Loading and relaxation dynamics of a red blood cell
Fabio Guglietta, Marek Behr, Giacomo Falcucci, Mauro Sbragaglia
LL OA D I N G A N D R E L A X AT I O N DY NA M I C SO F A R E D B L O O D C E L L † Fabio Guglietta
Department of Physics & INFN,
University of Rome “Tor Vergata” ∗ Chair for Computational Analysis of Technical Systems (CATS),
RWTH Aachen University † Computation-based Science and Technology Research Center,
The Cyprus Institute ‡ [email protected] Marek Behr
Chair for Computational Analysis of Technical Systems (CATS),
RWTH Aachen University † Giacomo Falcucci
Department of Enterprise Engineering “Mario Lucertini,”
University of Rome “Tor Vergata" § Mauro Sbragaglia
Department of Physics & INFN,
University of Rome “Tor Vergata” ∗ A BSTRACT
We use mesoscale numerical simulations to investigate the unsteady dynamics of a single red bloodcell (RBC) subject to an external mechanical load. We carry out a detailed comparison betweenthe loading (L) dynamics, following the imposition of the mechanical load on the RBC at rest,and the relaxation (R) dynamics, allowing the RBC to relax to its original shape after the suddenarrest of the mechanical load. Such comparison is carried out by analyzing the characteristic timesof the two corresponding dynamics, i.e., t L and t R . For small intensities of the mechanical load,the two dynamics are symmetrical ( t L ≈ t R ) and independent on the typology of mechanical load(intrinsic dynamics); in marked contrast, for finite intensities of the mechanical load, an asymmetry is found, wherein the loading dynamics is typically faster than the relaxation one. This asymmetrymanifests itself with non-universal characteristics, e.g., dependency on the applied load and/or on theviscoelastic properties of the RBC membrane. To deepen such a non-universal behaviour, we considerthe viscosity of the erythrocyte membrane as a variable parameter and focus on three differenttypologies of mechanical load (mechanical stretching, shear flow, elongational flow): this allows toclarify how non-universality builds-up in terms of the deformation and rotational contributions inducedby the mechanical load on the membrane. Our results provide crucial and quantitative information onthe unsteady dynamics of RBC and on its membrane response to the imposition/cessation of externalmechanical loads. Red blood cells (RBCs) are biological cells made of a vis-coelastic membrane enclosing a viscous fluid (cytoplasm):their main features are the biconcave shape and the absence † Electronic Supplementary Information (ESI) and fourvideos showing the simulations performed are available in theAncillary Files. A brief description of the movies is provided inESI. of a nucleus and most organelles, that allow them to carryoxygen even inside the smallest capillaries [1, 2, 3]. Infact, during the circulation, RBCs deform multiple times,rearranging their shape to adapt to the physiologicalconditions of the blood flow. The mechanical propertiesof RBC’s membrane have been deeply investigated,both numerically [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]and experimentally [15, 16, 17, 18, 19, 20, 21, 22, 23].Research interest on the mechanical response of RBC’s a r X i v : . [ c ond - m a t . s o f t ] F e b embrane was prompted by several reasons: amongthe others, the link between its properties and theerythrocyte’s health conditions [11, 16], or the role playedby the membrane dynamics in the design of biomedicaldevices[24, 25, 26, 27]. A huge effort has been devoted tothe characterization of the time-independent properties ofthe membrane, towards the definition of the correspondingsteady-state configurations. In recent years, also thedynamical behaviour of RBCs has been investigated in sev-eral works, both numerically [28, 29, 30, 31, 8, 32, 33, 34]and experimentally [17, 19, 20, 35]. When dealing withtime-dependent properties of biological cells (or capsules,in general), the membrane viscosity plays a crucialrole [36, 37, 38, 33, 34, 39, 40, 41, 42, 43]. Evans [44]showed that the RBC relaxation time is affected byboth the membrane viscosity and the dissipation in theadjacent aqueous phases (i.e., cytoplasm and externalsolution); neglecting the membrane viscosity, i.e., µ m = 0 , he predicted a relaxation time t R ≈ × − s(also confirmed by numerical simulations [33]), aremarkably lower value compared to other works in theliterature [22, 19, 17, 45, 33]. Several works aimed at theprecise estimation of the value of the membrane viscos-ity [46, 18, 45, 47, 22, 48, 49, 17, 19, 32], finding that µ m roughly ranges between − m Pa s and − m Pa s.Such a variability may be ascribed to different factors,e.g., the different theoretical models used to infer µ m [19, 46], the different experimental apparatuses (suchas micro-pipette aspiration [46, 18, 45, 17], microchanneldeformation [49], or other setups [47, 22, 48]), etc. Asa matter of fact, although µ m is an essential parameterto quantitatively