Prediction of Flow Features in Centrifugal Blood Pumps · Clinic for Thoraic and Cardiovascular...

ECCM-2001European Conference on

Computational Mechanics

June 26-29, 2001Cracow, Poland

Prediction of Flow Features in Centrifugal Blood Pumps

Marek Behr and Dhruv Arora

Department of Mechanical Engineering and Materials ScienceRice University, Houston, TX, USA

e–mail: [email protected]

Sebastian Schulte-Eistrup

Heart Center NRWClinic for Thoraic and Cardiovascular Surgery, Bad Oeynhausen, Germany

e–mail: [email protected]

Key words: Computational Hemodynamics, Finite Element Method, Centrifugal BloodPumps

Abstract. We present preliminary results from numerical simulations of blood flow in a pro-totype of a centrifugal pump. The simulations involve incompressible Navier-Stokes govern-ing equations, stabilized space-time finite element formulations, and parallel implementationon distributed memory architectures. The examples illustrate our approach of custom hybridstructured-unstructured tetrahedral meshes, which accommodate the rotation of the pump im-peller with respect to the stationary housing, without assuming axisymmetry of either part.

Marek Behr, Dhruv Arora, Sebastian Schulte-Eistrup

1 Introduction

The development of implantable left ventricular assist devices (LVADs), in the form ofcontinuous-flow axial and centrifugal pumps, offers hope for many heart attack victims waitingfor donor hearts. These autonomous devices are intended as a medium-term bridge to trans-plant, or, if enough progress is made, even as a permanent solution. Some of the engineeringchallenges in the design of blood pumps that need to be addressed are:

Efficiency. Implanted blood pumps should have minimal power requirements while providingthe necessary blood flow volume. Another consideration is the size, which must make implan-tation, in children as well as in adults, possible.

Shear-stress levels. Excessive levels of shear-stress in the blood flow through the pump lead toplatelet, and then white cell, damage. Gradual and smooth acceleration of the fluid throughoutthe flow path is an important design goal.

Stagnation and clotting. Areas of persistent stagnation in the flow field may result in thrombusformation and catastrophic failure of the device. Pump components must be shaped to avoidrecirculation and large stagnant flow zones.

We are extending our CFD methodology in order to speed up and automate the analysis ofblood flow in centrifugal pumps. Our approach is based on a space-time weak form of theincompressible Navier-Stokes equations, and finite element discretization using unstructuredtetrahedral meshes [1]. We are employing a mixed velocity-pressure formulation, using equal-order interpolations for the velocity and pressure fields. This convenient choice of interpolationis made possible by consistent stabilization techniques, in particular the Galerkin Least-Squares(GLS) approach. Another GLS benefit is the inclusion of SUPG-like [2] terms in the formula-tion, which provide optimal stability for high element-level Reynolds numbers. The governingequations are recalled in Section2, and the weighted-residual formulation is presented in Sec-tion 3.

In the presence of large mesh deformations, conventional mesh moving techniques often leadto unacceptably stretched elements, and a breakdown of accuracy. This necessitatesremesh-ing, defined as creating new inter-element connectivity, and sometimes a new set of nodes.Every remeshing requires a projection of the solution from the old mesh to the new one, andtherefore introduces additional inaccuracies to the computation. In the case of e.g. blood pumpimpeller, it is clear that such remeshing would have to be done rather frequently, many timesper revolution. To circumvent this difficulty, we are employing the Shear-Slip Mesh UpdateMethod (SSMUM) [3], which involves generating rigid unstructured meshes attached to indi-vidual boundaries of the domain, and connecting them together via a regular layer of elements.The objective is to restrict remeshing to that small layer, and control projection errors by lim-iting remeshing to changes in inter-element connectivity only. With this approach, the cost ofremeshing becomes negligible, and consequently, such remeshing can be carried out frequently,even at every time step. As an alternative to slipping mesh approach and Chimera methods,SSMUM does not entail accuracy losses due to frequent interpolation, nor does it have the re-strictions on the time step size which are inherent in the slipping mesh technique. The methodis further elucidated in Section4, and examples based on actual pump geometries are given in

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ECCM-2001, Cracow, Poland

Section5.

Throughout this research, we are employing scalable, portable and efficient implementationsof the methods involved, utilizing iterative solution update techniques and standard message-passing libraries. The simulations are carried out on distributed-memory parallel computers,including CRAY T3E-1200 and IBM SP [4].

