A Tensor-based Measure for Estimating Blood · PDF fileA TENSOR-BASED MEASURE FOR ESTIMATING BLOOD DAMAGE 1003 Artif Organs, Vol. 28, No. 11, 2004 oped, and the conservation equations

Artificial Organs28(11):1002–1015, Blackwell Publishing, Inc.© 2004 International Center for Artificial Organs and Transplantation

1002

Blackwell Science, LtdOxford, UKAORArtificial Organs0160-564X2004 International Society for Artificial Organs281110021015Original ArticleA TENSOR-BASED MEASURE FOR ESTIMATING BLOOD DAMAGED. ARORA ET AL.

Received October 2003; revised April 2004.Address correspondence and reprint requests to Dr. Marek

Behr, Lehrstuhl für Numerische Mechanik, Technische Uni-versität München, D-85747 Garching, Germany. E-mail:[email protected] or Dr. Matteo Pasquali, Department ofChemical Engineering, MS 362, PO Box 1892, Houston, TX 77251-1892, U.S.A. E-mail: [email protected]

A Tensor-based Measure for Estimating Blood Damage

*†Dhruv Arora, *‡Marek Behr, and †§Matteo Pasquali

*Department of Mechanical Engineering and Materials Science; †Computer and Information Technology Institute (CITI), Rice University, Houston, TX, U.S.A.; ‡Lehrstuhl für Numerische Mechanik, Technische Universität München, Garching,

Germany; and §Department of Chemical Engineering, Rice University, Houston, TX, U.S.A.

Abstract: Implantable ventricular assist devices give hopeof a permanent clinical solution to heart failure. Thesedevices, both pulsatile- and continuous-flow, are presentlyused as medium-term bridge to heart transplant or recov-ery. While long-term use of continuous-flow axial and cen-trifugal pumps is being explored, the excessive level ofblood damage in these devices has emerged as a designchallenge. Blood damage depends both on shear stress andexposure time, and device designers have relied tradition-ally on global space- and time-averaged estimates fromexperimental studies to make design decisions. Measuringdistributions of shear stress levels and the blood cell’sexposure to these conditions in complex rotary pump flowis difficult. On the other hand, computational fluid dynam-ics (CFD) is now being used as a tool for designing viabledevices, offering more detailed information about the flowfield. A tensor-based blood damage model for CFD analy-

sis is proposed here. The model estimates the time- andspace-dependent strain experienced by individual bloodcells and correlates it to blood damage data from steadyshear flow experiments. The blood cells are modeled asdeforming droplets and their deformation is tracked alongthe pathlines of a computed flow. The model predicts thatblood cells in a rapidly fluctuating shear flow can sustainhigh shear stress levels for very short exposure time with-out deforming considerably. In the context of mechanicalmodeling of the implantable Gyro blood pump beingdeveloped at Baylor College of Medicine, this suggeststhat blood cells traversing regions of highly fluctuat-ing shear stress rapidly may not hemolyze significantly.Key Words: Ventricular assist device—Computationalfluid dynamics—Red blood cell—Hemolysis—Dropletdeformation.

Every year 800 000 new cases of heart disease arereported in the United States; by a conservative esti-mate, 50 000 patients can use a new heart, but only2500 donor hearts are available annually (1). Similarstatistics prevail worldwide. While a patient waits fora healthy donor heart, life-saving medical treatmentwith a ventricular assist device (VAD) is currentlythe only hope for the ailing heart. Since the firstsuccessful VAD implant in 1966 by DeBakey (1),VAD technology has progressed to a point wherethese devices are being used as a medium-termbridge to transplant. Presently, the development ofthe third generation VAD devices with magneticallysuspended no-contact bearings is underway (2), and

these devices can potentially provide an alternatetreatment to heart transplant.

The development of VADs poses a tremendouschallenge because the working fluid in these devicesis blood, which has a highly complex, flow-dependentphysical and chemical nature. Blood is a suspensionof formed elements—red blood cells (RBCs), whiteblood cells, and platelets—in a Newtonian liquid(plasma) of viscosity 1 mPa·s (cP). At normal con-centrations, blood behaves as a non-Newtonianfluid. Most traditional blood-handling devices weredesigned for Newtonian fluids and adapted for bloodflow applications after prolonged in vivo animal tri-als. Besides being time-consuming and laborious, theprocess was costly, and by necessity, involved only ahandful of candidate designs.

There are two aspects of VAD design—hydraulicand hematologic. The hydraulic design ensures thatthe pump delivers appropriate flow rates against agiven pressure head; the hematologic design aims atminimizing the blood damage in the pump. Hydraulicdesign methods for centrifugal pumps are well devel-

A TENSOR-BASED MEASURE FOR ESTIMATING BLOOD DAMAGE 1003

Artif Organs, Vol. 28, No. 11, 2004

oped, and the conservation equations governing theflow are known completely (3–9). Thus, the hydraulicdesign process of the VAD is well-formulated.Recently, computational fluid dynamics (CFD) hasemerged as a reliable hydraulic design tool for VADs.Since a 3-dimensional non-Newtonian CFD simula-tion of a blood pump is a formidable task, severalgroups have conducted Newtonian CFD analysis ofboth pulsatile- and continuous-flow blood pumpsover the past decade (5–9) and reported hydraulicparameters such as flow rate, pressure head, shearstress, velocity profiles, location and size of stagna-tion zones, and local shear rate. These parametersprovide crucial quantitative information for hydrau-lic design, and qualitative information for hemato-logic design. Burgreen et al. reviewed the CFD-basedholistic design process of rotary blood pumps andpointed to the need for coupling between hydraulicand hematologic design (10).

Blood damage (hemolysis) and blood aggregation(thrombosis) are two important features of hemato-logic design (11). How these processes occur in com-plex flow situations is not yet understood clearly. Asa result, unlike the hydraulic design process, thehematologic design problem is not well-formulated.Previous works on hematologic design have relied ontest-loop experiments with human or animal bloodto obtain actual measurements of hemolysis (12–14).The experimentally measured hemolysis values, whilebeing accurate, provide only global, time- and space-averaged estimates by treating the blood pump as onecomplete unit. It is not possible to obtain a localestimate of hemolysis in a section of the blood pumpin test-loop experiments. Flow visualization tech-niques have been used to estimate qualitatively localhemolysis characteristics of the pumps (15). Thus,understanding the effect of design changes on deviceperformance (e.g., minimal hemolysis) is a slow,expensive, and laborious process. On the other hand,CFD-based hemolysis prediction is quicker, cheaper,can easily show the effect of localized design modifi-cation on pump performance, and can also assess theaggregate effect of multiple design changes. However,in order to succeed, it requires a realistic model ofblood damage. While the flow in blood pumps is 3-dimensional and unsteady, blood damage models areavailable only for simple steady shear flows.

