Rheol ActaDOI 10.1007/s00397-015-0869-4

ORIGINAL CONTRIBUTION

Dynamic and rheological properties of soft biological cellsuspensions

Alireza Yazdani1 ·Xuejin Li1 ·George Em Karniadakis1

Received: 4 May 2015 / Revised: 6 August 2015 / Accepted: 6 August 2015© Springer-Verlag Berlin Heidelberg 2015

Abstract Quantifying dynamic and rheological propertiesof suspensions of soft biological particles such as vesi-cles, capsules, and red blood cells (RBCs) is fundamen-tally important in computational biology and biomedicalengineering. In this review, recent studies on dynamicand rheological behavior of soft biological cell suspen-sions by computer simulations are presented, consideringboth unbounded and confined shear flow. Furthermore, thehemodynamic and hemorheological characteristics of RBCsin diseases such as malaria and sickle cell anemia arehighlighted.

Keywords Blood · Dynamics · Rheology · Red bloodcell · Vesicle · Capsule

Introduction

Blood is a biological fluid that delivers necessary substancessuch as nutrients and oxygen throughout the human bodyand carries away metabolic waste products. Plasma, redblood cells (RBCs), white blood cells (WBCs), and plateletsare the four main components of whole human blood. Dueto its particulate nature, blood is considered as a complex

This paper belongs to the special issue on the “Rheology of bloodcells, capsules and vesicles.”

Alireza Yazdani and Xuejin Li contributed equally to this work.

� George Em KarniadakisGeorge Karniadakis@brown.edu

1 Division of Applied Mathematics, Brown University,Providence, Rhode Island 02912, USA

non-Newtonian fluid exhibiting intriguing dynamic and rhe-ological behavior depending on flow rates, volume fractionof suspending particles especially RBCs, and the vesselgeometries.

The deformability of an RBC is determined by the geom-etry, elasticity, and viscosity of its membrane. A healthyRBC has a biconcave shape when not subject to any exter-nal stress and is approximately 8.0 μm in diameter and2.0 μm in thickness (Fig. 1a) (Fung 1993). The mem-brane of a RBC consists of a lipid bilayer supported byan attached spectrin network that acts as a cytoskeleton(Fig. 1b) (Pietzsch 2004). The resistance of the lipid bilayerto bending elasticity is controlled by the bending rigidity,while the spectrin network’s resistance to shear strain ischaracterized by the in-plane shear modulus. Experimen-tal and numerical observations of RBC behavior in flowmimicking the microcirculation reveal dramatic deforma-tions and rich dynamics (Abkarian et al. 2007; Dupire et al.2012). The extreme deformability allows RBC to squeezewithout any damage when passing through narrow capil-laries in microcirculation. However, this feature of RBCscan be critically affected by genetic or acquired patholog-ical conditions such as malaria (Diez-Silva et al. 2010)and sickle cell anemia (SCA) (Barabino et al. 2010). Thus,as an important component of the blood, RBCs and theirmechanical properties play a crucial role in the dynamic andrheological behavior of blood in normal and disease states.

The other two and important particles in blood, i.e.,WBCs and platelets, have much smaller density comparedto RBCs. Alterations in blood flow and its rheology havea significant impact on the adhesion and activation mecha-nisms of platelets and WBCs (Lehoux et al. 2006). WBCsare relatively larger and less deformable particles than RBCsand play a significant role in our immune system. In orderto get to the site of inflammation, WBCs typically migrate

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Fig. 1 Simple representation of a healthy human RBC (a) and its com-plex membrane structure (b). The equilibrium shape of healthy humanRBC is a biconcave disk approximately 8.0 μm in diameter and 2.0 μmin width. Its cell membrane is made of a lipid bilayer reinforced on its

inner face by a flexible two-dimensional skeletal protein network. Thefluid-mosaic model of the RBC membrane is adapted with permissionfrom Pietzsch (2004)

to the vessel walls and perform a rolling motion. Plateletsin their unactivated resting form are nondeformable bicon-vex discoid structures of 2–3 μm in diameter. Platelets playan important role in hemostasis and thrombus formationat the site of injured endothelium. Since RBCs are moredeformable, they migrate toward the vessel center, creat-ing a depletion or cell-free layer (CFL) close to the wall.This phenomenon was first found by Fahræus and Lindqvist(1931) in capillaries. The less deformable cells such asWBCs and platelets, however, tend to migrate toward thewall. A process called margination, which is essential forthe cells to reach to the site of action.

Other types of soft particles that share several intri-cate mechanical properties of RBCs are vesicles andcapsules. Vesicles are viscous liquid drops enclosedby bilayer membranes prepared from compounds suchas phospholipids (Ryman and Tyrrell 1979; Lipowsky1991) and amphiphilic block copolymers (Discher andEisenberg 2002; Li et al. 2009, 2013a, 2014c), whereasa capsule membrane is made of extensible polymershell (Pozrikidis, 2003; Barthes-Biesel 2010, 2011). Arti-ficial vesicles and capsules can be used for targeted drugdelivery and cell and hemoglobin encapsulation. Both parti-cles reproduce several dynamical behavior known for RBCsunder flow. Thus, vesicles and capsules have gained pop-ularity as model systems to study the RBC dynamics andmembrane biophysics in general.

Computational modeling and simulations can tacklea broad range of dynamics and rheology problems rel-evant to blood. Several computational models, includ-ing continuum-based fluid-structure interaction models(Rehage et al. 2002; Peskin 2002; Doddi and Bagchi 2009;Zhao et al. 2010; Veerapaneni et al. 2011a; Kumar and