characterize the time dynamics ofRBCs [34, 37], its precise value has not been accu-rately determined so far, which warrants a parametricinvestigation. Moreover, some earlier studies [50]proposed to use an increased apparent viscosity ratio toaccount for the energy dissipation due to the presenceof a viscous membrane: even though this assumptionprovides a qualitative description of the effects due to thepresence of the membrane viscosity, it does not accountfor a quantitative characterisation, as shown by recentstudies [37, 34, 41, 40]. Our previous work [33] aimedto investigate the effect of membrane viscosity µ m onthe relaxation dynamics of a single RBC, and we foundthat increasing the value of µ m , as well as increasing theintensity of the loading strength, leads to a fast recoverydynamics. Moreover, we simulated two experimentalsetups, i.e., the stretching with optical tweezers and thedeformation due to an imposed shear flow, and we founda dependency on the kind of mechanical load for finitestrengths of the load.The relaxation dynamics, however, gives only a partialcharacterization of the time-dependent response of RBCsto external forces. Obviously, the loading process shouldbe considered, as well, as already pointed out in earlierliterature papers. Chien et al. [18] experimentally studiedboth the loading and the relaxation dynamics of RBCmembrane through micro-pipette aspiration, providingevidence that the two dynamics are not symmetrical in certain conditions; however, a systematic studyinvolving different stress values and different typologiesof mechanical loads was not performed. Diaz et al. [43]studied the dynamics of a pure elastic capsule with ahyperelastic membrane deformed by an elongational flow:they focused on both loading and relaxation, findingan asymmetry. However, their model did not take intoaccount the membrane viscosity and the asymmetrywas not studied for different typologies of mechanicalloads. Thus, although previous literature points to twodistinct dynamics for loading and relaxation [18, 43, 42],a comprehensive parametric study on the effects of µ m for different typologies of mechanical loads and flowconditions (such as simple shear flow or elongationalflow) has never been attempted, so far. This paper aimsat filling this gap with the help of mesoscale numericalsimulations. Indeed, for this kind of characterization,numerical simulations can be thought of as the appropriatetool of analysis [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14],due to the obvious experimental difficulties in carryingout such systematic investigation [17, 18, 19, 21, 23].We provide a quantitative characterisation of loadingand relaxation dynamics exploring three typologies ofmechanical loads. To do this, we built three differentsimulation setups: the stretching simulation (STS), whichsimulates the deformation with optical tweezers [16](see Fig. 1, panel (a)); the shear simulation (SHS), i.e.,the deformation in simple shear flow (see Fig. 1, panel(c)); the four-roll mill simulation (FRMS), where thedeformation is induced by an elongational axisymmetricflow made by the rotation of four cylinders (see Fig. 1,panel (e)). These three numerical setups are chosen toinspect the different roles that the membrane rotationand/or membrane deformation have in the time-dependentdynamics. This information is summarised in Tab. 1.We systematically study the characteristic times of boththe loading ( t L ) and the relaxation ( t R ) processes andtheir ratio ˜ t = t L /t R , as a function of the load strengthand membrane viscosity µ m . For small strengths, thetwo characteristic times are essentially equal and setby the value of µ m ; however, for finite strengths, theloading dynamics is found to be faster than the relaxationdynamics, leading to a non-universal behaviour whilechanging the typology of the mechanical load. Suchnon-universal contributions are further characterized interms of the importance of rotation and deformation of themembrane, according to the different load mechanisms.Some useful parametrizations for both t R and t L as afunction of the membrane viscosities are also provided.The paper is organised in the following way: in Sec. 2 weprovide some details on the numerical method used tosimulate both the fluid and the membrane of the RBC; inSec. 3 we analyse the three simulated loading mechanisms(the stretching simulation, Sec. 3.1; the shear simulation,Sec. 3.2; the four-roll mill simulation, Sec. 3.3); a detaileddiscussion section with comparisons between the loadingmechanisms is provided in Sec. 4; finally, conclusions arereported in Sec. 5. a) ST retching S imulation (STS) (b) Stretching simulation (STS): deformation.(c) SH ear S imulation (SHS) (d) Shear simulation (SHS): deformation.(e) F our- R oll M ill S imulation (FRMS) (f) Four-roll mill simulation (FRMS): deformation. Figure 1: Loading-relaxation (L-R) experiments for red blood cell (RBC) at changing the typology of mechanicalload.