In our previous work on computational analysisof implantable ventricular assist devices (3,4), wereported hydraulic measures; here we propose amodel for predicting hemolysis based on instanta-neous deformation of RBCs in a general flow. Thismodel accounts for blood cell properties, and is tunedwith the experimental data on flow-induced RBC

deformation and hemolysis. In this article, we reviewthe literature on hemolysis in the next section. Thegoverning equations describing the blood flow arediscussed in the following section. A blood damagemodel based on RBC deformation can be developedonly with a clear understanding of RBC behavior inshear flow, which has been reported in several exper-imental studies; thus, the next section details theRBC behavior in steady shear flow as observed inviscometric experiments reported in the literature,and discusses a hemolysis-stress-exposure correla-tion. Then we set out the tensor-based blood celldeformation model; the parameters of this model arethen tuned with available experimental data. Theinstantaneous blood cell deformation is related tohemolysis prediction. Implementational details ofhemolysis prediction in homogeneous and inhomo-geneous flows are given, followed by numericalexperiments, which compare the new hemolysis pre-dictions with the traditional methods. Finally, we con-clude with a summary and future directions.

LITERATURE REVIEW: HEMOLYSIS IN BLOOD PUMPS

RBCs are the largest constituent of blood, andhemolysis refers to premature damage of RBCs,which otherwise have a normal life span of 120 daysin a healthy person (16). While the body can adaptto a moderately abnormal depletion rate of RBC,excessive hemolysis may lead to a low RBC countand a state of hemolytic anemia. Apart from lowRBC count response, the plasma free hemoglobin istoxic for the kidneys (2), and can eventually lead tomultiple organ failure. Hemolysis can happen due tovarious pathological conditions or external factors,which can all be reduced to four basic mechanisms:colloid osmotic lysis, perforation of a cell, excessivedeformation or fragmentation of RBCs, and eryth-rophagocytosis. Of these, deformation and fragmen-tation of RBCs due to shearing, i.e., mechanicalhemolysis, is the dominant mechanism of hemolysisin VADs. Mechanical hemolysis starts when an RBCdeforms excessively in response to high shearing, andleaks part of its hemoglobin content into the blood-stream through small reversible openings in its mem-brane. Catastrophic hemolysis occurs when the RBCmembrane ruptures. For over four decades, steady-shear hemolysis studies have been performed tounderstand mechanical hemolysis (17–20). Some ofthe early studies advocated the importance of wallshear stress and wall area (21), but it is well acceptednow that mechanical hemolysis is primarily a bulkphenomenon, which depends on the shear stress and

1004 D. ARORA ET AL.


exposure time (16,22). Although most studies ofmechanical hemolysis have focused on the effects ofsteady shearing, a few unsteady-shearing experi-ments have also been reported (23).

Despite extensive experimentation, there is a lackof consensus on a model for mechanical hemolysis.Moreover, the timescale of most reported mechanicalhemolysis experiments was three orders of magnitudegreater than the timescale in a VAD. The typical flowrate in a VAD matches that of a human heart (5 L/min); the characteristic residence time of an RBC ina VAD is approximately 500 ms. Thus, most steadyshear hemolysis data cannot be used for developinga blood damage model for VAD applications. Heuserand Opitz developed a Couette viscometer to mea-sure hemolysis at short exposure times (24), andhuman-blood hemolysis experimental data from thissetup were reported by Wurzinger et al. (25). Gier-siepen et al. (26) developed a correlation (discussedbelow) for steady shear hemolysis based on theseexperiments. These hemolysis data and correlationare the most relevant for VAD applications, and areused here for developing a blood damage model.

Whereas the hemolysis experimental data arefor steady shear loading, the flow in VADs is 3-dimensional and unsteady. Previous works attemptedto relate the 3-dimensional flow effects to steady shearloading through a single scalar parameter. A fewscalar-parameter-based hemolysis models haveappeared in the literature over the past decade (e.g.,Bludszuweit proposed a representative instantaneousone-dimensional stress parameter obtained from thesix components of the deviatoric stress tensor) (27).These models relating the complex VAD flow tosteady shear flows through a representative instanta-neous scalar stress parameter for hemolysis predic-tion are hereafter called “stress-based” and arefurther discussed below. It is also important to notethat the instantaneous stress, or instantaneous strain,models assume an instantaneous one-to-one relation-ship between local stress and RBC deformation; suchrelationship holds only in Lagrangian steady flows,i.e., flows where the velocity gradient is constant alongstreamlines (e.g., steady shear flow). The RBC mem-brane is viscoelastic, and the viscoelastic lag of bloodcell membrane to the applied shear rate can be takeninto account by considering accumulated straining.Another concern in the long term use of VADs is therepeated exposure to mechanical straining of theRBCs flowing through VAD implanted circulation.Yeleswarapu et al. developed a scalar damage accu-mulation model which incorporates the aging ofRBCs (28). This model requires a damage function,which is unknown in complex flow situations. Thus,

short exposure instantaneous stress-based models arepresently used for hemolysis predictions.

GOVERNING EQUATIONS FOR BLOOD FLOW

Blood flow is modeled by the momentum and massconservation equations for an incompressible fluid,

(1)

(2)

where r is the blood density (1058 kg/m3), u is thevelocity, p is the pressure, s is the extra stress, and fdenotes body forces per unit mass (e.g., gravity). Theproblem is closed by prescribing an appropriate con-stitutive equation for the stress. Blood is a shear-thinning viscoelastic fluid; Yeleswarapu et al. showedthat a generalized Oldroyd-B constitutive equationcan describe well the rheological behavior of bloodin shear flows (29). In the current work, we treatblood as a Newtonian liquid:

(3)

where m is dynamic viscosity of blood at high shearrate i.e., 3.5 mPa·s (cP). Thus, Eqs. 1 and 2 reduce tothe incompressible Navier–Stokes equations. How-ever, we develop the hemolysis model in a generalcontext that can be applied to flow calculations per-formed with other constitutive models.