Graham 2012a), and particle-based spectrin-level and mul-tiscale models (Boal et al. 1992; Discher et al. 1998;Noguchi and Gompper 2005b; Li et al. 2005, 2007; Pivkinand Karniadakis, 2008; Fedosov et al., 2010; Pan et al.,2010; Peng et al., 2013), have been developed. Continuum-based models treat the cell membrane and embeddingfluids as homogeneous materials, using immersed bound-ary method (IBM) (Peskin 2002; Doddi and Bagchi 2009;Yazdani and Bagchi 2011; Fai et al. 2013), boundary inte-gral method (BIM) (Ramanujan and Pozrikidis 1998; Lacet al. 2004; Zhao et al. 2010; Veerapaneni et al. 2011a),and fictitious domain method (FDM) (Shi et al. 2014; Haoet al. 2015). Particle-based models treat both the fluid andthe membrane as particulate materials, using multiparti-cle collision dynamics (MPCD) (Noguchi and Gompper2005b; McWhirter et al. 2009), coarse-grained molecu-lar dynamics (CGMD) (Kotsalis et al. 2010; Hanasakiet al. 2010; Li and Lykotrafitis 2012b; 2014a), dissipa-tive particle dynamics (DPD) (Pivkin and Karniadakis 2008;Fedosov et al. 2010; Pan et al. 2010; Peng et al. 2013),Smoothed Dissipative Particle Dynamics (SDPD) (Fedosovet al. 2014a), and smoothed particle hydrodynamics (SPH)(Hosseini and Feng 2012; Wu and Feng 2013). Currently,there is a wide variety of modeling approaches since thereis no universal solution for all blood-flow-related problems.Eventually, the cross-fertilization between continuum- andparticle-based models leads to practical yet physicallyand biologically accurate, and computationally efficientmethods that can tackle a broad range of dynamics andrheological problems relevant to biological applications.This is a very active research area, see for recent review(Barthes-Biesel 2010, 2011; Li et al. 2013b; Fedosov et al.2014b; Freund 2014; Winkler et al. 2014). In this article,

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we overview the applications of both continuum and dis-crete computational approaches on the modeling of dynamicand rheological properties of healthy and diseased RBCs aswell as other soft particles with focus on the most recentcontributions.

Dynamics of deformable particles in dilutesuspensions

Computational modeling and simulations help in predictinghow deformable particles behave in shear flow and pro-vide insights into how viscous flow transforms shapes ofthese deformable particles and how they distort the sur-rounding flow. Here, we assume a dilute suspension ofisolated particles where particle interactions are negligi-ble. The hydrodynamic stress, �b, for the bulk flow of aNewtonian fluid is given by

σ = −pI + η(∇v + (∇v)T), (1)

where p is the pressure, η is the fluid viscosity, I is the unittensor, and v is the fluid velocity. The surrounding fluidexerts tractions, �f, on the membrane that are balanced bythe tensions developed in the particle’s membrane

�f = (σ ex − σ in) · n = f κ + f el , (2)

where n is the unit normal vector, f κ and f el are the ten-sions due to the membrane bending and elastic resistances,respectively. It should be mentioned that RBC and vesiclepossess an inextensible bilayer membrane, whereas cap-sules normally do not conserve the area. The membranetensions can be evaluated using appropriate constitutivelaws, the most famous of which are the Helfrich’s bend-ing energy for lipid bilayer (Ou-Yang and Helfrich 1989),Skalak law for elastic and area-conserving cytoskeleton(Skalak et al. 1973), and new-Hookean hyperelastic law forcapsules (Dimitrakopoulos 2012). Also note that the RBCmembrane more precisely behaves as a viscoelastic materialfollowing a power law, and a more complicated viscoelastictension f vis can replace f el in Eq. 2 (Yazdani and Bagchi2013). Here, we summarize the generic behavior of vesicles,capsules, and RBCs in shear flow. Specifically, we focuson two situations relevant to blood circulation, i.e., uniformshear flow and confined (tube or channel) flow.

Vesicles, capsules, and RBCs in uniform shear flow

Vesicles, capsules, and RBCs in uniform unbounded shearflow have been observed to exhibit two primary typesof dynamics (see Fig. 2): (1) Tank treading (TT) motion(Fischer et al. 1978; Kantsler and Steinberg 2005) wherethe soft particle deforms into an elongated ellipsoidal shape

and its inclination angle θ remains constant with time; (2)Tumbling (TB) motion (Kantsler and Steinberg 2006) wherethe soft particle undergoes a periodic flipping motion and θ

is periodic in time. More complicated intermediate type ofdynamics such as vacillating-breathing (VB) (Misbah 2006)also called alternatively trembling (TR) (Deschamps et al.2009a, b) or swinging (SW) motion (Noguchi and Gompper2007) have also been observed both numerically and exper-imentally for these soft particles.

For a vesicle with constant volume V and constant sur-face area A, its dynamics in shear flow is fully characterizedby three parameters: (i) reduced volume, v = 3V/(4πR3

0),where R0 = √

A/4π is the radius of a sphere with thesame area, (ii) viscosity ratio, λ = ηin/ηout , between inter-nal and external fluid viscosity, and (iii) Capillary number,Caκ = ηγR3

0/κ , defined based on the bending stiffness κ

and shear rate γ . The classic Keller-Skalak theory (Kellerand Skalak 1982) predicts only TT and TB motions exist ifthe vesicle shape is fixed (Kraus et al. 1996; Noguchi andGompper 2004, 2005a). Allowing the vesicle shape to freelyevolve has led to VB motion (Kantsler and Steinberg 2006;Biben et al. 2011; Zhao and Shaqfeh 2011a; Yazdani andBagchi 2012).

Due to the shear elasticity of their membranes, cap-sules exhibit very similar dynamics as RBCs in shear flowsharing the same TT-SW-TB dynamics. There are exten-sive numerical and analytical studies on the dynamics ofspherical and nonspherical capsules in shear flow (e.g.,Bagchi and Kalluri, 2009; Barthes-Biesel and Rallison,1981; Pozrikidis, 2003; Sui et al., 2008). A VB-like dynam-ics similar to vesicles was found for nonspherical cap-sules in the numerical study (Bagchi and Kalluri 2009),where a large-amplitude oscillation in capsule shape anda strong coupling between the shape deformation and ori-entation dynamics were observed. The mechanism of theSW motion between vesicles and capsules/RBCs is different(Veerapaneni et al. 2011b): For capsules/RBCs, SW motionarises from a periodic variation in the elastic membraneenergy during TT motion. For vesicles, SW motion is dueto a periodic vesicle deformation as the TB inclusion pusheson the membrane.

