Simulating Cardiovascular Fluid Dynamics by the

Immersed Boundary Method

Boyce E. Griffith∗

New York University School of Medicine, New York, NY 10016

David M. McQueen† and Charles S. Peskin‡

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012

The immersed boundary method is both a general mathematical framework and a par-ticular numerical approach to problems of fluid-structure interaction. In this paper, wedescribe the application of the immersed boundary method to the simulation of cardiovas-cular fluid dynamics, focusing on the fluid dynamics of the aortic heart valve (the valvewhich prevents the backflow of blood from the aorta into the left ventricle of the heart)and aortic root (the initial portion of the aorta, which attaches to the heart). The aor-tic valve and root are modeled as a system of elastic fibers, and the blood is modeledas a viscous incompressible fluid. Three-dimensional simulation results obtained using aparallel and adaptive version of the immersed boundary method are presented. Theseresults demonstrate that it is feasible to perform three-dimensional immersed boundarysimulations of cardiovascular fluid dynamics in which realistic cardiac output is obtainedat realistic pressures.

Nomenclature

U physical domainx = (x, y, z) ∈ U Cartesian (physical) coordinatesu(x, t) fluid velocityp(x, t) fluid pressuref(x, t) Eulerian force density applied by the structure to the fluidδ(x) = δ(x) δ(y) δ(z) three-dimensional Dirac delta functionδh(x) = δh(x) δh(y) δh(z) three-dimensional regularized Dirac delta functionΩ Lagrangian coordinate domain(q, r, s) ∈ Ω Lagrangian (material) coordinatesX(q, r, s, t) physical position of Lagrangian (material) point (q, r, s) at time tF(q, r, s, t) Lagrangian force density applied by the structure to the fluid

I. Introduction

The immersed boundary method1,2 is a general mathematical framework for problems in which a rigid orelastic structure is immersed in a fluid flow, and it is also a numerical approach to simulating such problems.Fluid-structure interaction problems which may be simulated using the immersed boundary method includethe dynamic interaction of a flexible heart valve leaflet and the blood in which it is immersed, the flappingof a flexible insect wing, and the flow over a rigid airfoil, as well as many others.

∗Assistant Professor, Leon H. Charney Division of Cardiology, Department of Medicine, New York University School ofMedicine, 522 First Avenue, New York, NY 10016

†Research Professor, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY10012

‡Silver Professor, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012

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Generally speaking, the immersed boundary method treats fluid-structure interaction problems by cou-pling an Eulerian description of the fluid to a Lagrangian description of the structure. In the continuoussetting, interaction between fluid and structure variables is mediated by integral transforms with Dirac deltafunction kernels. When the continuous immersed boundary formulation is discretized for computer simula-tion, the Dirac delta function kernels are replaced by regularized versions of the delta function. This approachallows the fluid variables to be treated using efficient Cartesian grid methods while allowing for a fully La-grangian treatment of the immersed structure. Moreover, the immersed boundary approach does not requirethe generation of body conforming meshes. Consequently, the immersed boundary method is well-suited forproblems in which the structure undergoes significant movement or is subject to large deformations.

Although the immersed boundary method is broadly applicable to problems of fluid-structure interaction,in the present paper, we focus on the application for which the immersed boundary method was originallydeveloped,3,4 namely cardiovascular fluid dynamics. Consequently, we shall restrict our attention to thecase in which the structure is elastic and immersed in a viscous incompressible fluid.a In the remainderof the paper, we provide an overview of the immersed boundary approach to modeling and simulatingfluid-structure interaction, briefly describing recent extensions of this methodology. We also present three-dimensional simulations of cardiovascular fluid dynamics obtained using a parallel implementation of anadaptive version of the immersed boundary methodology. In earlier work, we have presented whole-heartsimulations of cardiac fluid dynamics, first with a uniform grid and shared-memory parallelization8–10,b andmore recently with an adaptive grid and distributed parallelization.11,12,c In this paper, we present resultsfrom the application of this methodology to the simulation of the fluid dynamics of the aortic heart valve(the valve which prevents backflow from the aorta into the left ventricle of the heart) and aortic root (thebase of the aorta, which attaches to the heart), including both the opening and closing phases of the valve,obtaining three-dimensional immersed boundary simulations of cardiac ejection in which realistic cardiacoutput is obtained at realistic blood pressures.

