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

Abstract

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.

On the Volume Conservation of the Immersed Boundary Method

“…It may be possible to follow their approach to derive modified staggered-grid finite-difference operators that further improve the volume-conservation properties of the staggered-grid IB method. We anticipate, however, that the unmodified staggered-grid approach will prove more useful in practice both because it is likely to be easier to implement and also because its extensions to cases involving adaptive mesh refinement [23,25,26,28,29] or physical boundary conditions [25,27,29] are significantly more straightforward. Such extensions, which we and others are actively developing, are needed in the context of many of the applications of the IB method to challenging problems of fluid-structure interaction.…”

“…Consequently, unlike collocated schemes, staggered-grid discretizations do not suffer from spurious pressure modes. Moreover, it is straightforward to use staggered-grid schemes for problems with nontrivial physical boundary conditions [27], to develop adaptive staggered discretizations that yield discretely divergence-free velocity fields [28,29], and to solve the discrete equations via efficient methods like multigrid [27,29].…”

Emerging Trends in Heart Valve Engineering: Part IV. Computational Modeling and Experimental Studies

“…Thus, IB models of the heart valves readily handle contact between leaflets without using an auxiliary contact model. 58,62 The conventional limitations associated with fictitious domain and IB-type methods are that they yield relatively low accuracy at fluid-solid interfaces. Accordingly, well-resolved three-dimensional simulations require high spatial resolution that presently necessitates the use of high-performance computing resources.…”

Fluid-Structure Interaction Model of Human Mitral Valve within Left Ventricle

“…Because the aortic valve is not included in the MV-LV model, the aortic tract is either fully open or fully closed, determined by the pressure difference between the value inside LV and the aorta. During the systolic ejection, a three-element Windkessel model [7] is connected to the outflow tract to provide a physiological pressureflow boundary condition, the systolic phase ends when the LV no longer pumps blood out. The end-diastolic pressure (8 mmHg) is maintained in the inflow boundary until the end-systole.…”

“…Over the last few years there have been a number of FSI valvular models using the immersed boundary (IB) approach [6,7,13,18]. However, none of these MV models included the effect of the LV motion, hence the flow field is not physiological.…”