2 Governing Equations

We consider blood as a viscous, incompressible fluid occupying at an instantt ∈ (0, T ) abounded regionΩt ⊂ R

nsd , with boundaryΓt, wherensd is the number of space dimensions. Inthe DSD/SST formulation, the spatial domain may change with respect to time, and the subscriptt indicates such time-dependence. The symbolsu(x, t) andp(x, t) represent the velocity andpressure. The external forces (e.g., the gravity) are represented byf(x, t). The momentum andmass balance equations can be written as follows:

ρ

(∂u

∂t+ u · ∇u− f

)− ∇ · σ = 0 onΩt, (1)

∇ · u = 0 onΩt, (2)

where the densityρ is assumed to be constant. We consider only the Newtonian fluid model, forwhich the deviatoric component of the stress tensorσ is related linearly to the strain rate tensor:

σ(u, p) = −pI + 2µε(u), ε(u) =1

2

(∇u + (∇u)T

), (3)

whereµ is the dynamic viscosity. The Dirichlet and Neumann-type boundary conditions are:

u = g on (Γt)g , (4)

n · σ = h on (Γt)h , (5)

where(Γt)g and(Γt)h are complementary subsets of the boundaryΓt. Statement of the fluidflow problem is completed by imposing suitable divergence-free initial condition for the veloc-ity field.

Remark 1

Blood can be modeled as a Newtonian fluid only if shear-stress levels in the flow are moderate,and the clearances involved arelarge when compared with blood cell diameter. For higher-fidelity simulations, a non-Newtonian constitutive model may be adopted. For a review of thebody of knowledge in this field see [5]. Another recent work thoroughly examines the appli-cability of continuum models, including viscoelastic ones, to blood as a liquid [6]. We alsoanticipate thatmicrostructure-based models [7] will play an important role in blood flow simu-lations.

Because the typical meshes are not able to resolve the flow features well enough to fully cap-ture turbulence effects at high Reynolds numbers, introduction of a turbulence model becomes

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necessary. In the present study, a simple Smagorinsky turbulence model is employed. In thismodel, the kinematic viscosityν is augmented by an eddy viscosityνt which is defined as

νt = (Ch)2 (2ε(u) :ε(u))12 , (6)

whereC = 0.15 andh is the element length defined here as the maximum of the edge lengthsfor the element.

3 Weak Form

To construct the finite element function spaces for the space-time method, we partition the timeinterval(0, T ) into subintervalsIn = (tn, tn+1), wheretn andtn+1 belong to an ordered seriesof time levels0 = t0 < t1 < · · · < tN = T . Let Ωn = Ωtn andΓn = Γtn . We will define thespace-time slabQn as the domain enclosed by the surfacesΩn, Ωn+1, andPn, wherePn is thesurface described by the boundaryΓt ast traversesIn. As it is the case withΓt, surfacePn isdecomposed into(Pn)g and(Pn)h with respect to the type of boundary condition (Dirichlet andNeumann) being applied.

After introducing suitable trial solution spaces for the velocity and pressure [1], (Shu)n and

(Shp )n, and test function spaces,(Vh

u)n and (Vhp )n, the stabilized space-time formulation of

Equations (1) and (2) is written as follows: given(uh)−n , find uh ∈ (Shu)n andph ∈ (Sh

p )n

such that∀wh ∈ (Vhu)n and∀qh ∈ (Vh

p )n:

∫Qn

wh · ρ(

∂uh

∂t+ uh · ∇uh − f

)dQ +

∫Qn

ε(wh) : σ(uh, ph)dQ

+

∫Qn

qh∇ · uhdQ +

∫Ωn

(wh)+n · ρ

((uh)+

n − (uh)−n)dΩ

+

(nel)n∑e=1

∫Qe

n

τMOM

1

ρ

[ρ

(∂wh

∂t+ uh · ∇wh

)− ∇ ·σ(wh, qh)

]

·[ρ

(∂uh

∂t+ uh · ∇uh − f

)− ∇ · σ(uh, ph)

]dQ

+

(nel)n∑e=1

∫Qe

n

τCONT∇ · wh ρ∇ · uhdQ =

∫(Pn)h

wh · hhdP. (7)

The following notation is being used in Equation (7):

(uh)±n = limε→0

u(tn ± ε),

∫Qn

. . . dQ =

∫In

∫Ωh

t

. . . dΩdt,

∫Pn

. . . dP =

∫In

∫Γh

t

. . . dΓdt. (8)

In most applications of our method, both velocity and pressure fields are represented usingpiecewise linear continuous interpolation functions. The stabilization parametersτMOM andτCONT follow definitions given in [1].