II. The immersed boundary method

II.A. Mathematical formulation

In the immersed boundary approach to fluid-structure interaction, the viscous incompressible fluid is de-scribed using the incompressible Navier-Stokes equations written in Eulerian form, whereas a Lagrangianformulation is used to describe the elasticity of the immersed elastic structure. In the present work, weassume that the fluid possesses a uniform mass density ρ and dynamic viscosity μ, and that the immersedelastic structure is neutrally buoyant.d Letting x = (x, y, z) ∈ U denote Cartesian (physical) coordinates,where U is the physical domain, letting (q, r, s) ∈ Ω denote Lagrangian (material) coordinates attached tothe structure, where Ω is the Lagrangian coordinate domain, and letting X(q, r, s, t) ∈ U denote the physicalposition of material point (q, r, s) at time t, the equations of motion are

ρ

(∂u∂t

+ (u · ∇)u)

+ ∇p = μ∇2u + f , (1)

∇ · u = 0, (2)

f(x, t) =∫

Ω

F(q, r, s, t) δ(x − X(q, r, s, t)) dq dr ds, (3)

∂X∂t

(q, r, s, t) =∫

U

u(x, t) δ(x − X(q, r, s, t)) dx, (4)

F(·, ·, ·, t) = F [X(·, ·, ·, t)]. (5)

aRigid immersed structures can be treated by the standard immersed boundary method5 and also by specialized variants ofthe basic immersed boundary methodology.6 The immersed boundary method has also been used to treat the interaction of arigid structure with a compressible fluid.7

bSee animations available online at http://www.math.nyu.edu/faculty/peskin/myo3D.cSee animations available online at http://www.cims.nyu.edu/∼griffith/heart anim.dSeveral alternative extensions of the immersed boundary method have been developed to treat the case in which the mass

density of the immersed structure is different from that of the fluid,1,13–18 and work underway at the Courant Institute andelsewhere aims to develop a version of the method which treats the case that the viscosity of the structure is different from thatof the fluid.

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Eqs. (1) and (2) are the incompressible Navier-Stokes equations, which are written in terms of the fluidvelocity u(x, t) and pressure p(x, t), along with a body force f(x, t) which is the Eulerian elastic forcedensity (i.e., the elastic force density with respect to the Cartesian coordinates x = (x, y, z)) applied bythe structure to the fluid. Eq. (5) indicates that the Lagrangian elastic force density (i.e., the elastic forcedensity with respect to the curvilinear coordinates (q, r, s)) generated by the elasticity of the structure isdetermined by a time-independent mapping of the configuration of the immersed structure.e

Next, we turn our attention to the two Lagrangian-Eulerian interaction equations, Eqs. (3) and (4).Both equations employ integral transformations which use the three-dimensional Dirac delta function δ(x) =δ(x) δ(y) δ(z) to convert between Lagrangian and Eulerian quantities. Eq. (4) states that the structure movesat the local fluid velocity, i.e.,

∂X∂t

(q, r, s, t) = u(X(q, r, s, t), t), (6)

which is the no-slip condition of a viscous fluid. The delta-function formulation in Eq. (4) is equivalent toEq. (6) but has the advantage that it can be discretized to obtain an interpolation formula, see Eq. (13),below. In this context, the no-slip condition is used to determine the motion of the immersed elastic structurerather than to constrain the motion of the fluid. Note that the no-slip condition will also appear below as aphysical boundary condition for the fluid along the boundary of the physical domain U .

