Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries

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Abstract

Flow and pressure waves emanate from the heart and travel through the major arteries where they are damped, dispersed and reflected due to changes in vessel caliber, tissue properties and branch points. As a consequence, solutions to the governing equations of blood flow in the large arteries are highly dependent on the outflow boundary conditions imposed to represent the vascular bed downstream of the modeled domain. The most common outflow boundary conditions for three-dimensional simulations of blood flow are prescribed constant pressure or traction and prescribed velocity profiles. However, in many simulations, the flow distribution and pressure field in the modeled domain are unknown and cannot be prescribed at the outflow boundaries. An alternative approach is to couple the solution at the outflow boundaries of the modeled domain with lumped parameter or one-dimensional models of the downstream domain. We previously described a new approach to prescribe outflow boundary conditions for simulations of blood flow based on the Dirichlet-to-Neumann and variational multiscale methods. This approach, termed the coupled multidomain method, was successfully applied to solve the non-linear one-dimensional equations of blood flow with a variety of models of the downstream domain. This paper describes the extension of this method to three-dimensional finite element modeling of blood flow and pressure in the major arteries. Outflow boundary conditions are derived for any downstream domain where an explicit relationship of pressure as a function of flow rate or velocities can be obtained at the coupling interface. We developed this method in the context of a stabilized, semi-discrete finite element method. Flow rate and pressure distributions are shown for different boundary conditions to illustrate the dramatic influence of alternative boundary conditions on these quantities.

Introduction

In normal and disease states in children and adults, quantification of three-dimensional flow phenomena and pressure fields is important to understand the response of the cardiovascular system to biomechanical forces [1]. In the development of the circulatory system in normal subjects and congenital heart disease patients, flow modulates diameter and pressure controls wall thickness [2], [3]. In older subjects, the initiation and progression of atherosclerotic and aneurismal disease is directly affected by the complex three-dimensional fluid mechanical environment of the major arteries [4], [5]. Moreover, the ability to adequately simulate flow and pressure is needed to model the performance of devices such as heart valves, LVADs (left ventricular assist devices), filters, stents and stent-grafts. Non-invasive three-dimensional flow and pressure data can also provide important information to determine the significance of an obstruction or predict the outcome of a procedure [6], [7].

In recent years, remarkable progress has been made in simulating blood flow in realistic anatomical models constructed from three-dimensional medical imaging data. Arguably, accurate anatomic models are of primary importance in simulating blood flow. However, as we demonstrate in this paper, realistic boundary conditions are equally important in computing velocity and pressure fields. Yet, this subject has received far less attention than image-based model construction for three-dimensional simulations. In contrast, significant progress has been made in devising outflow boundary conditions for solving the one-dimensional equations of blood flow in elastic vessels. For example, Stergiopulos solved the non-linear one-dimensional equations of blood flow in a comprehensive model of the arterial system using a lumped parameter model of the vasculature downstream of each branch in his numerical model [8]. Several groups have developed and analyzed the coupling of one-dimensional equations with lumped models [9], [10], [11], [12]. In contrast to methods coupling the one-dimensional equations of blood flow to lumped parameter models, Olufsen developed a distributed downstream model based on calculating the input impedance of an asymmetric binary fractal tree using Womersley’s linear wave theory [13], [14] and an algorithm for computing the impedance of a vascular network first proposed by Taylor [15]. Olufsen’s approach was to generate a fractal tree for each outlet starting from a vessel that matched the diameter of the outlet and diminished in size with each successive generation of vessels until a fixed terminal vessel size was attained. With this method an impedance for each outlet of the upstream numerical model was computed naturally from linear wave theory and branching laws. Olufsen’s distributed model of the downstream vasculature enabled the representation of more realistic flow and pressure waveforms than those obtained with lumped parameter models [16]. Steele and Taylor used a modified version of Olufsen’s impedance boundary condition to model blood flow at rest and during simulated exercise conditions [17]. In this case, vascular networks were assigned to the outlets of a model of the abdominal aorta, modified to represent the resting flow distribution of 11 different subjects and then dilated to simulate the effects of lower limb exercise. Vignon and Taylor developed a multidomain approach to couple one-dimensional equations to different lumped and one-dimensional boundary conditions [18]. While these one-dimensional methods can be used to compute flow rate and mean pressure, by design, they cannot be used to simulate complex three-dimensional flow phenomena and pressure losses. Three-dimensional numerical methods have been used to compute velocity fields and quantify shear forces acting on the surface of blood vessels. However, since most three-dimensional models of blood flow use zero or constant pressure, zero traction, or prescribed velocity profiles as outlet boundary conditions, blood pressure is not computed accurately and notably absent from reports of hemodynamic investigations [19], [20], [21], [22], [23], [24].