Left panels : the three different L-R experiments investigated in the paper are sketched: grey arrows refer to themechanical load, either an applied force F or an applied velocity U w , while the RBC membrane forces ( F mem ) aresketched with green arrows. In all simulation, the deformation D ( t ) , i.e., the ratio between the difference and the sumof the two longest diameters (see Eq. 11), is used to fit the loading and relaxation times ( t L and t R , respectively; seeEqs. (12) (13) (14)). Right panels : we report the deformation D ( t ) (see Eq. (11)) as a function of time for two valuesof membrane viscosity µ m . Panels (a-b) : we simulate the stretching with optical tweezers [16] (STS), in which twoforces with the same intensity and opposite direction stretch the membrane in two areas at the end of the RBC (seeSec. 3.1); the deformation D ( t ) is reported for F = 90 × − N. Panels (c-d) : deformation induced by simple shearflow (SHS), with U w = ( ± ˙ γ/ , , , where ˙ γ is the shear rate (see Sec. 3.2); the deformation D ( t ) is reported for ˙ γ = 86 s − . Panels (e-f) : in the four-roll mill simulation (FRMS) we simulate four rotating cylinders to reproduce anelongational flow that deforms the membrane [51] (see Sec. 3.3); the deformation D ( t ) is reported for F roll = 80 s − .Four videos showing these simulations are available (see Ancillary Files†). oad Type Rotation Direct ForcingSTS NO YESSHS YES NOFRMS NO NOTable 1: Summary of the main characteristics of the threekinds of applied mechanical load (see Fig. 1). For eachload type, we specify if the rotation is induced on the mem-brane while loading and if the forcing is directly appliedon the nodes of the mesh used to discretise the membrane(otherwise, the membrane is forced indirectly via hydrody-namic flow). We perform three-dimensional numerical simulations inthe framework of the Immersed Boundary – Lattice Boltz-mann method (IB–LBM) [52, 53]. The methodology, aswell as the membrane model, are the same already usedand validated in [33]: here we report an essential summary.The equation of motion for a fluid with viscosity µ is givenby Navier-Stokes (NS) equation ρ (cid:18) ∂ u ∂t + ( u · ∇∇∇ ) u (cid:19) = −∇∇∇ p + µ ∇∇∇ u + F , (1)where ρ and u are the density and the velocity of the fluid,respectively; p is the isotropic pressure; F is an externalbody force. If the fluid is incompressible (as in the presentwork) the condition ∇∇∇ · u = 0 holds.In the LBM, instead of directly solving NS equation byintegrating Eq. (1), the fluid is represented by the so-calledpopulations f i ( x , t ) , that stand for the density of fluidmolecules moving with velocity c i at position x and time t .The populations evolve according to the Lattice Boltzmannequation: f i ( x + c i ∆ t, t + ∆ t ) − f i ( x , t ) == − ∆ tτ (cid:16) f i ( x , t ) − f ( eq ) i ( x , t ) (cid:17) + f ( F ) i , (2)in which ∆ t is the discrete time step, τ is the relaxationtime, f ( F ) i is the source term which takes into account theforce density (it has been implemented according to the“Guo” scheme [54]), and f ( eq ) i is the equilibrium distribu-tion function (we refer back to [52, 53] for the details).The fluid density ρ and the velocity u are given by: ρ ( x , t ) = X i f i ( x , t ) ,ρ u ( x , t ) = X i c i f i ( x , t ) . (3)The link between NS and LB equations (Eq. (1) and Eq. (2),respectively) is given by the following relation: µ = ρc s (cid:18) τ − ∆ t (cid:19) , (4) where c s = ∆ x/ ∆ t √ is the speed of sound. In the fol-lowing, we considered both the lattice spacing ∆ x and thetime interval ∆ t equal to 1.We simulate two fluids: one outside the membrane (theplasma, with viscosity µ out = 1 . × − Pa s) and oneinside it (the cytosol, with viscosity µ in = 6 × − Pa s).The viscosity ratio is given by λ = µ in µ out , (5)providing λ = 5 . We implemented the parallel Hoshen-Kopelman algorithm to recognise which lattice sites areinside or outside the membrane (see [55] for details).The RBC membrane is described as a 3D triangular meshof ≈ elements, whose shape at rest is given by [56] z ( x, y ) = ± r − x + y r ·· C + C x + y r + C (cid:18) x + y r (cid:19) ! , (6)with C = 0 . × − m, C = 7 . × − m and C = − . × − m; r = 3 . × − m is the largeradius.The membrane is characterised by a resistance to sheardeformation, area dilation and