RED BLOOD CELLS IN SHEAR FLOW AND HEMOLYSIS

A blood damage model based on RBC deforma-tion should account for RBC properties and defor-mation in shear flow, and it should be related to arelevant hemolysis data set. An RBC at rest is abiconcave disc of a viscoelastic membrane filled witha Newtonian liquid with a viscosity of approximately6 mPa·s (cP). The RBC membrane has a relaxationtime of approximately 200 ms which depends on theage of the cell (30), and it can support an estimatedareal strain of 6% before rupturing (16). Owing to itsbiconcave shape, an RBC has 40% excess surfacearea compared to a sphere of the same volume. Theexcess surface area enables RBCs to undergo defor-mations that preserve both volume and surface area.In a flow, the RBCs behave as neutrally buoyantmicrocapsules with high deformabilty but small arealstretchability (31,32). The RBCs at rest aggregateinto coinstack shaped structures called rouleaux.These structures break as shear stress increases, andRBCs become dispersed. As reported by Schmid-

r∂∂u

u u ft

p+ ◊ — -ÊË

ˆ¯ = -— + — ◊ s,

— ◊ =u 0

s = — + —( )m u uT ,



Schönbein and Wells (31), the dispersed RBCs pre-serve their biconcave shape and tumble in a flow withshear stress below ª 0.1 Pa. The tumbling graduallyreduces and the cells begin to align with the flow atª 0.2 Pa. The RBCs deform into ellipsoidal shape,orient with the flow and show tank-treading at shearstress greater than 1 Pa. Tank-treading is a phenom-enon in which the cell membrane rotates around theenclosed fluid. It is not clear if the RBCs deform intoprolate or general ellipsoids, but most experimentalstudies refer to the general ellipsoidal state. Thetank-treading and deformation into ellipsoidal shapehas been confirmed in several later experimentalstudies (17,33). Leverett et al. found that above150 Pa, extensive hemolysis occurs due to shearstress alone (19). This shear stress corresponds toshear rate of ª 42 000 s-1 for whole blood. The RBCmembrane is believed to reach its 6% areal strainlimit at this shear rate, so that any further stretchingof the membrane leads to catastrophic hemolysis.Figure 1 (top) shows these configurations of RBC inshear flow. The leftmost state corresponds to no shearand shows rouleaux, and the rightmost state shows ahemolyzing RBC with pores on the surface.

Giersiepen et al. developed a correlation forsteady-shear hemolysis at short time scales relevantto flow in a VAD based on experimental results byWurzinger et al. (25,26). The correlation is:

(4)

where DHb/Hb is the ratio of plasma free hemoglo-bin to the total hemoglobin in the sample (i.e., the

DD

HbHb

t= ¥ -3 62 10 7 2 416 0 785. ,. .s

plasma free hemoglobin plus hemoglobin enclosed inRBCs), s is the shear stress (Pa) and Dt is exposuretime (s). A plot of percentage hemolysis versusapplied shear stress and exposure time is shown inFig. 2(a). These physical properties, deformation con-figurations, and hemolysis data are used hereafter fordeveloping the deformation based blood damagemodel.

Hemolysis predictions based on Eq. 4 have beenrecently reported for rotary blood pumps (34–36).De Wachter and Verdock used Eq. 4 to calculatehemolysis in hemodialysis cannulae (37). In thesestudies, the rate of hemolysis is integrated along thepathlines with an instantaneous scalar measureof stress s to compute accumulated hemolysis. Therate of hemolysis,

(5)

is shown in Fig. 2(b). The hemolysis accumulationdecreases with exposure time. It is important to notethat the correlation (4) was developed in steadyshear hemolysis experiments, where the shear stresss is time independent: however, later studies ofblood pumps used in Eq. 5 a scalar quantity derivedfrom the instantaneous deviatoric stress tensor,

. As mentioned before, this method of

computing hemolysis is here called “stress-based.”While DHb/Hb has been the measure of choice in allthe above mentioned works, Normalized Index ofHemolysis (NIH) is the standard clinical index used

ddt

HbHb

tD

DÊË

ˆ¯ = ¥ - -2 8417 10 7 2 416 0 215. ,. .s

s = -12

s s:

FIG. 1. Red blood cell and droplet defor-mation in shear flow: (top row) configura-tions of RBCs in increasing shear flow;(bottom row) spherical droplet deforminginto a general ellipsoid.

Rouleaux Biconcave- monodispersed- tumbling

Ellipsoidal

- oriented- tank-treading

Hemolyzing- pores- tank-treading- highly stretched

Spherical droplet Ellipsoidal

- oriented- tank-treading

Highly stretched ellipsoid

Increasing shear rate

sam

e su

rfac

e ar

ea



to report hemolysis (38) in flow loop tests. TheNIH per single pass through a test loop is related toDHb/Hb as:

(6)

where Hct is the blood hematocrit (45% for a healthyperson) and k is the hemoglobin content of blood(150 g/L for a healthy person). In this work bothDHb/Hb and NIH values are reported.

TENSOR-BASED MODEL OF RED BLOOD CELL DEFORMATION

An RBC in a general flow can be approximatelyviewed as a neutrally buoyant liquid droplet (31).Thus, the deformation of an RBC in shear flow canbe approximated by that of a liquid droplet, by takingthe physical properties of an RBC into consideration.This approach has been followed by Barthès-Bieseland Rallison (39), who derived a constitutive equa-tion for the deformation of a droplet by balancingthe interfacial tension with fluid stresses on eitherside of the interface. Maffettone and Minale pro-posed a droplet deformation equation in terms of asymmetric, positive-definite morphology tensor Sthat represents the shape of the droplet (40). Theequation takes into consideration the competingaction of interfacial tension on the droplet surface,which recovers the spherical shape of the droplet,and the force exerted by the surrounding liquid. Theequation is frame-invariant (which makes it applica-ble to complex flows), and it accounts for nonaffinedroplet deformation. Though RBCs can be modeledas droplets, the tank-treading motion shown by

NIH L bloodgHb

HbHct100 100 1( ) = ¥ ¥ -( ) ¥

Dk,

RBCs is absent in droplets. On the fluid interface ofa droplet the internal and external fluid velocitiesmatch. For an RBC with tank-treading motion, theinterface is a lipid bilayer that rotates aroundthe enclosed liquid with a velocity proportional tothe local fluid shear rate (33); the local velocities ofboth internal and external fluids match the rotationvelocity of the membrane. In what follows, we modifythe original equation to account for the tank-treadingphenomenon peculiar to RBCs in shear flow, whilepreserving the frame invariance.