Shear flow dynamics of RBCs is richer than capsulesand vesicles due to their unique mechanical properties andbiconcave shapes. Several new dynamics such as rolling,kayaking, and hovering were observed both in the experi-ments and simulations (Dupire et al. 2012; Cordasco andBagchi 2014). Similar to vesicles and close to the TB-to-TTtransition border, breathing-like motion for RBCs in whichthe membrane major axis oscillates about the flow directionas the shape exhibits large-amplitude deformations was alsoobserved in numerical study (Fig. 2) (Yazdani and Bagchi2011), whereas experimental observations suggested thatthe breathing dynamics could not exist for RBCs. More

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Fig. 2 Vesicles and RBCs in unbounded shear flow. a Trembling (alsoknown as VB or SW), and tumbling vesicle at slightly higher λ. bInclination angle vs. time and vs. deformation show the transition fromtrembling to tumbling as λ increases from 6 to 7. Here, Caκ = 6 andreduced volume v = 0.75. Figure is reproduced with permission from

(Spann et al. 2014). c Breathing dynamics of an RBC, and d tumblingdynamics. Vectors show the in-plane membrane TT velocities, wherethe contours present their magnitudes. t∗ is the dimensionless timedefined as γ t . Reprinted with permission from Bagchi and Yazdani(2012)

recent numerical studies have shown that the RBC stress-free shape can have considerable effects on its dynam-ics, stability of TT modes, and transition from TT to TB(Cordasco et al. 2014; Peng et al. 2014, 2015). These stud-ies suggest that RBC’s stress-free shape must be closerto a sphere rather than the biconcave shape confirmingthe experimental results (Dupire et al. 2012). Furthermore,breathing dynamics were no longer observed if the spherewas chosen as the stress-free shape. In addition, near

the TB-to-TT transition, there exists a narrow intermittentregion where theory predicts an instability such that TBmotion of RBCs can be followed by TT motion and viceversa. This intermittent dynamics were first reported in thenumerical simulations of Fedosov et al. (2010). A morerecent front-tracking study gives a detailed analysis of theintermittency and synchronous dynamics of RBCs in shearflow as well as their off-shear plane dynamics (Cordascoand Bagchi 2014).

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Although there is a number of experimental studiesaiming to address the viscoelastic properties of RBCmembrane and to measure the membrane viscosity (e.g.,Hochmuth et al. 1979; Puig-de-Morales-Marinkovic et al.2007), there are very few numerical studies incorporat-ing membrane viscosity in their models. Fedosov et al.utilized a worm-like-chain spectrin model with additionaldissipative forces to account for the lipid bilayer vis-cous effects (Fedosov et al. 2010). Comparing the simu-lated TT frequencies of RBCs with experiments showedthat a purely elastic RBC results in an over-predictionof TT frequencies. Similar conclusion was also made ina continuum study of capsule dynamics under the influ-ence of membrane viscosity (Yazdani and Bagchi 2013).They found that increasing the membrane viscosity isequivalent to increasing the viscosity of the internal fluidin regards to TT frequencies and TT-to-TB transitionaldynamics.

Vesicles, capsules, and RBCs in bounded flow

Deformable particles suspended in fluid flowing throughconduits are often encountered in many biological pro-cesses, and in biomedical devices. Examples are the motionof blood cells through blood vessels, flow chambers, andcell separation devices. Simulations of suspensions of bloodcells in a tube or a channel mimic many properties of bloodflow in vessel in vivo. Here, we summarize some of thesimulation studies on the dynamic behavior of these softparticles in tube and channel flow.

A widely studied flow is the confined Poiseuille flow,in which the dynamic behavior of vesicles, capsules, andRBCs is often considered in both continuum-based simu-lations (Secomb et al. 2007; Coupier et al. 2008; Dankeret al. 2009; Coupier et al. 2012) and particle-based simula-tions (Noguchi and Gompper 2005b; McWhirter et al. 2009,2011; Deng et al. 2012). In a Poiseuille flow at low Reynoldsnumbers, RBCs and vesicles in narrow channels or narrowvessels deform themselves into parachute-like shapes (seeFig. 3). For example, Pan and Wang studied the shape mem-ory ability of the RBC membrane by a spring model (Panand Wang 2009). They found that the parachute-likeshaped RBCs return back to initial biconcave shapes after

Fig. 3 Sequence of RBC shapes from biconcave to parachute inPoiseuille flow in microtube (unpublished work)

stopping the flow. A similar phenomenon was simulatedby Pivkin and Karniadakis using a multiscale RBC (MS-RBC) model based on DPD (Pivkin and Karniadakis 2008).RBC deformed into the parachute-like shape under theflow condition while it recovered its equilibrium biconcaveshape after the fluid returned to rest. McWhirter et al. usedMPCD to study the shape changes and clustering of RBCsunder flow in cylindrical microcapillary (McWhirter et al.2011). They found that the RBCs transition from disco-cyte shape at low velocity to parachute-like shape at highvelocity.

Vesicles have gained popularity as well-defined modelsto study the RBC dynamics. For example, Noguchi andGompper have employed MPCD to study the shape tran-sitions of fluid vesicles in capillary flows (Noguchi andGompper 2005b). Different from the shape transition ofRBCs, they found that the fluid vesicle transits its shapefrom discocyte to prolate ellipsoid with the increase of flowvelocity. Danker et al. employed BIM to investigate theeffect of viscosity ratio on vesicle migration in a Poiseuilleflow (Danker et al. 2009). They predicted the coexistence oftwo types of shapes: bullet-like and parachute-like shapes.Vesicles and RBCs flowing in microvessels often exhibitparachute-like shapes usually attributed to an increaseof hydrodynamic constraint; however, in a recent study,Coupier et al. showed that the presence of a fluid mem-brane leads to reverse phenomenon (Coupier et al. 2012).They found that the parachute-like to bullet-like shapescrossover depends on the capillary number but not on theconfinement.

Variability in deformation of different soft particles is acrucial factor in the variation of their lateral (lift) force andvelocities affecting their near-wall dynamics. Lateral migra-tion of capsules is studied numerically in planar Poiseuilleflow (Doddi and Bagchi 2008), square duct with a cor-ner (Zhu and Brandt 2014), and in bounded shear flow(Singh et al. 2014). For vesicles, there are experimental andnumerical studies addressing different and interesting shapetransformations as the vesicles migrate towards the center ofmicrochannel (e.g. Coupier et al. 2008; Danker et al. 2009;Zhao et al. 2011).