bending; the viscoelastic be-haviour is implemented, as well. The first two terms formthe strain energy W S are described by Skalak model [57]: W S = X j A j (cid:20) k S (cid:0) I + 2 I − I (cid:1) + k α I (cid:21) , (7)where k S = 5 . × − N m − [16] and k α = 50 k S [3]are the surface elastic shear modulus and the area dilationmodulus, respectively; I = λ + λ − and I = λ λ − are the strain invariants for the j -th element, while λ and λ are the principal stretch ratios [57, 3]. We adoptthe Helfrich formulation to compute the free-energy W B related to the resistance to bending [58]. Following [3], wediscretise the bending energy as: W B = k B √ X h i,j i (cid:16) θ ij − θ (0) ij (cid:17) , (8)where k B = 2 × − N m [59] is the bending modulus;the sum runs over all the neighbouring triangular elements,and θ ij is the angle between the normals of the i -th and j -th elements ( θ (0) ij is the same angle in the unperturbedconfiguration). Once we have the total free-energy W = W S + W B , we compute the force acting on the i − th nodeby performing the derivative of W with respect to thecoordinates of the node x i : F i = − ∂W ( x i ) ∂ x i . (9)Note that we are implementing neither area nor volumeconservation: in fact, as stated in [33], we checked thatboth area and volume were conserved, even without an xplicit area or volume conservation law (see ElectronicSupplementary Material in [33]).Regarding the viscoelastic term, we implemented the Stan-dard Linear Solid (SLS) model [60, 36]. The viscous stresstensor is given by τττ ν = µ s (cid:16) E − tr ( ˙ E ) (cid:17) + µ d tr ( ˙ E ) , (10)where E is the strain tensor (see [60, 33]); µ s and µ d arethe strain and dilation membrane viscosity: in this work,we assume µ s = µ d = µ m [33]. We refer to [60, 33] forthe computation of the stress tensor Eq. (10) as well as forthe nodal force F i .Finally, once we have the nodal force F i for each node i ofthe 3D mesh, we spread this force to the lattice nodes viathe IBM (see [52] for details). We adopt the same schemeas in [33]. In this section, we quantitatively study the loading time t L and the relaxation time t R with three different loadingmechanisms (see Fig. 1): the stretching with optical tweez-ers (STS, see Sec. 3.1), the deformation in simple shearflow (SHS, see Sec. 3.2) and the deformation in an elonga-tional flow (FRMS, see Sec. 3.3). These three simulationsdiffer mainly for two aspects (summarised in Tab. 1): thefirst one is that the membrane can be deformed by an ex-ternal force which acts directly on the membrane (like inthe STS) or by the viscous friction with the fluid (SHS andFRMS); moreover, the membrane can rotate (like in theSHS) or not (STS and FRMS). The idea underlying thechoice of these three different setups is to catch which ofthe aforementioned characteristics affects the loading andrelaxation dynamics.In order to quantify the loading time t L and the relaxationtime t R , we define the deformation parameter D ( t ) = d A ( t ) − d T ( t ) d A ( t ) + d T ( t ) , (11)where d A and d T represent the length of the axial andtransversal diameters, i.e., the greater and medium eigen-values of the inertia tensor (see [33]). In our computationaldomain, d A and d T lie in the x − y plane. We define theaverage deformation D av , i.e., the value of the deformation D such that lim t →∞ D ( t ) = D av in the loading simulationand D (0) = D av in the relaxation simulation.Qualitatively, the loading time t L is the characteristic timethe deformation D ( t ) takes to reach D av ; the relaxationtime t R is the characteristic time needed to relax to theinitial shape, after the arrest of the mechanical load. Quan-titatively, we can get t L and t R via a fit of D ( t ) /D av withthe following functions: L ( t ) = 1 − exp ( − (cid:18) tt L (cid:19) δ ) , (12) L ( t ) = 1 − exp ( − (cid:18) tt L (cid:19) δ ) cos (cid:18) tt cosL (cid:19) , (13) R ( t ) = exp ( − (cid:18) tt R (cid:19) δ ) , (14)where L is used to fit the loading time for the STS andthe FRMS (see Sec. 3.1-3.3, respectively); L is used tofit the loading time for the SHS (see Sec. 3.2); R is usedto fit the relaxation time t R for all three simulations; δ is aparameter introduced to improve the fit [32] (see [33] forsome more details).Note that we propose two different functions to fit dataduring the