Consider a general flow with velocity gradient .The velocity gradient can be decomposed into thesymmetric rate of strain tensor E and the antisym-metric vorticity tensor W, as:

(7)

The droplet deformation equation as proposed byMaffettone and Minale (40) is:

S∞ = f2 [E · S + S · E] - f1 [S - g(S)I], (8)

where

(9)

and II and III are second and third invariants of S,respectively, defined as:

(10)

The parameters f1 and f2 depend on the physicalproperties of the liquid droplet. The first term on theright-hand side represents how the flow acts todeform the droplet, and the second term accounts for

—u

— = + — + —( ) — - —( )u E W E u u W u u; ; .=12

=12

T T

gIIIII

S( ) =3

,

II III= ( ) - ( )( ) = ( )12

tr tr2 2S S S; det .

FIG. 2. Hemolysis correlation by Giersiepen et al. (26): (a) hemolysis versus exposure time Dt and stress s; (b) rate of change ofhemolysis correlation given by Eq. 5.

050

100150

200250

00.2

0.40.6

0.81

0

0.05

0.1

0.15

0.2

0.25

(N/m 2)Dt (s)

∆H

bH

b

050

100150

200250

00.2

0.40.6

0.810

0.1

0.2

0.3

0.4

0.5

0.6

0.7

σ (N/m2)Dt (s)

d dtD

Hb

Hb

(a) (b)

(

)

σ



the resisting effect of interfacial tension. The left-hand side term,

(11)

represents the Jaumann derivative, and indicatesthat the droplet is rotating with the vorticity of theexternal fluid. This term makes the equation frame-invariant. Roscoe showed that for a tank-treadingellipsoidal particle the effective vorticity seen by theparticle reduces in proportion to the tank-treadingfrequency (41). This result holds true for a stationaryellipsoid that maintains a fixed orientation with theflow. A tank-treading ellipsoidal RBC in unsteadyflow has varying orientation with the flow, and onlysees a reduction in relative vorticity, i.e., the differ-ence between the vorticity W and the rate of rotationof the principal axes of the ellipsoidal particle. Therelative vorticity becomes apparent, if we consider anorthogonal frame of reference rotating with respectto the fixed orthonormal frame as:

(12)

where ei and are the orthonormal vectors of fixedand rotating frames, and Q is the volume-preservingrotation matrix (Q ◊ QT = I). The velocity gradients inthe two frames are related as:

(13)

This can also be written as:

(14)

Since is a skew symmetric tensor, taking trans-pose of the above equation, and adding and subtract-ing to itself we get:

(15)

where is the rotation rate of the rotatingframe with respect to the fixed one. To account forthe instantaneous rotation of the tank-treading cellwe consider the rotating frame defined by the uniteigenvectors of S; the rotation W is hence com-puted as:

(16)

Thus, Maffettone and Minale (40) equation of drop-let deformation in a frame of reference rotating withthe cell becomes:

(17)

where

SS

W S S WS S

u So ,∫ - ◊ - ◊[ ] =∂∂

+ ◊ —ddt

ddt t

;

˜ ,e Q ei it= ( ) ◊

ei

˜ ˜ ˙ .— = — + ◊v v Q QT

˜ ˜ ˙E W E W Q Q+ .( ) = +( ) + ◊T

Q QT ◊

˜ ˜ ˙ ,E E W W Q Q W= ; = + ◊ ∫ -T W

- = ◊W Q QT

ei

W = =∂∂

+ ◊ —ÊË

ˆ¯

˜˜

˜˜

˜ .ee

ee

u eii

ii

iddt t

S S S I E S S E W S S Wo += - - ( )[ ]+ ◊ ◊[ ]+ ◊ - ◊[ ]f g f f1 2 3˜ ˜ ˜ ˜ ,

(18)

It is assumed that the shape of the droplet remainsellipsoidal at all times. It can be easily verified thatEq. 17 preserves the volume of the droplet (i.e., dIII/dt = 0). The first term on the righthand side of Eq. 17models the shape recovery of the droplet in theabsence of shear stress. The second term representsthe nonaffine deformation of the droplet, and thethird term accounts for the tank-treading motionwhich reduces the relative vorticity seen by the drop-let. Such reduction in vorticity was also applied toRBCs by Keller and Skalak, while keeping the RBCat a fixed orientation (42). The eigenvalues of S arethe square lengths of the three axes of the shapeellipsoid. Equation 17 is an implicit equation in Sbecause the rotation rate W depends on S (unless theflow is steady, where dS/dt = 0 and W = 0). Figure 1(bottom) shows the deformed droplet configurationas represented by our model in comparison to theactual shape of a deformed RBC in shear flow. Thedroplet is spherical at no shear (S = I), and becomesellipsoidal as the shear increases. The ellipsoidalshape and orientation is identified by the eigenvaluesof S.

PARAMETERS

Equation 17, in its general form, is applicable to amicrocapsule that shows tank-treading. The equationis made specific for capturing RBC deformation bysetting its parameters f1, f2, and f3. The steady shearexperimental observations reported in the literatureare used to choose the appropriate values of theseparameters. In the absence of flow, a deformed drop-let relaxes to its natural configuration according to:

(19)

For small deformations S ª IIII + eA; consequently,Eq. 19 reduces to dA/dt = –f1A (Appendix A). Thesolution to this approximate equation isS = I + eA0exp(–f1t), and has 1/f1 as the relaxationtime. The relaxation time of RBCs is approximately200 ms (30). Thus, we set:

f1 = 5.0 s-1, (20)

so that the droplet and the RBC have the same relax-ation time. It is important to note that, althoughdroplets and RBCs recover to different shapes, theyboth deform into ellipsoidal shapes under shearing.