RBCs are highly deformable allowing them to travelthrough capillaries with diameter smaller than their size.When RBCs flow through capillaries, they undergo dras-tic deformation by shear stress. Accordingly, microflu-idic channels are used to mimic human capillaries andstudy RBC deformability. It has been found that RBCsundergo a continuous and severe transition from their nor-mal biconcave shapes to ellipsoidal shapes by the elonga-tion of their sizes in the flow direction (longitudinal axis)and shortening of their sizes in the cross-flow direction(transverse axis). Figure 4a shows a qualitative compar-ison of the simulation results from the two-component

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Fig. 4 a Shape characteristics of a RBC traversing across microflu-idic channels from experimental (left) and simulation (right) data. bBilayer-cytoskeletal detachment for an RBC traversing the microflu-idic channel at different bilayer-cytoskeletal elastic interactions. The

lipid bilayer and the cytoskeleton are shown as dark and light grey (redand blue in the online version) triangular networks, respectively, andonly half of the triangular network of the lipid bilayer is shown forclarity. Reproduced from Li et al. (2014b), by permission

RBC model based on the DPD approach (Li et al. 2014b)with experimental measurements (Quinn et al. 2011). Thedynamic observations of the deformation of RBC traver-sal across the microfluidic channel were in accordancewith the experimental phenomena. The simulations alsodemonstrated that the bilayer-cytoskeletal elastic interac-tion coefficient, kbs , plays a key role in the RBC traversingnarrow microfluidic channel as shown in Fig. 4b. Whenkbs is large, there is a strong coupling between the lipidbilayer and the cytoskeleton, i.e., the RBC behaves as ifit was an effective membrane. If kbs is small, the bilayer-cytoskeletal coupling is weak, and the detachment of thelipid bilayer from the cytoskeleton is much more likely tooccur.

Dynamics of platelets and WBCs

As mentioned in the previous section, confined blood flowin tubes and channels exhibit an intricate and rich dynamicsof its constituents, namely RBCs, platelets, and WBCs. Lat-eral migration of RBCs leads to the CFL formation adjacent

Fig. 5 Margination of platelets in whole blood simulation at 35 %Hct in microchannel with 75 % degree of stenosis. These simulationsare based on DPD to model RBCs, platelets, and plasma (unpublishedwork)

to the vessel wall. Reduced local viscosity in this layer helpsreducing the resistance to flow in small vessels, and is crit-ical for blood flow in microcirculation. Furthermore, RBCstend to push out various solutes from the central region intothe CFL (Fig. 5). This process leads to margination of lessdeformable particles such as platelets and WBCs to the wall,which is essential to their functions.

A margination theory for binary suspensions of sphericalcapsules in a dilute or semi-dilute suspension was proposedin (Kumar and Graham 2012b). The key idea is the bal-ance between wall-induced hydrodynamic migration andshear-induced pair collisions (also known as shear-induceddiffusion) of particles. In a similar approach, Narishmanet al. developed a low-order, kinetic theory to describethe concentration distribution of deformable particles inwall-bounded Couette flow (Narsimhan et al. 2013).

WBC margination has been studied both experimentallyand numerically in detail (Fedosov et al. 2012; Freund 2007;Sun et al. 2003). It is found that besides WBC’s sphericaland less deformable shape, several flow parameters con-tribute significantly to its margination as well. These factorsinclude flow hematocrit (Hct ) level, flow rate, RBC aggre-gation, and the vessel geometry. Fedosov et al. in a 2D studyfound that WBC margination becomes weaker as the cellsbecome more deformable, whereas RBC aggregation andhigher Hct proved to enhance WBC margination (Fedosovet al. 2012).

Similar to WBCs, it is extremely crucial for plateletsfunction in hemostasis and thrombosis. An increased con-centration of platelets near the walls has been confirmedin in vivo (Woldhuis et al. 1992) and in vitro (Tilles andEckstein 1987) studies. To describe experimental observa-tions of the margination of platelets, a phenomenologicaldrift-diffusion model for platelet motion in blood flow

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was developed (Eckstein and Belgacem 1991). The drift-diffusion equation results in a good qualitative descriptionof platelet margination. Here, the drift can be considered tobe due to wall-platelet hydrodynamic interactions (althoughthere is no rigorous physical foundations for that), whereasthe shear-induced diffusivity of platelets is due to theircollisions with RBCs. There is a growing number of numer-ical studies on platelet margination and its mechanisms(AlMomani et al. 2008; Tokarev et al. 2011; Zhao andShaqfeh 2011b; Reasor Jr et al. 2013; Vahidkhah et al.2014; Vahidkhah and Bagchi 2015). Crowl and Fogelsonemployed a Fokker-Planck approach coupled with 2Dnumerical simulations to model platelet transport in bloodflow (Crowl and Fogelson 2011). The anisotropy, and three-dimensionality in platelet margination as well as theirinteractions with RBCs necessitate 3D whole blood simu-lations (Zhao and Shaqfeh 2011b; Reasor Jr et al. 2013;Vahidkhah et al. 2014; Vahidkhah and Bagchi 2015). Theeffect of particle shapes on its margination has been shownto be substantial in the numerical study of Reasor Jr et al.(2013); they found that spherical particles marginate moreeffectively than those having ellipsoidal shapes. A recentstudy of microparticles in blood have shown that the shapeof the particles affects their dynamics differently in themargination than the adhesion. While oblate ellipsoidal par-ticles of moderate aspect ratio show the highest near-wallaccumulation, very long elongated particles are more likelyto form the initial wall contact. Furthermore, the adhesionefficiency of oblate discocyte shapes similar to platelets wasproved to be the highest (Vahidkhah and Bagchi 2015).

Dynamics of diseased RBCs

Blood disorders can affect any or all of the three types ofblood cells, i.e., RBCs, WBCs, and platelets. Blood disor-ders can also affect the plasma, i.e., the liquid portion ofblood. Numerical simulations have been used for qualitativeand quantitative interpretation and predictions of mechani-cal properties and dynamic behavior of RBCs in malaria andother hematological diseases (Fedosov et al. 2011a; Quinnet al. 2011; Lei and Karniadakis 2013). Below, we highlightsome of the research being performed to study the changesin mechanical and dynamic properties of RBCs in malariaand SCA.

Malaria-infected RBCs

Malaria is one of the most severe parasitic diseases. Whenthe malaria parasite is inside an RBC, the cell is stiffer thannormal. Thus, blood flow may be significantly affected byaltered RBC structural and mechanical properties occurringin malaria. The computational RBC models are ideal tools

to study the RBC dynamics in malaria. For example, Imaiet al. employed SPH to study the microvascular hemody-namics arising from the malaria infection (Imai et al. 2010,2011). They examined flow in microtubes and found thatthe hydrodynamic interaction between healthy RBCs andmalaria-infected RBCs causes train formation. Ye et al. usedDPD to simulate the flow dynamics of healthy RBCs andmalaria-infected RBCs (Ye et al. 2014). They observed adownward inclusion phenomenon where malaria-infectedRBCs cannot undergo the dynamic motion at the levelhigher than that exhibited by the healthy RBCs.