loading (i.e. Eq. (12) and Eq. (13)); this is dueto the different deformation process of the RBC: in the STSand FRMS, D ( t ) is a monotonic increasing function withan asymptote in D = D av (see Fig. 1, panels (b) and (f));in the SHS, D ( t ) oscillates around D av , and the amplitudeof the oscillations varies with the value of the membraneviscosity µ m (see Fig. 1, panel (d)). These oscillations havebeen also observed for viscoelastic capsules[38, 60, 33].Since we want to compare the loading time t L and therelaxation time t R , we define the ratio ˜ t = t L t R . (15)In all the following simulations, the membraneviscosity ranges between µ m ∈ [0 , . × − m Pa s [46, 18, 45, 47, 22, 48, 49, 17, 19, 32].In the STS, the applied force is in the range F ∈ [5 , × − N; in the SHS, we simulatedshear rates in the range ˙ γ ∈ [1 . , s − ; finally, inthe FRMS, we simulated F roll ∈ [1 , s − (see Eq. (16)). In order to simulate the stretching with optical tweezers,we apply two forces with the same intensity F and oppositedirections at the ends of the RBC (see Fig. 1, panel (a)).Simulations are performed in a 3D box L x × L y × L z =(28 , , × − m. In Fig. 2, we report the loadingtime t L (panel (a)) and the relaxation time t R (panel (b)) asa function of F , for different values of membrane viscosity µ m . In both cases, increasing the loading strength (as wellas decreasing the value of membrane viscosity µ m ) resultsin a faster dynamics. It is interesting to compare t L and t R : in Fig. 2, panel (c), we report the ratio ˜ t (see Eq. (15)).As expected, decreasing the value of F , t L and t R tend tocoincide; on the other hand, increasing F leads to a loadingtime t L that is smaller than t R . In the shear simulation, the deformation is due to a lin-ear shear flow with intensity ˙ γ . We set the wall velocityU w = ( ± ˙ γ/ , , , and the RBC is oriented as reportedin Fig. 1, panel (c): we chose this orientation to allow fora precise estimation of the loading and relaxation times:indeed, without such a choice one would meet compli-cations coming from the emergence of other dynamicalmodes [61, 29]. Simulations are performed in a 3D box (a) (a) (b) . . . . . (a) (b) (c) (a) (b) (c)(d) (a) (b) (c)(d) (e) . . . . . (a) (b) (c)(d) (e) (f) (a) (b) (c)(d) (e) (f)(g) (a) (b) (c)(d) (e) (f)(g) (h) . . . . . (a) (b) (c)(d) (e) (f)(g) (h) (i) t L [ − s ] F [ − N] STSSHSFRMS . . . µ m [ − m Pa s] t R [ − s ] F [ − N] ˜ t F [ − N] t R [ − s ] ˙ γ [s − ] t R [ − s ] ˙ γ [s − ] ˜ t ˙ γ [s − ] t R [ − s ] F roll [s − ] t R [ − s ] F roll [s − ] ˜ t F roll [s − ] Figure 2: Characteristic times t L (first column of panels) and t R (second column of panels) as well as the ratio ˜ t = t L /t R (third column of panels) are reported for the three simulations performed, i.e., stretching simulation (STS, , panels(a-c), Sec. 3.1), shear simulation (SHS, , panels (d-f), Sec. 3.2), four-roll mill simulation (FRMS, , panels (g-i),Sec. 3.3), for different values of membrane viscosity µ m (from lightest to darkest color): µ m = 0 . m Pa s ( , , ), µ m = 0 . × − m Pa s ( , , ), µ m = 1 . × − m Pa s ( , , ), µ m = 3 . × − m Pa s ( , , ). The reddashed line represents the reference value for the symmetric case, i.e., ˜ t = 1 . L x × L y × L z = (20 , , × − m. In Fig. 2, the load-ing time t L (panel (d)) and the relaxation time t R (panel(e)) as functions of ˙ γ for different values of membraneviscosity µ m are reported, as well as the ratio ˜ t (panel (f)).While the relaxation time t R shows a similar behaviourcompared to the STS (see Fig. 2, panel (b)), a few morewords are needed regarding the loading time t L . Unlikethe STS, now we have two characteristic times, that are t L and t cosL (see Eq. (13)): t L measures the time the mem-brane takes to reach the average deformation D av , while t cosL measures the period of the oscillations. Data for t cosL are reported in ESI†, Fig. 1. In contrast to the STS, theloading time t L first decreases, and then it does not changemuch with the intensity of the mechanical load. In this case, we simulate the effect of four cylinders rotat-ing [51], as shown in Fig. 1, panel (e), in order to create aflow similar to a pure elongational one. Simulations are per-formed in a 3D box L x × L y × L z = (48 , , × − m.The idea is to simulate a loading mechanism that is a mix- .