In Eq. 17, the eigenvalues of the morphology ten-sor show oscillatory transients at varying shear rateswhen f2 π f3. Such shape fluctuations have not been

SS

S So ∫ - ◊ - ◊[ ]ddt

W W .

ddt

f gS

S S I= - - ( )( )1 .



observed experimentally; thus, we restrict f2 = f3. In asteady shear flow,

(21)where G is the intensity of the flow. At steady state,the droplet remains at fixed orientation to the flow(W = 0), and the steady-state droplet deformationequation is:

(22)

The eigenvalues l1, l2, and l3 of S are computedstraightforwardly from Eq. 22 as:

(23)

(24)

(25)where L, B, and W are three semiaxial lengths of thedroplet. The major axis of the droplet is oriented tothe flow at an angle:

(26)

As shown in the rightmost states of Fig. 1 (bottom),the droplet surface area is matched with the hemo-lyzing RBC (6% areal strain at 42 000 s-1). The areaof a general ellipsoidal droplet is computed by a con-vergent series used by Keller and Skalak (42). Anundeformed RBC has 40% excess surface area withrespect to a droplet of same volume, and it undergoes6% areal strain before hemolyzing; therefore, an ellip-soidal droplet such as that used in our model with thesame volume as an RBC should stretch to 1.4 ¥ 1.06times its original surface area at “hemolysis.” Match-ing the droplet area with hemolyzing RBC area gives:

f2 = f3 = 1.25 ¥ 10-3. (27)

Thus, the three parameters f1, f2, and f3 togetherincorporate relaxation time, long-lived shape oscilla-tions, tank-treading, and critical areal strain limit intothe hemolysis model. Equation 17, with these param-eter values, can be used to estimate RBC deforma-tion in a general flow.

— =È

Î

ÍÍÍ

˘

˚

˙˙˙

=È

Î

ÍÍÍ

˘

˚

˙˙˙

+ -È

Î

ÍÍÍ

˘

˚

˙˙˙

u

E W

0

0

0

0

0

0

0

0

0

2

0

2

0

0

0

0

0

0

2

0

2

0

0

0

0

0

G

G

G

G

G

/

/

/

/

,

1 244 344 1 244 344

f g f T1 2= - ( )( ) = — ◊ + ◊ —( )S S I u S S u .

Wf

f f G2

112

12

22 2

1 3

= =+

ÊËÁ

ˆ¯l

/

,

Lf

f f Gf f G Gf f f G

f2

212

12

22 2

1 312

22 2

2 12

22 2

12

= =+

ÊËÁ

ˆ¯

+( )+ +ÈÎÍ

˘˚

l/

,

Bf

f f Gf f G Gf f f G

f2

312

12

22 2

1 312

22 2

2 12

22 2

12

= =+

ÊËÁ

ˆ¯

+( )- +ÈÎÍ

˘˚

l/

,

q =+ +

ÊËÁ

ˆ¯

-tan .1 1

22

22

12

f

Gf G f f

HEMOLYSIS PREDICTION

In a strain-based hemolysis model, it is reasonableto assume that the instantaneous rate of leaking ofhemoglobin through an RBC membrane—i.e., therate of hemolysis—depends on the instantaneousshape distortion of the RBC. Equation 17 gives theinstantaneous shape of the droplet in a general flow;the instantaneous shape distortion D = (L - B)/(L + B) is computed using the axial lengths of thedroplet (the eigenvalues of S). In a steady shear flowthere is a unique relationship between the distortionand the intensity Geff, as well as the correspondingsteady shear stress seff Equations 23–25 yield:

(28)

(29)

Because in steady shear there is a one-to-one corre-spondence of shear stress and distortion, we con-struct a strain-based hemolysis model by requiringthat the strain-based and stress-based models yieldthe same results in steady shearing; using Eq. 4 as inthe stress-based model, results in a strain-basedrelationship:

(30)Below, we compare the strain-based and stress-basedhemolysis prediction for homogeneous and inhomo-geneous flows, after discussing the numerical aspects.

NUMERICAL METHOD

The blood flow in a complex domain is obtainedby solving the governing equations with the stabi-lized space-time finite element method, and the mov-ing boundaries are handled with a sliding meshtechnique. A complete description of the finite ele-ment method for 3-dimensional analysis of the Gyrocentrifugal blood pump has been previously reported(3,4). The velocity and pressure data from theCFD simulation are postprocessed to obtain RBCdeformation.

A least-squares recovery procedure is used toextract accurate velocity gradients from the velocitydata. In the case of a general inhomogeneous flow,the RBC deformation is computed along pathlines.This enables a Lagrangian computation of RBCmorphology given by Eq. 17 The coordinates ofpathlines are computed using forward Euler integra-tion as:

Gf D

D feff =

-( )12 2

2221

,

s meff blood eff= G .

DHbHb

f DD f

t= ¥-( )

ÊËÁ

ˆ¯

-3 62 101

7 12 2

222

2 416

0 785. ..

.mblood



xn+1 = xn + unDt, (31)

where n is the temporal index of the trace point,and xn and xn+1 are positions of the trace point attime tn and tn+1 = tn + Dt, respectively. Figure 3(a)shows two trace points, xn and xn+1, inside an ele-ment formed by mesh nodes—1, 2, 3, and 4. Thevelocity un is interpolated at location xn using piece-wise linear shape functions over the finite elementmesh. Unlike in a steady problem, velocity data inunsteady cases are available only at discrete time-steps. For accurate pathline tracing, the step-ratio(ratio of step size |xn+1 - xn| to maximum velocity|u|max in the domain) is set to ~10-5. Due to such asmall step-ratio, the number of trace points alongthe pathline is much larger than the number of dis-crete time steps of the CFD data. The data betweenCFD time-steps are interpolated linearly to obtainvelocities at the pathline time-steps. Figure 3(b)shows the temporal discretization with CFD dataon the left and pathline steps on the right. Thus,both spatial and temporal interpolations arerequired for the unsteady problems whereas onlyspatial interpolation is required for the steadycases.