The remaining interaction equation, Eq. (3), converts the Lagrangian elastic force density F into theequivalent Eulerian elastic force density f . It is important to note that generally F(q, r, s, t) �= f(X(q, r, s, t), t);however, F and f are nonetheless equivalent as densities. This can be seen as follows. Let V ⊆ U be anarbitrary region in U , and let X−1(V, t) = {(q, r, s) : X(q, r, s, t) ∈ V } denote the set of all material pointswhich are physically located in region V at time t. We verify that F and f are equivalent densities bycomputing

∫V

f(x, t) dx =∫

V

(∫Ω

F(q, r, s, t) δ(x − X(q, r, s, t)) dq dr ds

)dx (7)

=∫

Ω

(∫V

F(q, r, s, t) δ(x − X(q, r, s, t)) dx)

dq dr ds (8)

=∫X−1

(V,t)

F(q, r, s, t) dq dr ds. (9)

Thus, the total force applied to the region V at time t is equal to the total force generated by the materialpoints which are physically located in the region V at time t.

To complete the description of the problem, we must describe the boundary and initial conditions. Alongthe boundary of the physical domain U , we impose no-slip boundary conditions for the tangential componentsof the fluid velocity along with either no-penetration boundary conditions or normal traction boundaryconditions. Since the fluid is incompressible and Newtonian, the combination of no-slip tangential velocityboundary conditions and normal traction boundary conditions can be seen to be equivalent to prescribingboundary conditions for the pressure p. It is convenient to use such prescribed-pressure boundary conditionsto connect the model of the aortic root to reduced models of the heart (at the upstream boundary) andthe systemic arterial tree (at the downstream boundary). To determine appropriate initial conditions, weassume that the fluid is initially at rest, that the structure is initialized in an unstressed configuration, andthat there is no applied normal traction along the boundary at the initial time t = t0. Consequently, u ≡ 0at time t = t0. Although the pressure p is not a state variable for an incompressible fluid and does notrequire an initial condition, in this case it follows that p ≡ 0 at time t = t0.

II.B. A fiber-based model of tissue elasticity

In the present work, we describe the properties of the aortic valve and aortic root in terms of a system ofelastic fibers. In the case of the valve, these fibers can be viewed as representing the collagen fibers whichsupport the significant pressure load borne by the closed valve when the left ventricle is relaxed, and in thecase of the vessel, these fibers can be viewed as representing the smooth muscle, elastin, and collagen thatgive the vessel wall its strength.

eTo simulate the more complicated case of the contraction and relaxation of cardiac muscle tissue, it is necessary that F be atime-dependent mapping. Heart valves are passive structures, however, and therefore their elastic properties may be consideredto be time independent.

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Suppose that the Lagrangian coordinates (q, r, s) have been chosen so that each fixed value of (q, r) labelsa particular fiber. Thus, for a fixed value of the pair (q, r) = (q0, r0), the mapping s → X(q0, r0, s) is aparametric representation of the fiber labeled by (q0, r0). Let τ = ∂X/∂s/ |∂X/∂s| denote the unit tangentvector aligned with the fiber direction, and let T = σ (|∂X/∂s| ; q, r, s) denote the fiber tension, which isdefined so that Tτ dq dr is the force generated by the fiber bundle dq dr. In this case, it is possible todemonstrate19 that the Lagrangian elastic force density is expressed by

F =∂

∂s(Tτ ) . (10)

Note that although fiber-based elasticity models are convenient for describing anisotropic materials suchas the leaflets of the aortic heart valve or cardiac muscle tissue, the immersed boundary framework does notrequire the use of fiber-based models. For instance, it is straightforward to use with the immersed boundarymethod any elasticity model which computes nodal forces (or force densities) from nodal positions. Inparticular, it is possible to employ a nodal finite element model to describe the elasticity of the structureand to compute F from X.f

II.C. An overview of the numerical scheme

In a typical numerical treatment of the immersed boundary method, the Eulerian variables are discretizedon a Cartesian grid and the Lagrangian variables are discretized on a curvilinear mesh which moves withthe structure. Let the computational domain U be a rectangular Lx × Ly × Lz box which is discretizedon a regular Nx × Ny × Nz Cartesian grid with grid spacings Δx = Δy = Δz = h. Suppose that the fluidvariables, including the three components of the velocity and the pressure, are defined at the centers of thecells of the Cartesian grid. Let (i, j, k) label the individual Cartesian grid cells for integer values of i, j, andk, where 0 ≤ i < Nx, 0 ≤ j < Ny, and 0 ≤ k < Nz, and let xi,j,k =

((i + 1

2

)h,

(j + 1

2

)h,

(k + 1

2

)h)

denotethe physical location of the center of cell (i, j, k). Similarly, let (q, r, s) label an individual curvilinear meshnode, and let the curvilinear mesh spacings be denoted by Δq, Δr, and Δs.