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12 0 . (a) (b) (c) t L [ − s ] σ [Pa] STSSHSFRMS . . . µ m [ − m Pa s] t R [ − s ] σ [Pa] ˜ t σ [Pa] Figure 3: Comparison between the characteristic times t L (panel (a)), t R (panel (b)) and ˜ t = t L /t R (panel (c)) asfunctions of the stress σ (see Sec. 4) for the three simulations performed, i.e., stretching simulation (STS, , Sec. 3.1),shear simulation (SHS, , Sec. 3.2), four-roll mill simulation (FRMS, , Sec. 3.3), for different values of membraneviscosity µ m (from lightest to darkest color): µ m = 0 . m Pa s ( , , ), µ m = 0 . × − m Pa s ( , , ), µ m = 1 . × − m Pa s ( , , ), µ m = 3 . × − m Pa s ( , , ). The red dashed line represents the referencevalue for the symmetric case, i.e., ˜ t = 1 .ture of stretching with optical tweezers and deformationin simple shear flow (see Tab. 1): in fact, in this case, themembrane does not rotate (like in the STS) and the defor-mation is caused by the flow (like in the SHS).To create such a flow, we impose a force density [51] F ( x, y ) = 2 kµ F roll sin( kx ) cos( ky ) − cos( kx ) sin( ky )0 ! , (16)where k = 2 π/L x , µ is the local fluid viscosity, and F roll isused to tune the load strength. We multiplied Eq. (16) by k to make the velocity gradient independent of the size ofthe fluid domain ∗ : ∂ u ∂ x = F roll (cid:18) cos( kx ) cos( ky ) − sin( kx ) sin( ky )sin( kx ) sin( ky ) − cos( kx ) cos( ky ) (cid:19) , (17)where we have reported the only x and y components, i.e.,the components in the plane of the shear. Note that Eq. (16)gives a pure elongational flow only in x = π/ , π/ andin y = π/ , π/ .In Fig. 2 we report the loading time t L (panel (g)) and therelaxation time t R (panel (h)) as a function of F roll . Asfor the STS and the SHS, both t L and t R decrease whenthe loading force increases or the membrane viscosity µ m decreases. In Fig. 2, panel (i), the ratio ˜ t is reported. In our simulations, the intensity of the three mechanicalloads is changed by varying different quantities, i.e., F for ∗ The following result is valid in a homogeneous fluid withdynamics viscosity µ . the STS, ˙ γ for the SHS and F roll for the FRMS. To facilitatea comparison between them, we first consider the charac-teristic times t L and t R as well as the ratio ˜ t as functions ofthe characteristic simulation stress σ (Fig. 3). To evaluatethe stress σ for the STS, we computed the area A at theend of the RBC where the force F is applied. Then, thestress is given by σ STS = F/A ; for the SHS, we wrotethe stress as σ SHS = 2 ˙ γµ out . Finally, for the FRMS, thestress is given by the stress-peak σ FRMS = µ out F roll . In allthree simulations, the loading and relaxation times ( t L and t R , respectively) show qualitatively the same behaviour,i.e., they decrease when the loading strength increases orwhen the membrane viscosity µ m decreases (see Fig. 3,panels (a) and (b)); the ratio ˜ t = t L /t R is reported in Fig. 3,panel (c). For small forces ( σ → ) we observe a cleartendency towards symmetry between loading and relax-ation ( ˜ t ( σ, µ m ) → ), meaning that the characteristic times t L and t R tend to be equal. This is the limit where oneexpects to recover the intrinsic dynamics of the membrane,which depends only on the value of membrane viscosity µ m [33, 19].On the other hand, for finite force strengths, loading and re-laxation dynamics are asymmetrical , i.e., ˜ t = 1 for σ = 0 .As already noticed elsewhere [42], this asymmetry couldbe explained by energetic considerations: in fact, duringthe loading phase, the deformation is driven by the externalload (i.e., an external source of energy), while during therelaxation, the membrane provides the whole energy. Be-yond these qualitative considerations, results in Fig. 3 pro-vide a systematic characterization of the relaxation times,as a function of either the stress σ or the membrane viscos-ity µ m : an important message conveyed by our analysis isthat the asymmetry is not universal , i.e., on equal values ofmembrane viscosity µ m , the ratio ˜ t depends on the kind of . . . . . . .
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12 0 . (c) − − (c) D a v σ [Pa] . . . µ m [ − m Pa s] φ t [ − s] STSSHSFRMS − − Loading Relaxation (b)
Figure 4:
Panel (a) : snapshots of RBC at selected times.A point is selected on the membrane (green sphere) toperform a Lagrangian tracking and determine the timedependency of the angle φ that the point direction formswith the y axis in the deformation plane. Panel (b) : wereport the angle φ (see panel (a) above) as a function oftime for the stretching simulation (STS, , Sec. 3.1), shearsimulation (SHS, , Sec. 3.2), four-roll mill simulation(FRMS, , Sec. 3.3), for µ m = 3 . × − m Pa s.The red and blue shades represent loading and relax-ation regions, respectively. Panel (c) : average deforma-tion D av (see text for details) as a function of the stress σ for the three simulations performed (STS, SHS andFRMS). SHS data are displayed for different values ofmembrane viscosity µ m (from lightest to darkest