The droplet morphology is also solved using for-ward Euler integration along a pathline as:

Sn+1 = Sn + DSn, (32)

(33)All the nth quantities are computed at tn. Unlike therate of strain and vorticity tensors, the rotation ratetensor Wn is computed using the information from the(n - 1)th step:

DD

S S S I E S S E

W S S W S Sn n n n n n n

n n n n n n n n

f g f

f t

= - - ( )[ ]+ ◊ + ◊[ ]( +◊ - ◊[ ]+ ◊ - ◊[ ])

1 2

3

˜ ˜

˜ ˜ .W W

(34)

where (ei)n and (ei)n-1 are eigenvectors of Sn andSn-1, respectively. The orientation of the eigenvectorsis checked at each step to ensure correct computationof the rotation rate.

An instantaneous distortion Dn is computed fromthe eigenvalues of Sn, and is used to compute anequivalent effective steady shear flow intensity (Geff)n

and shear stress (seff)n. Hemolysis accumulated alongthe pathlines is computed by integrating the rate ofhemolysis correlation in Eq. 5.

In the case of homogeneous shear flows, the rateof strain and vorticity tensors are known analytically;thus, the equation can be simply integrated withouttracing pathlines. In the next section we considerhomogeneous steady and sinusoidal shear flows, andan inhomogeneous flow in a 2-dimensional bloodpump.

NUMERICAL RESULTS

Steady shear flowFor steady shear flow of intensity G, the velocity

gradient and droplet deformation are given byEqs. 21 and 22, respectively. The eigenvalues com-puted in Eqs. 23–25 are used to compute the distor-tion D, which when used in Eq. 28 gives:

Geff = G; seff = s, (35)

as expected. Because the effective membranestress is the same as the fluid stress, the hemoly-sis prediction by both stress- and strain-basedmodels match exactly in a steady shear flow. Forexample, both stress- and strain-based hemolysismodels predict DHb/Hb = 0.306% for G = 5000 s-1

after 15 s.

Wn i ni n i n

i t= ( ) ( ) - ( ) -Â e

e e 1

D,

FIG. 3. Interpolation of velocity data: (a) spatial interpolation—the dotted line represents a pathline running through an element; (b)temporal interpolation—pathline data in the right are interpolated from CFD data on the left.

1 2

34

tn

tn+1

xn

xn+1

CFD data Pathline data

t = 0.0 s

t = 1.0 s

t = 2.0 s

t = 0.25 s

t = 0.50 s

t = 0.75 s

t = 1.25 s

t = 1.50 s

t = 1.75 s

t

(a) (b)



Sinusoidal shearConsider a steady shearing flow of intensity Go

superimposed with sinusoidal shearing G1 sin(wt).The intensity G(t) of the model flow is:

(36)

A similar flow is observed in VADs, where baselineshearing is superimposed with a periodic impulsecaused by rotating impeller or a pulsating chamber.Such a flow may also be observed for devices con-nected to the heart. Since the model flow is homoge-neous and the velocity gradient is known, we canintegrate Eq. 17 in a Lagrangian frame by theexplicit Euler method to get the morphology tensorS.

To study hemolysis in the case of sinusoidal shear-ing we define 3 nondimensional parameters:

a = G0f2t, (37)

(38)

g = wt, (39)

where t is the relaxation time of RBC membrane(200 ms), a is a measure of baseline shearing, b is thestrength of sinusoidal pulse, and g is the nondimen-sional frequency. To account properly for the effectof exposure time we compute the hemolysis accumu-lated in 20 s of shearing after steady state is achieved.A range of a, b, and g is selected and hemolysisresponse is studied. The nondimensional frequency gis varied from 0.4 to 10, such that there are enoughcycles in 20 s at the lowest frequency. The range of aand b ensure that the maximum shear rate never

G tdudy

G G tx( ) ∫ ∫ + ( )0 1 sin .w

b =GG

1

0,

exceeds the critical strain rate limit of 42 000 s-1. Asexpected, the hemolysis scales as:

(40)

over the complete range of a, i.e., the effect of thebaseline shearing in a superimposed steady plus sinu-soidal shearing flow is the same as the effect of theconstant shearing in a steady shearing flow. Figure 4shows a plot of percentage (DHb/Hb)/a2.416 versus band g.

For g < 1.0, i.e., for a low frequency sinusoidalimpulse, the time period of the sinusoidal shearing iscomparable to the relaxation time of RBC. Thus, thecell deformation closely follows the sinusoidal shear-ing, and the effective stress seff closely matches theinstantaneous fluid shear stress s. As a result, bothstress- and strain-based models predict similar levelsof hemolysis, and match each other exactly as g tendsto zero (which corresponds to the steady shear case).As the frequency increases, the time period of sinu-soidal shearing becomes smaller than the RBC relax-ation time. Consequently, the RBC does not deformcompletely in response to rapidly varying sinusoidalshearing. Thus, the contribution of the sinusoidalcomponent of shearing to the cell deformation dimin-ishes with increasing frequency. This result is consis-tent with Hashimoto’s experimental observation ofdecrease in RBC destruction when exposure time atlarger shear rate (>500 s-1) is interspersed withsmaller shear rates (<300 s-1) (23). For g-value as highas 10, the time period of sinusoidal shearing is 20 ms,and the sinusoidal component of shearing does notcause any significant cell deformation. Consequently,the hemolysis is caused by the baseline shearing only.

DHbHb

µ a 2 416. ,

FIG. 4. Sinusoidal shear flow: % (DHb/Hb)/a2.416 versus b and g.

100

101

100

10110–1

100

% D

Hb

Hb

a2.41

6

b g

(

)/



The nondimensional amplitude of sinusoidalshearing b, determines the contribution of sinusoidalimpulse to cell deformation. For b < 1, the predictedhemolysis grows with increasing b; as expected, thisenhanced contribution to hemolysis is significantwhen g < 1. As the magnitude of the fluctuating com-ponent of the shear rate approaches the baselineshearing value (b ª 1) at low frequency (g < 1), thecell deformation follows closely the sinusoidal shear-ing; thus, the cell shape oscillates between a com-pletely relaxed state and a highly stretched one. Afurther increase in b causes a reversal in the defor-mation of the cell during part of the period of oscil-lation. This yields a higher predicted effectivebaseline shearing. While the effect of flow reversal,i.e., b > 1, and consequent increase in baseline shear-ing is gradual when g < 1, there is a sharper increasewhen g > 1.