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Remark 2

Both viscoelastic continuum, e.g. Oldroyd-B, and microstructure-based non-Newtonian modelsrequire significant modifications to the weak form given above. Either the stress, or the confor-mation tensor, join the velocity and pressure fields as independent unknowns. For a review offinite element approaches to the modeling of viscoelastic fluids, including the popular Elastic-Viscous Stress Splitting (EVSS) approach, see [8]. A stress-velocity-pressure GLS extension ofthe formulation (7) for Oldroyd-type fluids has been investigated [9].

The nonlinear system given by Equation (7) is solved with the Newton-Raphson method withan analytic Jacobian. At each nonlinear iteration, a sparse linear system is solved using theGMRES iterative mesh update technique [10].

4 Deforming Domain Issues

The mesh update methods developed in conjunction with the DSD/SST formulation and appliedto complex flow problems include special-purpose and automatic mesh moving techniques. Inmost cases, the automatic mesh moving needs to be combined with remeshing. Remeshing,defined as creating new inter-element connectivity and sometimes new set of nodes, could bea bottleneck, depending on factors such as how frequently and how locally remeshing is re-quired and whether an automatic mesh generator needs to be called at each remesh. Limitingthe frequency of remeshing would limit the applicability to problems with modest changes inthe spatial domain. Generous use of remeshing would lift such limitations, but would result inexcessive costs in terms of automatic mesh generation and projection of the solution from theold mesh to the new one. Furthermore, these projections introduce additional inaccuracies to thecomputation, which, depending on the frequency and scope of remeshing, could be excessive.

The Shear-Slip Mesh Update Method (SSMUM) [11, 3] was designed to accommodate somespecific classes of problems which could be handled with the DSD/SST formulation, with-out limitations on the frequency of remeshing. These problems are characterized by large, butlinear or rotational, motion of some parts of the boundary with respect to other parts. This sit-uation occurs when one object forming the computational boundary undergoes translation orrotation with respect to other boundaries. In the SSMUM, we introduce a regular, typically“hand-made”, thin layer of elements; at each time step let the elements in this layer to undergo“shear” deformation; restrict remeshing to that layer as changes in inter-element connectivity;and thus control projection errors and cost of remeshing. Because the remeshing in the thinlayer is accomplished by simply changing the inter-element connectivity, no nodal projectionis needed. As a special case of this approach, the nodal positions for the old (deformed) andnew (good-quality) meshes can be matched, so that the new element shapes are very compa-rable to the shapes prior to the ”shear” deformation. A typical sequence of events is shown inFigure1. This approach is related to the one used previously in the context of finite-differenceturbomachinery simulations [12].

Applications of the SSMUM are based on special-purpose mesh designs which combine re-gions of rigid, non-deforming elements with layers of the shear-absorbing elements. A trans-

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Deformation Reconnect

Figure 1: Shear-slip layer concept in 2D (top) and 3D (bottom): the sequence of deformationfollowed by reconnecting.

lating object may be embedded in a strip (in 2D) or a tube (in 3D) of rigid elements that move“glued” to that object. Similarly, a rotating object may be embedded in a disk of rigid elementswhich rotate “glued” to that object. These non-deforming regions are immersed in another set ofnon-deforming elements spanning the exterior boundaries, and connected with the previouslydescribed shear-slip layer. This concept, which was discussed in more detail in [3], is illustratedfor the 2D case in Figure2.

Figure 2: Shear-slip layer concept: special-purpose meshes. Regions of deforming elements areshown in grey, and regions of rigid elements are shown in white.

5 Numerical Examples

We now concentrate our attention on a specific family of geometries, representing centrifugalblood pumps. A typical configuration is schematically shown in Figure3. It includes an inflowtube positioned above the impeller, and an outflow tube on the cylindrical part of the housing.

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The impeller and its shaft are held loosely in place by two fluid-filled hemispherical bearings.The impeller has a small number of large primary top vanes, typically six, and also a lessernumber of secondary bottom vanes, typically two. The main purpose of the lower vanes is notthe transfer of the momentum to the fluid, but washing out the fluid from the low-clearance areaunder the impeller.

top vanes (6)

bottom vanes (2)

bearing

Figure 3: Centrifugal blood pumps: typical configuration.