color): µ m = 0 . m Pa s ( ), µ m = 0 . × − m Pa s ( ), µ m = 1 . × − m Pa s ( ), µ m = 3 . × − m Pa s( ). STS data ( ) and FRMS data ( ) data are only reportedfor µ m = 3 . × − m Pa s. mechanical load. Just to give some numbers, the differencebetween the values of ˜ t for the STS and FRMS is roughlyconstant ( ≈ σ ;for the SHS the situation is a bit more complex because ˜ t depends on µ m . However, if we compare SHS against STSfor µ m = 3 . × − m Pa s, we find a difference of lessthen 30% for small values of σ (i.e., σ < . Pa), whilesuch a difference goes over the 50% for large values of σ (i.e., σ > . Pa).If we think that the asymmetry comes from the pres-ence of a mechanical load with finite strength [43, 42],it comes naturally to expect a non-universality and a de-pendency on the details of the loading mechanism. Thanksto our analysis, we are in a condition to further char-acterise this non-universality: indeed, we observe thatwhile ˜ t does not depend on µ m for the STS and FRMS( ˜ t STS = ˜ t STS ( σ ) , ˜ t FRMS = ˜ t FRMS ( σ ) ), it actually does inthe SHS ( ˜ t SHS = ˜ t SHS ( σ, µ m ) ). The collapse shown by ˜ t in the STS and FRMS (see Fig. 3, panel (c)) suggests afactorisation of the loading and relaxation times in twocontributions: one depending on the membrane viscosity µ m and one on the load intensity σ : t K L ( σ, µ m ) ≈ t ∗ K L ( σ ) t K ( µ m ) for K = STS, FRMS , (18) t K R ( σ, µ m ) ≈ t ∗ K R ( σ ) t K ( µ m ) for K = STS, FRMS , (19)where the superscript K stands for the kind of mechanicalload. Given this factorisation, we have ˜ t K ( σ ) = t ∗ K L ( σ ) t ∗ K R ( σ ) for K = STS, FRMS . (20)To make progress, we investigated what is the physical in-gredient at the core of the factorization given in Eqs. (18)-(19), or, alternatively, why the SHS does not factorize asin Eqs. (18)-(19). We argue that such a difference betweenSHS and STS/FRMS is mainly due to the different dynam-ics that are induced by the mechanical load. First of all,as already stated in Sec. 3, in the SHS there are two char-acteristic times in the loading process (see Eq. (13)), onerelated to the time the deformation takes to reach the steadyvalue D av , and one related to the period of the oscillation( t L and t cosL , respectively). The main difference betweenSHS and STS/FRMS lies in the rotation of the membrane(see Tab. 1) that is present only in the SHS. In fact, one cansplit the velocity gradient ∂ u ∂ x in the symmetric (rotational)and antisymmetric (elongational) parts: ∂ u ∂ x = (cid:18) γ (cid:19) = (cid:18) ˙ γ γ (cid:19) + (cid:18) ˙ γ − ˙ γ (cid:19) , (21)where the only two components in the shear plane arereported. The rotational part causes the rolling motion ofthe membrane (see Fig. 4, panels (a) and (b)), while theelongational one tends to deform the RBC and pushesthe main diameter to an angle of π/ with respect to theshear direction; an increase in the membrane viscositycauses an increase in the time needed for the membrane toadapt to the flow and to deform; meanwhile, the rotational . . . . (a) . . . . (a) (b) . . . . (a) (b) (c) . . . . (a) (b) (c)(d) . . . . (a) (b) (c)(d) (e) . . . . (a) (b) (c)(d) (e) (f) t L [ − s ] µ m [m Pa s]STSSHSFRMS Linear fit: σ = 0.001 PaLinear fit: σ = 0.01 PaLinear fit: σ = 0.1 Pa .
001 0 .
01 0 . σ [Pa] t L [ − s ] µ m [m Pa s] t L [ − s ] µ m [m Pa s] t R [ − s ] µ m [m Pa s] t R [ − s ] µ m [m Pa s] t R [ − s ] µ m [m Pa s] Figure 5: Characteristic times t L (panels (a-c)) and t R (panels (d-f)) as functions of the membrane viscosity µ m (seeSec. 4) for the three performed simulations: stretching simulation (STS, , Sec. 3.1), shear simulation (SHS, ,Sec. 3.2), four-roll mill simulation (FRMS, , Sec. 3.3), for different values of stress σ (from lightest to darkest color): σ = 0 . Pa ( , , ), σ = 0 . Pa ( , , ), σ = 0 . Pa ( , , ).component promotes a rotation of the main diameter.Overall, the increase in membrane viscosity µ m leadsto a decrease of the average deformation D av (see alsoFig. 4, panel (c)). To make these arguments clearer, wehave made two videos available in the Ancillary Files†:in one we show the simulation with ˙ γ = 1 . s − and µ m = 3 . × − m Pa s, while in the other one the sim-ulation with ˙ γ = 1 . s − and µ m = 0 m Pa s is reported.In both cases, the tank-treading motion of the membraneappears, but, for µ m = 0 m Pa s, the membrane deformsmore than in the case with µ m = 3 . × − m Pa s.In Fig. 4, panel (c), we report the average deformation D av as a function of the stress σ for all three mechanicalloads at changing the membrane viscosity µ m . Since D av is not sensitive to the value of membrane viscosity µ m forthe STS and FRMS, for these two cases we report onlypoints for µ m = 3 . × − m Pa s. It emerges that,in the SHS, the average deformation D av saturates at aconstant value: an increase in the shear rate ˙ γ causes aninitial increase of the average deformation D av ; then, D av reaches a plateau and increasing the shear rate ˙ γ beyond acertain value does not result in an increased deformation.The