These results indicate that a low baseline shearingwith superimposed high frequency (or short-lived)spikes is less damaging than a constant shearing ofintermediate intensity. In the context of centrifugaldevices, this suggests that conical centrifugal pumpswith constant, uniform shear should yield higherhemolysis than vaned centrifugal pumps, where theblood cells experience short-lived high shear levelsinterspersed with low shear baseline values. This hasalso been found in recent experimental studies byKawahito and Nosé (12). It is important to stress thatthe difference between hemolysis prediction withstress- and strain-based model increases with the fre-quency of the fluctuating shear component.

A particular case with a = 0.5, b = 0.75 and g = 4 isobserved for 1 s, and shown in Fig. 5. The 1-s windowof observation was selected only after the end of thetransient response to sinusoidal shearing. Figure 5(a)shows the maximum eigenvalue lmax of S and sinuso-idal shear flow intensity G(t) with respect to time. The

eigenvalue lags the sinusoidal impulse due to theviscoelastic nature of the RBC. The phase differencebetween the sinusoidal shearing and the eigenvalueresponse is directly proportional to the sinusoidalfrequency. Figure 5(b) shows the accumulated hemol-ysis as predicted by the strain- and stress-based mod-els in the observed time period. The stress-basedmodel assumes equivalence of the instantaneous fluidstress and RBC membrane stress. On the other hand,the strain-based model computes an equivalentsteady shear stress corresponding to an RBC defor-mation. Hence, the stress-based model predictshigher hemolysis than the strain-based model. Bothstress- and strain-based hemolysis predictions startfrom zero, but the stress-based hemolysis has a higheraverage slope than the strain-based model. Becauseover longer exposure times the average slope deter-mines the hemolysis, the difference between thestress- and strain-based hemolysis increases with theexposure time. The accumulation of hemolysis overa period of 20 s is shown in Fig. 5(c).

Hemolysis in 2-dimensional pumpThe steady and sinusoidal shearing experiments

involve homogeneous simple shear flows; the flow inVADs is inhomogeneous and 3-dimensional. A sim-plified 2-dimensional version of the 3-dimensionalGyro centrifugal blood pump (43) is constructed totest the hemolysis model. Figure 6(a) shows thegeometry of the blood pump. It has a chamber diam-eter of 5.6 cm, and inlet and outlet ports are 0.8 cmin width. A 4-vane impeller rotates at 600 rpm andthe flow rate (per unit depth) in the pump is 2.59 (L/cm)/min. The geometric parameters and flow rate aresimilar to the Gyro operating conditions. Specifically,the flow rate is chosen so that the average residencetime of a blood cell traveling through the 2-dimensional blood pump is the same as the typical

FIG. 5. Sinusoidal shear flow (G0 = 2000 s-1; G1 = 1500 s-1; w = 20 s-1): (a) maximum eigenvalue and flow intensity versus time (f1 =5/s, f2 = f3 = 1.25 ¥ 10-3); (b) stress- and strain-based hemolysis accumulation versus observation time of 1 s; (c) stress- and strain-basedhemolysis accumulation versus observation time of 20 s.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1.6

1.7

1.8

1.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1000

2000

3000

4000

G(t)

lmax

Time (s)

G(s

–1)

l max

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6¥10–3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

Time (s)

%D

Hb

Hb

NIH

(g/

100

L)

stress-based

strain-based

(a) (b)

0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 5 10 15 200

1

2

3

4

5

Time (s)

%D

Hb

Hb

NIH

(g/

100

L)

stress-based

strain-based

(c)



residence time in the Gyro blood pump operatingunder normal conditions; the range of shear rates iscomparable.

Figure 6(b) shows the structured/unstructuredfinite element mesh, with 10 506 nodes and 9498 ele-ments. A moving unstructured mesh is attached tothe impeller, and another stationary mesh isattached to the chamber walls. These two unstruc-tured meshes are stitched together with a structuredmesh. The rotation of the impeller mesh is accommo-dated using the shear-slip mesh update method. Theimpeller completes one revolution in 160 time stepsof 0.625 ms each. The velocity and pressure data aresaved at every step. Figure 7(a) shows the velocityvectors in the pump after 10 revolutions. The dropletdeformation along the pathlines is computed withthe Euler integration scheme by our postprocessor.A piecewise linear interpolation of velocity is usedover the finite element mesh to obtain the data at thetrace points. Because the flow is unsteady, a linearinterpolation in the temporal direction is also usedto obtain the nodal velocities at the trace point timesteps. The components of the velocity gradients are

recovered by a least-squares recovery procedure ateach trace point time step. Figure 7(b) shows twopathlines traced in the 2-dimensional pump flowusing the velocity data. Figure 8 shows the changesin eigenvalues of the morphology tensor and thehemolysis prediction along these pathlines. Asexpected, the eigenvalue of the morphology tensoralong the first pathline shows similarity between thesinusoidal shearing and 2-dimensional blood pumpflow. The similarity is distinctly visible in the stress-based hemolysis predictions where the stress-basedhemolysis characteristic along the first pathline issimilar to the corresponding sinusoidal shearingflow. As in the case of sinusoidal shearing, the strain-based model predicts lower hemolysis than thestress-based one. This implies that the RBCs movingalong these pathlines are affected by the baselineshearing only, and the short-lived high shear rates donot have any significant effect. We calculated theaverage hemolysis by scattering 100 uniformly dis-tributed tracer points over the inflow section andfollowing them for up to 1.0 s (or until they exit thedevice). The total average hemolysis computed with

FIG. 6. Two-dimensional pump: (a) geometry; (b) hybrid mesh.

(a) (b)

FIG. 7. Two-dimensional pump: (a) velocity after 10 revolutions;(b) pathlines.

1 2

(a) (b)

FIG. 8. Two-dimensional pump: (a)eigenvalue of morphology tensor; (b)hemolysis for two pathlines shown inFig. 7(b).