Different cell models have also been employed to studythe adhesion dynamics of malaria-infected RBCs underflow. For example, Fedosov and coworkers performed ambi-tious DPD simulations of the adhesion and flipping dynam-ics of malaria-infected RBCs (Fedosov et al. 2011a). Theirsimulation results revealed several types of cell dynam-ics such as firm adhesion and intermittent slipping. Theyalso modeled the effect of solid parasite inside the malaria-infected RBCs and found that the presence of a rigid bodyinside the RBC induces higher variances in cell motionand pronounces flipping of the malaria-infected RBC as ittethers to the surface.

Quantifying the dynamic cell deformability for variousstages of malaria-infected RBCs and other types of bloodcells is significant. Bow et al. developed a microfluidicdevice with periodic obstacles to blood flow to character-ize the biomechanical properties of malaria-infected RBCs(Fig. 6b) (Bow et al. 2011). They employed the MS-RBC model to translate the experimental measurementsinto quantitative data describing the mechanical propertiesof individual RBCs. Their simulations were able to cap-ture the effect of the changes of RBC properties arisingfrom parasitization on the movement of healthy RBCs andmalaria-infected RBCs quite accurately (Fig. 6d). Aingaranet al. used finite elements to investigate the deformabilityproperties of malaria-infected RBCs during sexual develop-ment (Aingaran et al. 2012). Their simulations suggestedthat along with deformability variations, the morpholog-ical changes of the parasite may play an important rolein tissue distribution in vivo. In a recent study, Wu andFeng used SPH to simulate the traverse of malaria-infectedRBCs through a converging microfluidic channel (Wu andFeng 2013). They examined the influences of RBC struc-tural and mechanical changes due to malaria infection onthe RBC dynamics in microfluidic channels and found thatRBC gradually lost its deformability with progression of themalaria infection.

Most blood test analysis in medical laboratories is oftenperformed on cell-free samples, therefore, blood-plasmaseparation needs to be achieved. The blood-plasma sepa-ration depends on cell deformability. The malaria-infected

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Fig. 6 a Illustration of the flowcytometer device. bExperimental images ofring-stage infected (red arrows)and uninfected (blue arrows)RBCs in the channels. c Thecomputational RBC modelconsists of 5000 particlesconnected with links. Theparasite is modeled as a rigidsphere inside the cell. d DPDsimulation images ofmalaria-infected RBCs travelingin channels of converging (left)and diverging (right) poregeometries. Reproduced fromBow et al. (2011), by permission

RBCs lose their deformability and in turn affects bloodflow. The motion of RBC flow in bifurcating microfluidicchannel was simulated using a low-dimensional RBC (LD-RBC) model (Li et al. 2012d). The separation efficiency ofmalaria-infected RBCs was found to be lower than those ofhealthy RBCs.

Dynamic behavior of RBCs in SCA

SCA is an inherited blood disorder exhibiting heteroge-neous cell morphology and abnormal rheology, especiallyunder hypoxic conditions (Chien et al. 1970; Kaul and Xue1991). SCA has been characterized as the first “molecu-lar disease” being linked to an alteration in the molecularstructure of hemoglobin. The symptoms of the disease havebeen traced to the polymerization of sickle hemoglobinupon hypoxic condition, forming long fibers that distort theshapes of RBCs and dramatically alter their mechanical andrheological properties. The affected RBCs become morerigid and sticky compared with normal RBCs, resulting ina difficulty when they flow through small blood vesselsand capillaries, hence depriving tissues and organ of oxy-gen.

Computational modeling and simulations of sickle RBCsand sickle blood are helpful in understanding how sick-ling occurs and how it affects RBC dynamics and bloodvaso-occlusion (Chiang and Frenette 2005). To this end,Dou and Ferrone developed a kinetic model for sicklehemoglobin polymerization to understand domain forma-

tion. Their qualitative results showed that the transformationof a single fiber into wheat-sheaf bundles and then tospherulitic domains (Dou and Ferrone 1993). Dynamic sim-ulations of self-assembly of sickle hemoglobin confirmedthat chain chirality is the main driver for the formationof sickle hemoglobin fibers (Li et al. 2012c). DifferentCGMD models have also been introduced to simulate thedynamic behavior such as zipping and unzipping dynam-ics of sickle hemoglobin fibers (Li and Lykotrafitis 2011;Li et al. 2012a). The simulation results showed that fiberfrustration and compression result in partial unzippering ofbundles of sickle hemoglobin fibers.

Decreased RBC deformability and increased cell rigidityin SCA have significant implications for microcirculatoryflow. Some simulation attempts have been made to under-stand the dynamic behavior of blood flow in SCA. Forexample, Dupin et al. studied sickle blood flow through anaperture of diameter less than the size of a single cell (Dupinet al. 2008). Lei and Karniadakis quantified the hemody-namics of blood flow with SCA under various physiologicalconditions (Lei and Karniadakis 2012). They found that theincrease of flow resistance by granular RBCs was greaterthan the resistance of blood flow with sickle-shaped RBCs.They also applied adhesion dynamics to the effect of SCAand found that blood flow exhibits a transition from steadyflow to partial/full occluded state, see Fig. 7. Their simu-lation results showed that the adhesion interaction betweencells and vessel walls plays a profound effect on the hemo-dynamics of sickle RBCs and addressed the conditions of

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sickle cell shape, adhesiveness, and elasticity that can causethe occlusion of a small blood vessel (Lei and Karniadakis2013).

Rheology of vesicles, capsules, and RBCs

While the dynamics of rigid particles in suspension andtheir rheological behavior have been investigated in depthand well understood, simulating the rheology of deformableparticles especially in non-dilute suspensions has becomepossible only recently. As a consequence of the deformablenature of soft particles, the suspension attains viscoelasticproperties and the effective viscosity (ηeff ) shows a pro-nounced dependence on shear rate. In the past decades,computational models have proven to be an important toolin predicting the macroscopic flow properties of a suspen-sion (e.g., shear viscosity, yield stress) from the mesoscopicproperties of these soft particles (e.g., deformability, mem-brane properties, and particle size). Recent progress on therheology of soft-particle suspensions by computer simula-tions is reviewed herein.