For the purpose of flow simulation, only the internal surfaces of the pump, such as the onesshown in Figure3, need to be represented. The actual pump is slightly larger, containing motorand control components directly beneath the housing; the impeller rotation is driven throughmagnetic couplings.

We have developed meshes for two models of such centrifugal pumps, and performed prelimi-nary flow simulations using one of these models.

5.1 KP305 Centrifugal Pump

The KP305 is a compact centrifugal pump design [13], with a chamber diameter of 40 mm,inflow tube diameter of 9 mm, and outflow tube diameter of 11.9 mm. This pump is designedto deliver the flow volume of 5 l/min at approximately 3000 rpm. The external CAD geometry,as well as the finite element representation of the internal surfaces are shown in Figure4. Alsovisible is the portion of the SSMUM layer, which surrounds the impeller. Due to typically lowclearances on the lower side of the impeller, the most natural layer design is an axisymmetricinverted pot with its main axis coinciding with the axis of rotation. The pot intersects the lower

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Figure 4: KP305 centrifugal blood pump: CAD model and the computational mesh.

surface of the housing and the upper tip of the impeller shaft. The outer diameter of the SSMUMlayer is 37.5 mm and its maximum thickness is 0.5 mm.

The mesh shown in Figure4 is generated in a semi-automatic process. At first, the internalsurface of the chamber and the external surface of the impeller is extracted from the CADmodel and made “watertight”. The model surface is then cut at locations where the SSMUMwill intersect it; this produces two disjoint sections of the model surface, one of which willbe stationary and another will be spinning. The layer surface elements (inner and outer) aregenerated and attached to the two sections, forming two watertight surfaces. An automatic meshgenerator [14] is then used to generate two unstructured meshes, which are then combined witha structured layer mesh to form the final mesh. In the example presented here, the total numberof elements is 488,689, and the number of space-time nodes is 172,152. The SSMUM layer has96 segments in circumferential direction, and 45 segments in the remaining direction.

The initial simulation involved 768 time steps, with a time step size of14∗96

seconds, and 4 timesteps elapsing between each SSMUM reconnect. Each time step involved 4 nonlinear iterations,and a Krylov space size of 100 in the GMRES solver. While the flow field is still developing, atthe end of the 768 time steps we were able to observe the pressure distribution on the impellersurface shown in Figure5, as well as the potential persistent areas of slow-moving fluid, whichpresent a clotting risk. These are in evidence in Figure6, which shows the isosurface of thevelocity magnitude close to the stagnation value, in a stationary reference frame (at 5 in/s), aswell as in a rotating reference frame attached to the impeller (at 10 in/s relative to the impeller).Each “bubble” which brings the isosurface away from the solid boundary is a stagnation area.While the stagnation areas in the outflow tube are observed to be merely transient, the areaswhere the inflow impinges on the upper portion of the shaft, and around the bottom portion of

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Figure 5: KP305 centrifugal blood pump: Pressure distribution on the impeller surface

the shaft, remain so for the duration of the simulation, and therefore are a cause for concern.

Figure 6: KP305 centrifugal blood pump: Stagnation areas in a stationary frame of reference(left) and in a rotating frame of reference (right).

5.2 PI710 Centrifugal Pump

The PI710 is a larger centrifugal pump design, with a chamber diameter of 2.24 in, and a uni-form inflow and outflow tube diameter of 0.354 in. The external CAD geometry, and the finiteelement representation of the internal surfaces are shown in Figure7. Also visible is the portionof the SSMUM layer, surrounding the impeller. Because of the significant slant of the upperimpeller surface and the housing lid, the top portion of the SSMUM “pot” is now a cone, be-coming flat only in the vicinity of the shaft. The outer diameter of the SSMUM layer is 2.12 inand its maximum thickness is 0.04 in.

As the development of the KP305 pump has been stopped, our simulations now focus on the

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Figure 7: PI710 centrifugal blood pump: CAD model and the computational mesh.

PI710 model. In addition to the stagnation area detection, the levels of shear-stress present inthe flow field are also being investigated.

6 Conclusions

We have presented aspects of a computational approach used to perform simulations ofunsteady, incompressible blood flow in a centrifugal pump. The method is based on theDeformable-Spatial-Domain/ Stabilized Space-Time (DSD/SST) finite element formulation ofthe Navier-Stokes equations. A numerical example involved a 3D computation of a complexunsteady flow in the KP305 centrifugal blood pump.

7 Acknowledgment

This research was supported in part by National Science Foundation through computing re-sources provided by the National Partnership for Advanced Computational Infrastructure.

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