0 0.1 0.2 0.3 0.4 0.50.85

0.9

0.95

1

1.05

1.1

1.15

lmax(2)

lmax(1)

S33(1)S33(2)

lmin(2)lmin(1)

Time (s)

Eig

enva

lues

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8¥105

0 0.05 0.1 0.15 0.2 0.250

0.5

1

1.5¥105

Pathline 1

Pathline 2 stress-based

stress-based

strain-based

strain-based

Time (s)

%D

Hb

Hb

(a) (b)



the stress-based model is 2.01 ¥ 10-5% per tracepoint, whereas the hemolysis predicted by the strain-based model is 1.45 ¥ 10-5%. (By using Eq. 6 this canbe translated to NIH values per single pass throughthe pump of 1.66 ¥ 10-3 and 1.19 ¥ 10-3 g/100 L forthe stress-based and strain-based models, respec-tively; although the 2-dimensional pump is a hypo-thetical one, these values fall in the typical range foractual centrifugal blood pumps, e.g., 1 ¥ 10-3 to4 ¥ 10-3 g/100 L ) (12,44).

CONCLUSIONS

The design cycle of VADs can be considerablyshortened by the use of CFD in the design process.CFD are frequently applied to obtain an appropriateflow rate, pressure head, velocity profile, etc., i.e., anefficient hydraulic design of VADs. However, pre-dicting mechanical hemolysis levels—hematologicdesign—requires a realistic model for blood damage.The experimental research on mechanical hemolysishas focused mainly on steady shear flows, and a cor-relation is available for such experimental data relat-ing hemolysis to exposure time and steady shearstress. In the absence of any unsteady data and cor-relation, CFD-based hemolysis predictions in thepast have used the steady correlation directly in theunsteady flow situations by replacing the steadyshear stress with a scalar quantity derived from theinstantaneous deviatoric stress tensor. Such amethod is here called “stress-based.”

This work proposes a tensor-based blood damagemodel, also referred to as “strain-based” method.The model is developed using an analogy betweenRBCs and droplets. In the model, an instantaneousmorphology of RBC is computed, which conse-quently provides an instantaneous deformation esti-mate of RBC. A steady shear stress corresponding tothe instantaneous deformation is used in the steadyshearing hemolysis correlation to predict hemolysis.The model incorporates experimentally observedphenomena—tank-treading and ellipsoidal defor-mation. The parameters of the model are set basedon experimentally measured quantities—relaxationtime of the RBC membrane, excess surface area ofthe RBC as compared to a droplet of the same vol-ume, and the strain rate threshold for catastrophichemolysis. The model also accounts for the estimatedareal strain limit of the RBC membrane.

The strain-based model is tested in both homoge-neous flows, where the velocity gradient is uniformthroughout the flow domain, and inhomogeneousflows. The stress-based model of Giersiepen et al.(26) and the strain-based model proposed in this arti-

cle predict equal hemolysis levels in steady simpleshear flow and agree equally well with the data ofWurzinger et al. (25). The stress-based model pre-dicts higher hemolysis when a sinusoidal shear flowis superimposed on a simple shear flow. This differ-ence suggests that the relative merit of stress-basedand strain-based models could be assessed bystudying short-time hemolysis in flows where a time-periodic straining is superimposed on a steady strain-ing; an apparatus for such studies could be built bymodifying the flow-through Couette device ofHeuser and Opitz (24) and driving the inner cylinderat a time-dependent rate.

The difference between the stress- and strain-based models grows as the frequency of sinusoidalshearing increases. This is consistent with the exper-imental observation of negligible hemolysis in ablood sample exposed for a few microseconds toshear stress as high as 1000 Pa (16). In the context ofblood pumps, this suggests that sustained moderatelevels of stress (typically encountered in conicalblood pumps) may be more damaging than a short-lived high shear stresses superimposed on a lowerbaseline shearing (typically seen in vaned centrifugalblood pumps). The RBC deformation is computed ina 2-dimensional blood pump, and hemolysis predic-tions by the two models are compared along path-lines. The stress-based model predicts a higherhemolysis than the strain-based one. Albeit in a sim-plistic way, the strain-based model accounts for thephysical phenomena of RBC membrane stretchingand is thus a suitable candidate for realistic hemolysispredictions.

Currently, a 3-dimensional implementation of thedeformation equation is under development; theresults from a complete 3-dimensional blood-pumpsimulation will be compared with experiments beingconducted at Baylor College of Medicine. Data fromthis study will further assess the model, and wouldbe immensely valuable for VAD designers. The pro-posed model can also be used to optimize VADdesign by minimizing the deviation of the morphol-ogy tensor from identity over a family of pathlines.

Acknowledgments: This work was supported bythe National Science Foundation under award CTS-ITR-0312764, and by Texas ATP grant 003604-0011-2001. Computational Resources were provided bythe National Partnership for Advanced Computa-tional Infrastructure (NPACI), by the Rice Compu-tational Engineering Cluster, funded by NSF throughMRI award EIA-0116289, and by the Rice TerascaleCluster funded by NSF under Grant EIA-0216467,Intel, and Hewlett-Packard.



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APPENDIX A

A deformed droplet returns to its natural configu-ration according to,

(A-1)

For small deformations S = I + eA. The eigenval-ues li of S are:

li = 1 + emi, (A-2)

ddt

f gS

S S I= - - ( )( )1 .



where mi are the eigenvalues of A. SubstitutingS = I + eA in Eq. A-1 yields:

(A-3)

where g(S) = 3III/II. Consequently, the second andthe third invariants of S become:

II = 3 + 2e(m1 + m2 + m3) + O(e2), (A-4)

III = 1 + e(m1 + m2 + m3) + O(e2), (A-5)

but the volume of droplet is preserved (dIII/dt = 0),thus tr(A) = m1 + m2 + m3 = 0. Neglecting the O(e2)

e eddt

f gA

I A S I= - + - ( )( )1 ,

terms leads to g(S) = 1. Substituting g(S) in Eq. A-3we get:

(A-6)

The solution to this equation is of the form:

A = A0exp(-f1t), (A-7)

where 1/f1 is the relaxation time. Because therelaxation time of RBC membrane is 200 ms we setf1 = 5 s-1.

ddt

fA

A= - 1 .

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A Tensor-based Measure for Estimating Blood · PDF fileA TENSOR-BASED MEASURE FOR ESTIMATING BLOOD DAMAGE 1003 Artif Organs, Vol. 28, No. 11, 2004 oped, and the conservation equations