Rheology of vesicle and capsule suspensions

To quantify the rheology of a suspension in uniform shearflow with the shear rate γ , we first evaluate the bulkstress of the suspension. Following Batchelor (Batchelor1970), the overall force-free suspension stress is split into acontribution of the pressure in the fluid volume, the fluidstress in the absence of particles, and the stress due to theparticles

σ b = −P I + 2ηE + σ p , (3)

where P , E, and σp are the volume-averaged scalar pres-sure, strain tensor, and the average particle stress, respec-tively. Assuming negligible inertia, the particle contributionto the suspension stress can be calculated using the stressletformulation (Batchelor 1970) and Eq. 2

σp = 1

V

∑

N

∫

A

[1

2(�f r+r�f)+ηout (λ−1)(un+nu)

]dA,

(4)

where r and u are the position and velocity vectors on theparticle’s surface, V is the volume of the fluid, and thesummation is carried over all N particles. The effective sus-pension viscosity can then be computed using the shearcomponent of the stress tensor as ηeff = σxy/γ , whichalso gives the intrinsic viscosity due to particle contribu-tions ηI = (ηeff −ηout )/ηoutφ with φ being the suspensionvolume fraction.

The perturbation theory analysis of nearly-spherical vesi-cles in a dilute solution (φ <≈ 10 %) predicts ashear-rate-independent suspension rheology at leading order. Moreimportantly, the shear viscosity decreases with λ and attainsa minimum at λc corresponding to the TB-to-TT transi-tion (Misbah 2006; Danker and Misbah 2007), followedby an increase. This nonmonotonic behavior is in agree-ment with experiments (Vitkova et al. 2008) in semi-diluteregime as well. Interestingly, in a subsequent study usingboundary integral simulation of non-dilute vesicle suspen-sions, similar nonmonotic behavior for the effective vis-cosity was observed for volume fractions as high as 20 %(Zhao and Shaqfeh 2013). In the dilute limit, suspensions ofdeformable particles are shear thinning due to the increaseddeformation of the suspended phase. Similar to vesicles,numerical simulations of dilute elastic spherical capsulesin a tank-treading configuration demonstrate a minimum inthe suspension viscosity as λ increases (Bagchi and Kalluri2010).

Dense suspension of capsules has first been studied in2D (Breyiannis and Pozrikidis 2000). More recently, rhe-ology and microstructure of suspensions of initially spher-ical elastic capsules have been studied using a coupledlattice-Boltzmann/finite element (LB/FE) method for vol-ume fractions as high as 40 % (Clausen et al. 2011). Adetailed numerical study of suspension of aggregation-freecapsules with RBC-like membrane properties was carriedout in Gross et al. (2014), where φ up to 90 % and reducedshear rates of Cas = ηout γ R/Gs < 0.1 were considered,with Gs the capsule shear modulus. They found that for theeffective viscosity at small capillary numbers, a Newtonianplateau is present at low volume fractions, which goesover into a yield stress regime at high volume fractions;for large capillary numbers, the viscosity is strongly shear-thinning, following a power-law with ηeff ∼ Ca0.5

s . Theevidence of yield stress at low shear rates was found evenfor non-aggregating capsules. Achieving higher shear ratesstill remains a challenge.

Blood rheology

Blood cells are subjected to intense mechanical stimula-tion from both blood flow and vessel walls, and theirrheological properties are important to their effective-ness in performing their biological functions in micro-circulation. Computational modeling and simulations ofblood flow have improved considerably in recent years(Baskurt and Meiselman 2003; Zupancic Valant et al. 2011;Apostolidis and Beris 2014; Fedosov et al. 2014b). Forexample, dynamic simulations can model how blood flowbehaves in microfluidic channels and predict blood viscos-ity in silico (Fedosov et al. 2011c). Different cell models

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Fig. 7 a Sickle blood flow withadhesive dynamics. The greendots represent the ligands coatedon the vessel wall. The bluecells represent the “active”group of sickle cell exhibitingadhesive interaction with thecoated ligands. b Mean velocityof the sickle blood at differentstages of the adhesive dynamics.Reproduced from Lei andKarniadakis (2012), bypermission

have also been employed for various qualitative and quanti-tative interpretations as well as predictions of biomechanicalproperties of RBCs with hematological diseases. Examplesinclude dynamic cell deformability for various stages ofmalaria-infected RBCs (Fedosov et al. 2011a) and vaso-occlusion phenomena in SCA (Lei and Karniadakis 2013).Here, we draw attention to the computational hemorheologyof blood flow in health and disease.

Rheological behavior of normal blood

At the microcirculatory level, the particulate nature of theblood becomes significant. The rheological properties ofRBCs are key factors of the blood flow characteristicsin microvessels because of their large hematocrit (Hct ≈45 %) in the whole blood. Blood viscosity has been exten-sively investigated, and it is generally believed that fivefactors, namely hematocrit, cell deformability, cell aggrega-tion, plasma viscosity, and temperature, primarily determinethe rheological behavior of blood (Baskurt and Meiselman2003).

Dynamic simulation can predict the rheological and flowproperties of blood flow. Two different RBC models, theMS-RBC model (Pivkin and Karniadakis 2008; Fedosovet al. 2010) and the LD-RBC model (Pan et al. 2010), havebeen developed and employed to predict rheological proper-ties of blood flow. In particular, the LD-RBC model, whichmodels one RBC as a closed torus-like ring of ten colloidalparticles, allows the simulation of blood flow over a widerange of hematocrit levels at computational costs that areconsiderably below those for the MS-RBC model. Usingthese two cell models, Fedosov et al. accurately predictedthe dependence of blood viscosity on shear rate and hemat-ocrit (Fig. 8) (Fedosov et al. 2011c). The model predictionof blood viscosity was in agreement with experimental mea-surement (Merrill et al. 1963; Chien et al. 1966; Skalak et al.1981). Their cell suspension model also captured the effect

of aggregation on the viscosity at low shear rates and sug-gests that cells and molecules other than RBCs have littleeffect on the viscosity.