For simulations of blood flow in large arteries, the outlet boundary conditions represent the downstream vasculature including smaller arteries, arterioles, capillaries, venules and veins returning blood to the heart. Clearly, the vast extent and complexity of the circulation precludes a three-dimensional representation of the entire circuit, yet ignoring the effect of the downstream circulation results in grossly inaccurate predictions of velocity and pressure fields for many problems where the distribution of flow between the major arteries is unknown. If zero or equal pressures or tractions are used for different outlets, the flow split will be dictated solely by the resistance to flow in the branches of the domain of interest, neglecting the dominant effect of the resistance of the downstream vascular beds. An alternative approach is to utilize three-dimensional models for the major arteries where high-fidelity information is needed, and reduced-order models to represent the remainder of the system. While closed-loop models are optimal, a simpler approach is to directly represent the vasculature of the small arteries and arterioles using zero-dimensional or one-dimensional models. These models can be terminated at the level of the capillary vessels where an assumption of constant pressure is reasonable. Several groups [6], [25], [26], [27], [28], [29], [30], [31], [32], [33] have successfully coupled three-dimensional models to either resistances or more sophisticated zero-dimensional models (lumped models), but this coupling has been performed iteratively, and generally applied to geometries with few outlets and low resistances (as seen in the pulmonary vasculature). A further limitation of these methods to couple three-dimensional and zero-dimensional models is the fact that there is no direct relationship between the anatomy of the downstream vascular bed and the lumped parameters resulting in difficulties in specifying these parameters and relating them to subsequent physiologic or pathophysiologic changes in the downstream vasculature. In addition, for many simulations based on three-dimensional imaging data, anatomic information is available for vessels downstream of the primary region of interest. The incorporation of such data would improve the accuracy of the models of the downstream vasculature.

Methods to couple three-dimensional and one-dimensional models were first described by Formaggia et al. [34], [35]. While great progress was made in these landmark papers, the coupling was performed for simple geometries and iteratively. Based on our experience, implicit coupling significantly improves convergence, especially for models with multiple outlets. In addition, these papers did not include coupling between three-dimensional domains and complex vascular networks as have been incorporated in one-dimensional numerical solutions of blood flow in arteries. Since the vascular bed from the major arteries to the capillaries can include tens of millions of blood vessels, non-linear one-dimensional models would be intractable. However, using linear wave propagation theory the input impedance of the downstream vascular bed can be computed for large complex vascular trees. A method to prescribe the impedance (calculated using linear wave theory) of these downstream vascular beds at the outlets of three-dimensional models would enable the specification of realistic boundary conditions for three-dimensional simulations of blood flow and pressure.

While inadequate outflow boundary conditions and rigid-wall models are the main impediments to the realistic prediction of pressure in three-dimensional blood flow simulations, in this paper we focus on the first issue and do not address the issue of wall deformability in the three-dimensional domain. We describe a new method to prescribe outflow boundary conditions in the context of the finite element method as this method is particularly well suited for handling complex geometries and boundary conditions inherent in modeling blood flow [24]. The approach we describe is based on the Dirichlet-to-Neumann (DtN) [36] and the variational multiscale [37] methods and is an extension of the 1D coupled multidomain approach we successfully applied with a variety of models of the downstream domain [18]. For one-dimensional problems, we demonstrated that a DtN map can be calculated for the impedance of complex vascular trees and that this approach incorporates naturally occurring wave reflections from a downstream bed. Wave propagation in transient and periodic states was simulated and the importance of selecting appropriate boundary conditions was demonstrated for one-dimensional simulations of blood flow. We also noted that the best boundary condition for cardiovascular applications is not the one that exhibits no wave reflection, since wave reflections naturally arising from downstream beds (from bifurcations, tapering, and variations in wall properties) should propagate back upstream into the numerical domain. We concluded that, at present, impedance-based boundary conditions are the best approach for incorporating natural sites of wave reflection in the downstream vasculature.

In this paper we present a coupled multidomain approach for three-dimensional finite element simulations of blood flow and pressure. Given the computational expense of three-dimensional numerical methods and the resolution limits imposed by current imaging technologies, we constrain the three-dimensional domain to the major arteries, and model the downstream domains with simpler models (Fig. 1). The outlet boundary conditions are implemented implicitly resulting in good stability and convergence properties at physiologic pressures. The organization of the paper will begin with the description of our coupled multidomain method in three-dimensions and its specialization to resistance and impedance boundary conditions. We then demonstrate this new method on a straight, cylindrical blood vessel, a bifurcation model with a stenosis on one side, and a subject-specific model of the human abdominal aorta.

Section snippets

Governing equations (strong form)

The method described can be applied to conservative as well as advective formulations of the incompressible Navier–Stokes equations, and was successfully implemented in both cases. We proceed by defining the spatial domain as Ω and its boundary as Γ. The three-dimensional equations for the flow of an incompressible Newtonian fluid consist of the three momentum balance equations and the continuity equation (written here in advective form) subject to suitable initial and boundary conditionsρv,t+ρ

Results

In the examples below, the profile of the inlet velocity was chosen to be parabolic for subsequent comparisons with one-dimensional analysis simulations [18]. For a prescribed input flow, we do not expect that the pressure fields will be significantly different between time-varying parabolic and Womersley inlet boundary conditions. Furthermore, while the focus of this paper is on the outlet boundary conditions, the method described could be applied in a similar fashion for inlet boundary

Conclusions

We have successfully developed and implemented a method to prescribe outflow boundary conditions intended for three-dimensional finite element simulations of blood flow based on the Dirichlet-to-Neumann and variational multiscale methods. As long as the effect of the downstream domain can be represented by an explicit function of pressure as a function of flow rate or velocity, the methods described can be used to couple the upstream three-dimensional numerical domain with the downstream

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. 0205741 and a predoctoral fellowship from the American Heart Association. The authors gratefully acknowledge Dr. Mette Olufsen for the use of her methods to compute input impedance of vascular trees and Dr. Farzin Shakib for his linear algebra package (http://www.acusim.com). Finally, the authors gratefully acknowledge the assistance of Dr. Nathan Wilson for assistance with software development, Dr.

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