The discretizations of the Lagrangian-Eulerian interaction equations, Eqs. (3) and (4), both employthe same regularized version of the Dirac delta function, denoted δh(x). We take the three-dimensionalregularized delta function δh(x) to be the tensor product of one-dimensional regularized delta functions, i.e.,

δh(x) = δh(x) δh(y) δh(z). (11)

In the present work, we define the three-dimensional regularized delta function δh(x) in terms of the four-point one-dimensional regularized delta function described in Ref. 1. The construction of this one-dimensionalregularized delta function must be altered near boundaries in the computational domain. In three spatialdimensions, δh(x) has a support of 4 × 4 × 4 = 64 grid cells.

Letting Xq,r,s denote the position of curvilinear mesh node (q, r, s), and letting Fq,r,s denote the discretiza-tion of the Lagrangian elastic force density evaluated at curvilinear mesh node (q, r, s), the discretization ofEq. (3) is

f i,j,k =∑q,r,s

Fq,r,s δh(xi,j,k − Xq,r,s)Δq Δr Δs, (12)

where f i,j,k denotes the discretization of the Eulerian elastic force density at the centers of the Cartesiangrid cells. Similarly, Eq. (4) is discretized as

ddt

Xq,r,s =∑i,j,k

ui,j,k δh(xi,j,k − Xq,r,s)ΔxΔy Δz, (13)

where ui,j,k denotes the discretization of the Eulerian fluid velocity at the centers of the Cartesian grid cells.Employing the same regularized delta function both to interpolate the fluid velocity from the Cartesian gridto the curvilinear mesh and to spread the Lagrangian elastic force density from the curvilinear mesh tothe Cartesian grid ensures that momentum, force, torque, and power are all conserved during Lagrangian-Eulerian interaction.1 Moreover, since in the present case the immersed structure is neutrally buoyant,energy is also conserved during Lagrangian-Eulerian interaction.1

fSee, e.g., lecture notes available on-line at http://www.math.nyu.edu/faculty/peskin/ib lecture notes/index.html.

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II.D. Adaptive mesh refinement and other implementation details

In the results presented in Sec. III, we employ a cell-centered Godunov-projection method to solve the incom-pressible Navier Stokes equations on an adaptively refined hierarchical Cartesian grid like that of Fig. 1 (seealso Fig. 6). The locally refined grid is constructed in a manner which ensures that the immersed structureremains covered throughout the simulation by the grid cells which comprise the finest level of the hierarchicalCartesian grid. To prevent complicating the discretization of the equations of Lagrangian-Eulerian interac-tion, additional Cartesian grid cells are added to the finest level of the locally refined grid to ensure that foreach curvilinear mesh node (q, r, s), the support of the regularized delta function centered about Xq,r,s isalso covered by the grid cells which comprise the finest level of the locally refined grid. Consequently, we areable to employ the standard discrete Lagrangian-Eulerian interaction equations, Eqs. (12) and (13), withoutmodification in the adaptive scheme.

Level 0

Level 1

Level 2

Figure 1. An adaptively refined hierarchical Cartesian grid comprised of three nested levels.

We have described a version of the Godunov-projection method employed in the present work, along withthe use of this incompressible flow solver in the context of the immersed boundary method, previously.2,11,12

