageal transport. We present three cases that differ in the material model and muscle fiber architecture. The overall transport features are consistent with those from the previous model. We remark that the continuum-based model can handle more realistic and complicated material behavior. This is demonstrated in our third case with spatially varying fiber architecture. We find this unique muscle fiber architecture could generate a so-called pressure transition zone. This suggests an important role of muscle fiber architecture in esophageal transport.

physics.comp-ph↗

arXiv2015

Immersed boundary-finite element model of fluid-structure interaction in the aortic root

It has long been recognized that aortic root elasticity helps to ensure efficient aortic valve closure, but our understanding of the functional importance of the elasticity and geometry of the aortic root continues to evolve as increasingly detailed in vivo imaging data become available. Herein, we describe fluid-structure interaction models of the aortic root, including the aortic valve leaflets, the sinuses of Valsalva, the aortic annulus, and the sinotubular junction, that employ a version of Peskin's immersed boundary (IB) method with a finite element (FE) description of the structural elasticity. We develop both an idealized model of the root with three-fold symmetry of the aortic sinuses and valve leaflets, and a more realistic model that accounts for the differences in the sizes of the left, right, and noncoronary sinuses and corresponding valve cusps. As in earlier work, we use fiber-based models of the valve leaflets, but this study extends earlier IB models of the aortic root by employing incompressible hyperelastic models of the mechanics of the sinuses and ascending aorta using a constitutive law fit to experimental data from human aortic root tissue. In vivo pressure loading is accounted for by a backwards displacement method that determines the unloaded configurations of the root models. Our models yield realistic cardiac output at physiological pressures, with low transvalvular pressure differences during forward flow, minimal regurgitation during valve closure, and realistic pressure loads when the valve is closed during diastole. Further, results from high-resolution computations demonstrate that IB models of the aortic valve are able to produce essentially grid-converged dynamics at practical grid spacings for the high-Reynolds number flows of the aortic root.

cs.CE↗

arXiv2015

A fully resolved active musculo-mechanical model for esophageal transport

Esophageal transport is a physiological process that mechanically transports an ingested food bolus from the pharynx to the stomach via the esophagus, a multi-layered muscular tube. This process involves interactions between the bolus, the esophagus, and the neurally coordinated activation of the esophageal muscles. In this work, we use an immersed boundary (IB) approach to simulate peristaltic transport in the esophagus. The bolus is treated as a viscous fluid that is actively transported by the muscular esophagus, which is modeled as an actively contracting, fiber-reinforced tube. A simplified version of our model is verified by comparison to an analytic solution to the tube dilation problem. Three different complex models of the multi-layered esophagus, which differ in their activation patterns and the layouts of the mucosal layers, are then extensively tested. To our knowledge, these simulations are the first of their kind to incorporate the bolus, the multi-layered esophagus tube, and muscle activation into an integrated model. Consistent with experimental observations, our simulations capture the pressure peak generated by the muscle activation pulse that travels along the bolus tail. These fully resolved simulations provide new insights into roles of the mucosal layers during bolus transport. In addition, the information on pressure and the kinematics of the esophageal wall due to the coordination of muscle activation is provided, which may help relate clinical data from manometry and ultrasound images to the underlying esophageal motor function.

physics.comp-ph↗