VALIDATION OF AN OPEN SOURCE FRAMEWORK FOR THE SIMULATION OF BLOOD FLOW

Tiziano Passerini Emory University Atlanta, GA, USA

Annalisa Quaini University of Houston

Houston, TX, USA

Umberto Villa Emory University Atlanta, GA, USA

Alessandro Veneziani Emory University Atlanta, GA, USA

Suncica Canic University of Houston

Houston, TX, USA

ABSTRACT We describe in this paper an open source framework for

the solution of problems arising in hemodynamics. The proposed framework is validated through comparison against experimental data for fluid flow in an idealized medical device with rigid boundaries; and verified with a numerical benchmark for flow in compliant vessels. The core of the framework is an open source parallel finite element library that features algorithms to solve both fluid and fluid-structure interaction problems. The computed results are in good quantitative agreement with experimental measurements and theoretical estimates.

INTRODUCTION Computational fluid dynamics (CFD) is nowadays a tool of

choice for the investigation of blood flow problems. This has been made possible by the development of effective numerical methods combined with the availability of powerful computing resources. The complete pipeline to design a computational model is composed of

i) a pre-processing phase, dealing with the identification of the computational domain, the generation of high quality volume meshes, and the identification of physics parameters describing the problem (e.g., the density and viscosity of the blood, the stiffness of the arterial wall);

ii) a processing phase, corresponding to the simulation of blood flow and vessel wall mechanics;

iii) a post-processing phase for the computation of quantities of interest (e.g., the wall shear stress).

Application of this pipeline has a potential clinical impact, through the association of hemodynamics features with initiation, development and outcome of cardiovascular diseases.

This motivates us to focus on the issues of validation (i.e., we check that the computational model solves the right equations) and verification (i.e., we check that it solves the equations correctly).

METHODS We use the unsteady, incompressible Navier-Stokes

equations to describe the motion of blood in a rigid or deformable vessel, over a time interval of interest. We approximate in time the equations by the backward differentiation formula of order 2 and we use the Galerkin/Finite Element method for the spatial discretization. We model the arterial wall as a linearly elastic, or Hookean, material. Its motion can be described by the elastodynamics equation. We discretize in time this equation using a method from the family of generalized-α schemes. The space discretization is obtained by applying the Galerkin/Finite Element method. The interaction between blood and the arterial wall is modeled by coupling the Navier-Stokes equations (written in ALE formulation) with the elastodynamics equations. At the fluid-structure interface, we prescribe two transmission conditions: the continuity of velocity and the continuity of stresses. The numerical solvers for the model equations are implemented in C++ using algorithms and data structures provided by the open source finite element library LifeV (www.lifev.org). LifeV-based applications depend on third-party libraries, in particular Trilinos (Sandia National Labs) and SuiteSparse (Tim Davis et al.) for the solution of linear systems; ParMETIS (Karypis Lab) for mesh partitioning; and BLAS/LAPACK libraries.

We run the simulations on different computing platforms including clusters installed on-campus at the University of

Proceedings of the ASME 2013 Conference on Frontiers in Medical Devices: Applications of Computer Modeling and Simulation

FMD2013 September 11-13, 2013, Washington, DC, USA

FMD2013-16125

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Houston and clusters from the Extreme Science and Engineering Discovery Environment (XSEDE) consortium.

RESULTS

Figure 1: Computed solution of the FDA benchmark,

in three flow regimes (top: laminar, middle: transitional, bottom: turbulent).

The benchmark proposed by the FDA ([1]) consists in

simulating the flow of an incompressible Newtonian fluid with prescribed density (1056 kg/m3) and viscosity (0.0035 Pa s) in an idealized medical device shaped like a nozzle (see Fig. 1). The system is studied in a variety of conditions: laminar, transitional and turbulent flow regimes. The numerical solution always matches qualitatively the available measurements. Using validation metrics proposed in [3], for several axial locations we evaluate the relative mismatch between computed and measured values. We obtain very good estimates of the flow rate (errors within 1%) and good estimates of the velocity at all axial locations, under all exercise conditions.

Figure 2: Computed solution of the fluid-structure

interaction benchmark. The vessel wall is colored by the magnitude of the radial displacement (in cm), the

fluid domain by the pressure (in dyn/cm2) The numerical fluid-structure interaction benchmark pro-

posed in [2] deals with the propagation of a pressure wave in a

fluid-filled elastic tube (see Fig. 2). The fluid filling the deformable tube has density 1000 kg/m3 and viscosity 0.004 Pa s, while the elastic shell has Young’s modulus 105 Pa and Poisson’s ratio 0.3. We compare the computed features of the propagating wave with the theoretical predictions, obtaining an excellent agreement (relative errors within 1% for both wave speed and frequency).

In both numerical experiments, we select the proper grid resolution with mesh independence studies.

CONCLUSIONS Using the proposed open source framework we set up

numerical experiments solving two published flow problems, representative of relevant scenarios in hemodynamics. We report high computational costs in particular for the simulation of transitional and turbulent flow regimes in the FDA benchmark. Highly energetic structures at the micro scale require very fine meshes and small time steps to be correctly captured in the simulations. The efficiency of the computing platform - as well as the careful selection of the discretization parameters - is crucial in this kind of applications. Based on our experiences, we observe that the level of uncertainty that is intrinsic to the direct observation of a physical phenomenon dominates the inaccuracy (for instance due to the resolution of the grid) of a properly designed computational model.

An important outcome of this work is the production of a suite of scripts and codes that are based on a completely open-source set of tools, and therefore will be readily shared with the community through the web portal www.lifev.org.

ACKNOWLEDGMENTS This work has been partly supported by NSF through grant

DMS- 1109189. Passerini has been partially supported by NIH grant R01 HL70531. This work used the XSEDE, which is supported by NSF grant number OCI-1053575. The authors wish to acknowledge technical support and access to computing resources provided by the TLC2 and the Texas Tech University’s HPCC. The authors wish to thank Prof. R. Glowinski for his advice on this project.

REFERENCES [1] Computational Fluid Dynamics: An FDA’s Critical

Path Initiative. https://fdacfd.nci.nih.gov/. [2] C. Greenshields and H. Weller. A unified formulation

for continuum mechanics applied to fluid-structure interaction in flexible tubes. Int. J. Numer. Methods Eng., 64:1575–1593, 2005.

[3] S. Stewart, E. Paterson, G. Burgreen, P. Hariharan, M. Giarra, V. Reddy, S. Day, K. Manning, S. Deutsch, M. Berman, M. Myers, and R. Malinauskas. Assessment of CFD performance in simulations of an idealized medical device: Results of FDA’s first computational inter laboratory study. Cardiovascular Engineering and Technology, 3(2):139–160, 2012.

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