This version of the immersed boundary method is only semi-implicit and therefore stability considerationsrequire the use of small timesteps. The primary differences between the scheme used in the present workand that described in Refs. 2,11,12 are summarized as follows: (1) We have found that the L-stable implicittreatment of the viscous terms in the incompressible Navier-Stokes equations used in Refs. 2,11,12 does notappear to yield any noticeable improvement in the stability of the scheme or in the quality of the numericalresults when compared to the simpler Crank-Nicolson scheme. Consequently, in the present work we employthe more computationally efficient Crank-Nicolson scheme, which is A-stable but not L-stable. (2) TheGodunov scheme employed in the present work employs the xsPPM7 method of Rider, Greenough, andKamm20 (a recent version of the piecewise parabolic method (PPM) of Colella and Woodward21) to performthe initial extrapolation instead of the piecewise linear scheme of Refs. 2, 11, 12. (3) Following Martin andColella,22 quadratic interpolation is employed at coarse-fine interfaces to compute cell-centered values locatedin the ghost cells located on the “coarse side” of coarse-fine interfaces. In the method of Refs. 11,12, simplerlinear interpolation was used at coarse-fine interfaces. (4) No-slip and no-penetration physical boundaryconditions for the fluid are specified using methods introduced by Brown, Cortez, and Minion,23 and normaltraction boundary conditions are specified using methods by Yang and Prosperetti.24

III. Simulating the fluid dynamics of the aortic heart valve

We have applied our adaptive version of the immersed boundary method to simulate the fluid dynamicsof the aortic valve and aortic root. Our construction of the aortic valve was based on the mathematicaltheory of the fiber architecture of aortic heart valve leaflets described by Peskin and McQueen25 (see Fig. 2),and the shape of our model aortic root was based on the idealized geometric description of Reul et al.,26

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A. B.

Figure 2. The initial configuration of the model valve as viewed from above (A) and from the side (B).

Figure 3. The model aortic root and proximal ascending aorta. The model aortic root is connected at its upstreamboundary (located at the bottom of the figure) via a semi-rigid tube to a prescribed time-dependent pressure sourcelocated at z = 0 cm, which serves as a simplified model of the left ventricle. At its downstream boundary (located atthe top of the figure), the model aortic root is connected at z = 15 cm to a Windkessel model which serves as a reducedmodel of the systemic arterial tree.

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A. B.

C. D.

Figure 4. Flow patterns around the model valve and in the model root and proximal ascending aorta. The motion ofthe blood is indicated by red markers which are passively advected by the flow. The valve is shown just before (A)and after (B) opening, and just before (C) and after (D) closing. The open valve offers low flow resistance, allowingthe model to generate physiological cardiac output. At the end of the cycle, the valve closes and allows only minorregurgitation.

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A. B.

z (cm)

p(m

mH

g)

2 4 6 8 10 12 14

0

50

100

z (cm)

p(m

mH

g)

2 4 6 8 10 12 14

0

50

100

C. D.

z (cm)

p(m

mH

g)

2 4 6 8 10 12 14

0

50

100

z (cm)

p(m

mH

g)

2 4 6 8 10 12 14

0

50

100

Figure 5. Pressure as a function of distance along the center-line ((x, y) = (5 cm, 5 cm)) of the model. The upstreamend (z = 0 cm) of the model is connected to a specified pressure source which serves as a simplified model of the leftventricle. The downstream end (z = 15 cm) of the model is connected to a Windkessel model which serves as a reducedmodel of the systemic arterial tree. Pressure boundary conditions determined by these reduced models are prescribedboth at the inlet and at the outlet. The valve itself is positioned at approximately z = 4 cm. A. At the beginningof the simulation, we pressurize the downstream end of the domain to 85 mmHg, ensuring that the valve is closedand supports a significant, physiological pressure load. B. After the initial loading of the closed valve, we increase thepressure at the upstream boundary to 120 mmHg. At the time shown in this panel, the upstream and downstreampressures are approximately equal, and the valve is about to open. C. After the valve opens, it offers very little flowresistance. Blood is accelerated as it passes through the open valve, causing a drop in pressure slightly downstreamfrom the valve, followed by pressure recovery near the outlet. D. At the end of the cycle, the upstream pressure isreturned to 0 mmHg. At the time shown in this panel, the valve has closed and is once again supporting a significantpressure load.

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A. B.