Under physiological conditions, blood is typically vis-coplastic, i.e., it exhibits a yield stress that acts as a mini-mum threshold for flow to begin. The yield stress is usuallycalculated by extrapolation of available viscometric data tozero shear rate on the basis of Casson’s equation (Casson1992):

τ1/2xy = τ

1/2y + η1/2γ 1/2, (5)

where τy is a yield stress and η is the suspension viscosity.In a recent study, Apostolidis and Beris simulated the rhe-ology of blood in steady-state shear flows (Apostolidis andBeris 2014). They showed that the Casson model emergesnaturally as the best non-Newtonian model for fitting avail-able shear stress vs shear-rate literature data. In an extensionstudy, they simulated blood flow rheology in transient shearflows (Apostolidis et al. 2015). They found that the mod-ified version of the “Delaware model” (Mujumdar et al.2002) is capable of predicting the time-dependent shearflow rheology of blood at low and moderate values of shearrate. Fedosov et al. computed shear stress for RBC suspen-sion and extrapolated it to zero shear rate using a polynomialfit in Casson coordinates (Fedosov et al. 2011c). Their sim-ulation results support the hypothesis that whole blood hasa non-zero yield stress due to cell aggregation.

Blood rheology in malaria

Malaria-infected RBCs show progressing alteration of theirmechanical and adhesive properties as the parasite devel-ops. These changes greatly affect rheological properties ofmalaria-infected RBCs and lead to obstructions of smallcapillaries. Numerical simulations have been performed to

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Fig. 8 a Non-Newtonianrelative viscosity of blood as afunction of shear rate atHct = 45 % from the MS-RBCmodel (black lines) andLD-RBC model (blue lines). bRelative viscosity of blood as afunction of hematocrit at twodifferent shear rates.Reproduced from Fedosov et al.(2011c), by permission

simulate infected RBCs in malaria (Fedosov et al. 2011b;Peng et al. 2013; Li et al. 2014b). The rheological proper-ties of RBCs in malaria have been studied and comparedwith those obtained by optical magnetic twisting cytometry(Puig-de-Morales-Marinkovic et al. 2007) and by moni-toring membrane fluctuations at room, physiological, andfebrile temperatures (Park et al. 2008, 2010). The simula-tion results matched well with the membrane fluctuationsof malaria-infected RBCs, and suggest that the interactionsbetween the lipid bilayer and the cytoskeleton of RBCs playkey roles in determination of cell membrane mechanical andrheological properties.

Progression through the stages lead to considerableincrease in flow resistance of blood in malaria as found inrecent experiments (Dondorp et al. 2004). The DPD compu-tational RBC models have also been employed to predict thebulk viscosity and adhesive dynamics of malaria-infectedRBCs (Fedosov et al. 2011a). The simulations quantitativelypredicted the increase of blood flow resistance in malaria(see Fig. 9) and demonstrated that adherence of malaria-infected RBCs to the vascular endothelium contribute to theblood flow resistance as well.

Blood rheology in SCA

In SCA, mechanically fragile and poorly deformable RBCscontribute to impaired blood flow and other pathophysiolog-ical aspects of the disease. When the flow of blood is rela-tively slow, cellular reactions that lead to adhesion of sickleRBCs to vascular endothelium, resulting in vaso-occlusionand consequent clinical manifestations such as organ dam-age, pain, and even death. Computational methods havebeen used to evaluate rheological properties, to help multi-factorial nature and pathogenesis of vaso-occlusion.

The sickle RBCs has decreased cell deformability andincreased cell rigidity, causing abnormal rheology in sickle

blood and eventually various complications of SCA. It isknown that the deformability of individual sickle RBCs isdecreased under fully oxygenation state and deteriorates fur-ther under deoxygenation. Previous studies have shown thatthe viscosity of sickle RBCs or sickle blood are above nor-mal under oxygenation and become further elevated withreductions in oxygen tension (Kaul and Xue 1991). Toverify the significant role of cell deformability in deter-mining the rheological property of sickle RBCs, numer-ical simulations have been performed at different shearmodulus values and at different oxygen tensions (Lei andKarniadakis 2012). The blood flow showed that an increase

Fig. 9 Flow resistance of blood in malaria for various parasitemialevels and tube diameters. Symbol “X” corresponds to the schizontstage with a near-spherical shape. Inset shows a snapshot of healthy(red) and malaria-infected RBCs (blue) in Poiseuille flow in a tubeof diameter D = 20 μm. Reproduced from Fedosov et al. (2011a), bypermission

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Fig. 10 Shear viscosity ofsickle RBC suspensions withdifferent cell morphology atdifferent shear rate (unpublishedwork)

in shear modulus leads to greater viscosity of sickle RBCsand the flow resistance further increase at deoxygenatedstate.

Sickle RBC suspensions exhibit different levels of vis-cosity for different cell morphologies, which are dependenton the rate of deoxygenation (Kaul and Xue 1991). Grad-ual deoxygenation is known to result in predominantlyelongated- and classic sickle-shaped RBCs, which areintrinsically more rigid and viscous. The increase in viscos-ity is much greater when deoxygenation is rapid (resultingin less distorted but highly viscous granular-shaped RBCs)than when it is gradual. The MS-RBC model has also beenemployed to simulate the rheological properties of sickleRBC suspensions. The simulation results confirmed that thesickle RBC suspensions exhibit different viscosity valuesfor different cell shapes: the granular RBC suspension isthe most viscous, while the shear viscosity of sickle RBCsuspensions containing elongated RBCs shows a dramaticdecrease (Fig. 10).

Computational models have also been employed to quan-tify the adhesive and dynamic properties of sickle RBCsuspensions in tube flows (Lei and Karniadakis 2012,2013). Available evidence indicates that adhesive interac-tions between the sickle RBCs and vascular endotheliumplay a key role in triggering vaso-occlusion in straightvessels.

Summary and outlook

Computational simulations are playing an increasinglyimportant role in enhancing our understanding of thedynamics and rheology of soft biological cell suspensionsand various local processes in blood (e.g., thrombosis,malaria, SCA). In this article, an overview on the current

progress in modeling of suspensions of synthetic and bio-logical particles such as vesicles, capsules, and RBCs wasgiven.

Blood flow in microcirculation normally falls in theStokes regime where the inertia is negligible, hence mak-ing boundary integral methods (BIM) a very good candidateto address suspension flows of soft particles. However, thefirst numerical studies on the dynamics of capsules usingboundary element method (with second-order or cubic poly-nomial basis functions) in linear shear flow suffered thenumerical instability at high deformation due to mesh degra-dation (Ramanujan and Pozrikidis 1998; Lac et al. 2004).Consequently, BIM formulations with higher order repre-sentations of the membrane using spherical harmonics werepresented that could achieve spectral accuracy (Zhao et al.2010; Veerapaneni et al. 2011a). More recently, a loop sub-division method was developed in the context of BIM topresent smooth and C1 continuous vesicle shapes with veryhigh deformations (Spann et al. 2014).