Figure 6. A. Hydrostatic pressure in the model vessel plotted on the plane y = 5 cm, which bisects one of the valveleaflets. Note that these plotted values correspond to those of Fig. 5C. Note that there is essentially no pressure dropacross the open valve, which is located at approximately z = 4 cm. See the caption of Fig. 5 for further details. B. Sameas A, but with the adaptive grid also shown. Note that there are three levels of refinement and that the finest levelcovers the entire structure.

which was derived from angiographic films from 206 healthy patients (see Fig. 3). The dimensions of themodel vessel were based on measurements of human aortic roots harvested after autopsy which had beenpressurized to a normal systolic pressure of 120 mmHg.27 The model valve was sized so that it fits withinthe model aortic root.

The upstream inlet of the aortic root is connected via a semi-rigid tube to the z = 0 cm boundary of thefluid domain, where time-dependent pressure boundary conditions are prescribed, and the downstream outletof the aortic root is connected to the z = 15 cm boundary of the fluid domain. The simulation proceedsas follows: During an initial pressurization phase, the pressure at the outlet is raised from 0 to 85 mmHg.Once the pressure at the outflow boundary reaches 85 mmHg, the pressure at that boundary is subsequentlydetermined via a 3-element Windkessel model which has been fit to human data28 (note that we use the3-element Windkessel model of Ref. 28, not the 4-element Windkessel model described in that same paper).Once the downstream portion of the vessel has been pressurized, the pressure at the inflow boundary risesfrom 0 mmHg (corresponding to diastole, when the heart is relaxed) to 120 mmHg (corresponding to systole,when the heart contracts) over the course of 0.1 s. The upstream pressure remains at 120 mmHg for 0.2 s,and then, over the course of 0.1 s, returns to a resting value of 0 mmHg. Simulation results are presentedin Figs. 4–7. Note that in Fig. 5 it is clear that there is essentially no pressure drop across the valve once itopens, and that after the valve closes, it maintains a significant pressure load.

The familiar “lub-dup” heart sounds are caused by the closure of the heart valves, with the “lub” resultingfrom the closure of the mitral and tricuspid valves early in systole, and the “dup” resulting from the closureof the aortic and pulmonic valves at the end of systole. Note that the “dup” heart sound is clearly visiblein the flow record presented in Fig. 7. It is also visible in the computed pressure waveforms (data notshown). Moreover, once the valve closes, it remains closed for the remainder of the simulation and allows noregurgitation. Stroke volume for this simulation is approximately 90 ml, which is within the physiologicalrange.29

IV. Conclusions

In this paper, we have described the immersed boundary method for simulating the interaction of anelastic structure and a viscous incompressible fluid, and we have presented results obtained by applyingthis method to the simulation of cardiovascular fluid dynamics. Moreover, we have demonstrated that it is

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A.

time (sec)

flow

(lit

er/m

inut

e)

0.2 0.4 0.6 0.8

-10

0

10

20

30

B.

time (sec)

flow

(lit

er/m

inut

e)

0.4 0.45 0.5-5

0

5

C.

time (sec)

flow

(lit

er/m

inut

e)

0.405 0.41-2

-1

0

1

2

Figure 7. Flow rate through the aortic valve (reported in liters per minute) as a function of time. Stroke volume isapproximately 90 ml, which is within the physiological range. Note that during the second half of the cycle, when thevalve is closed, the valve allows no backflow. The boxed region in panel A indicates the region of detail shown in panelB. Likewise, the boxed region in panel B indicates the region shown in panel C. Note that the oscillations in the flowrecord which appear to be quite sharp in panel A are actually smooth; see panels B and C.

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feasible to perform three-dimensional immersed boundary simulations of cardiac ejection in which realisticcardiac output is obtained at realistic blood pressures. Although such simulations require a significantamount of computing resources at the present time, we believe that the development of effective implicittime discretization schemes for the immersed boundary method will result in a significant improvement inefficiency. By building on recent work on implicit immersed boundary methods,30–32 we expect that it willbe possible to reduce substantially the computational cost of the immersed boundary method, increasingits utility not just for simulating cardiovascular fluid dynamics but for a broad range of fluid-structureinteraction problems.

Acknowledgments

This work was sponsored in part by a grant from the American Heart Association to BEG. Computationswere performed at New York University using computer facilities funded in large part by a generous donationby St. Jude Medical, Inc.

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