Among other continuum solvers, the immersed bound-ary method (IBM) has become very effective in addressingFSI problems (Peskin 2002; Fai et al. 2013). In this method,a force density is distributed to the Cartesian mesh in thevicinity of the moving boundary (front) in order to accountfor its effect. The main advantage of IBM is that the front isadvected in a fixed Eulerian grid which is not needed to beconformal. There are several numerical extensions of IBMdepending on the choice of the structural or fluid formula-tions. For example, the immersed-boundary/front-trackingmethod in (Doddi and Bagchi 2009; Yazdani and Bagchi2011) used a finite element triangulation of the cell’s mem-brane, while the full Navier-Stokes equation was solvedusing a projection splitting scheme. In addition, Lattice-Boltzmann method as the fluid solver has been used in

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Fig. 11 a Schematic of thehuman RBC membrane. b–d Amesoscale detailed membranemodel. b, c show the initialconfiguration of the cellmembrane from the top view (b)and side view (c), and (b) showsthe equilibrium configuration ofthe cell membrane from the sideview. The blue sphere representsa lipid particle and the red spheresignifies an actin junctionalcomplex. The gray sphererepresents a spectrin particle andthe black sphere represents aglycophorin particle. The yellowand green circles correspond toa band-3 complex connected tothe spectrin network and amobile band-3 complex,respectively. Reproduced fromLi and Lykotrafitis (2014a), bypermission

several studies (Sui et al. 2008; Clausen et al. 2011). Oneproblem with the IBM is the smoothing of the force densitytransferred to the fluid which renders the surface diffusiverather being sharp. More recently, fictitious domain meth-ods (FDM) for solving coupled fluid-structure systems hasbeen used (Shi et al. 2014; Hao et al. 2015). In the FDM,the fluid and solid domains are coupled by introducing aLagrange multiplier over the solid domain. This multiplieracts as a penalty term that imposes the kinematic constraintin the solid domain. It has to be mentioned that model-ing fluid-structure interaction at the microscopic level withcontinuum solvers requires consistent inclusion of thermalfluctuations; to address this issue, a stochastic formulationfor IBM has been given (Atzberger et al. 2007).

Particle-based methods (Noguchi and Gompper 2005b;McWhirter et al. 2009; Fedosov et al. 2010), wheremesoscopic particle-collision models are employed, pro-vide opportunities to significantly reduce computationalexpense on a per-cell basis. Being constructed from aCGMD approach, they can naturally include thermal fluc-tuations. Such formulations are compatible with coarse-grained descriptions of the membrane mechanics (Fedosov

et al. 2010) with the advantage of including the membraneviscosity with no additional cost. However, some issuesremain open for the particle-based methods. For exam-ple, the narrow regions between closely packed cells andconsequently the lubrication effect will not be capturedaccurately unless the particle density is increased signif-icantly.Imposing the no-slip condition on the cell surfaceis also a challenge. In a more recent study, an alterna-tive Lagrangian formulation for the hydrodynamics andcoarse-grained cell membrane based on smoothed dissipa-tive particle dynamics (SDPD) has been presented (Fedosovet al. 2014a).

The current computational models of soft biological par-ticles offer unique tools for the qualification and quantifica-tion of dynamics and rheological properties of healthy andpathological RBCs under select experimental conditions.However, they do not lend themselves to detailed whole-cell investigations of a wide variety of biophysical problemsinvolving the RBCs, such as the bilayer loss in heredi-tary spherocytosis due to defective protein attachments anduncoupling between the lipid bilayer and cytoskeleton insickle cell anemia due to sickle hemoglobin polymerization

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(Liu et al. 1991). For these reasons, there is a compellingneed to develop a more realistic RBC representation, e.g.,to endow the spectrin-based RBC models with more accu-rate structure, e.g., account separately for the lipid bilayerand cytoskeleton but also include explicitly the transmem-brane proteins. Recent efforts have been directed towardsthis approach. For example, Li and Lykotrafitis introduceda two-component RBC membrane model, where the lipidbilayer and cytoskeleton as well as the transmembrane pro-teins are explicitly represented by coarse-grained particles(Fig. 11) (Li and Lykotrafitis 2012b, 2014a). This detailedmembrane model has been successfully applied to studysome membrane-related problems in RBCs such as thedynamic behavior of RBC membrane under shear and thediffusion of transmembrane proteins. However, at present, itis computationally inefficient to be used for blood dynamicsand rheological studies involving large numbers of RBCsin flow. Recently, a two-component continuum-based wholecell model (Peng et al. 2010) and a two-component particle-based whole cell model (Peng et al. 2013), which treat thelipid bilayer and the cytoskeleton as two distinct compo-nents, have been developed. These whole cell models havebeen used in simulations of blood flow and quantified theexistence of bilayer-cytoskeletal slip for RBCs in patholog-ical state. However, they are not yet capable of modelingthe interactions between the spectrin filaments and trans-membrane proteins such as the band-3. Thus, a challengein computational modeling of blood flow which encompassall scales would be to develop hybrid model that integratesboth approaches. Such a model would provide a more reli-able method and an overall modeling framework to extractdynamic and rheological properties of RBCs.

To sum up, continuum and particle-based methods haveproved themselves to become more efficient and mature indealing with complicated problems of soft particle suspen-sions. Initially, they were used extensively to study the intri-cate underlying physics of dilute suspensions, and recentlyhave become ideal tools to study the rheological behaviorof dense suspensions in health and disease (e.g., malariaand SCA). The range of applicability of these methods isquite diverse, and they can be used in addressing other typesof biological problems involving microscopic deformablecells in moving or quiescent fluid. These problems includebut not limited to cell motility and adhesion of other typesof eukaryotic cells such as bacteria, dynamics of cancercells in microvascular networks, whole blood simulationsin complex geometries such as aneurysms and stenoses,mechanobiology of cells, and complex multiscale biologicalprocesses such as thrombus formation at the injury site.

Acknowledgments We acknowledge support from the NationalInstitutes of Health (NIH) grants U01HL114476 and U01HL116323.A.Y. acknowledges TACC/STAMPEDE resources through XSEDE

grant (TG-DMS140007), and X.L. acknowledges ALCF throughINCITE program for providing computational resources that have leadto the unpublished research results reported within this paper.

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