higher the membrane viscosity µ m , the lower is thevalue of ˙ γ for which the plateau is reached. Furthermore, when compared to the STS and FRMS, we can see thaton the same values of σ the average deformation D av ismuch smaller in the SHS. Again, this is due to the rotationof the membrane during the loading. In both STS andFRMS, the membrane does not rotate, so that the energyinjected by the flow is used to deform the membrane.These investigations reveal that it is impossible to predictthe loading and relaxation times if we only know thedeformation and have no information about the kind ofmechanical load.In view of the above considerations on the deformation, itappears also natural to study the characteristic times asfunctions of the average deformation D av . We performedthis analysis (see ESI†, Fig. 2), confirming the picturedisplayed in Fig. 3: again, ˜ t shows a collapse for theSTS and the FRMS and does not depend on the valueof the membrane viscosity µ m ; for the SHS ˜ t shows adependency on both D av and the membrane viscosity µ m .The results on t L ( D av ) and t R ( D av ) (Fig. 2 in ESI†, panels(a) and (b), respectively) further confirm that in generalthere is no correlation between the degree of deformationof the membrane and the characteristic times for differentkinds of mechanical loads.In our previous work [33], we have already seen that, for mall forces, t R is linear in µ m , in agreement with literaturepredictions [19]. Now we can go further, and we studythe dependency of both t L and t R as functions of µ m fordifferent values of σ . This will help further to determine towhat degree these two kinds of dynamics can be regardedas different dynamics [43, 42]. In Fig. 5, we reportboth t L ( µ m ) and t R ( µ m ) (first and second row of panels,respectively) for three values of σ spanning two orders ofmagnitude as well as their linear fit (whose coefficients arereported in ESI†, Tab. 1) for all three simulations (STSin panel (a) and (d); SHS in panel (b) and (e); FRMS inpanel (c) and (f)). In all three simulations, for a fixedvalue of σ , the linear approximation is reasonably good.For small values of σ (e.g, σ = 0 . Pa), both t L and t R are similar for all three simulations; that is not surprising,since at small values of stress σ the intrinsic properties ofthe membrane arise. Regarding the sensitivity of the lineartrend with respect to a change in σ , we observe differentbehaviours in the two dynamics. Regarding the loadingdynamics, we observe that for high values of σ , the threeload mechanisms provide similar linear fits, while inthe intermediate region of σ , the SHS shows a differentbehaviour with respect to the STS and FRMS. Regardingthe relaxation dynamics, the variability of the linear trendswith the value of the stress is more pronounced in presenceof hydrodynamical forces (i.e. SHS and FRMS), whilein the STS the linear behaviour of t R with respect to µ m is only slightly perturbed by a change in the stress ifcompared to the others. These quantitative observationsare summarized in Tab. 1 in ESI†.Before concluding this section, a few words must besaid about the parameter δ used to improve the fit (seeEqs. (12)-(13)-(14)): in general, in all three setups, δ is close to one, especially in the STS and FRMS (seeFig. 3 loading in the SHS, where the parameter δ seemsto tend asymptotically to δ ≈ . at increasing valuesof shear rate ˙ γ : this deviation from 1 reflects the effectof the kind of mechanical load also on δ , showing that,during the loading in the SHS, D ( t ) is not that close to anexponential function, and multiple loading times arise [33]. A comprehensive characterisation of the viscoelastic prop-erties of the RBC membrane, as well as the way themembrane responds to an external force, are crucialinformation in different fields, from the detection ofpathologies [11, 16, 19], to the design of biomedical de-vices [24, 25, 26, 27]. Having a more quantitative informa-tion on the time-dependent response of the RBC membraneat varying the intensity and the typology of the mechani-cal load is a key ingredient in designing some biomedicaldevices. One example comes from the VADs [62]: ifthe residence time inside the impeller (that is the regionwhere the RBCs experience a wide range of stress) is muchshorter then the loading time, RBCs do not have time todeform; on the contrary, a higher residence time leads to a finite deformation that can cause hemolysis.When an external force acts on a viscoelastic membrane,two main kinds of dynamics arise: the loading and the relaxation dynamics with associated times t L and t R . Ear-lier investigations pointed to the fact that these two kindsof dynamics are two distinct processes, since during therelaxation there is no external force to drive the mem-brane, in contrast to the loading [43, 42]. To the bestof our knowledge, however, an exhaustive comparativecharacterization of these two kinds of dynamics has neverbeen conducted. This motivated our work to investigatethese two kinds of dynamics with different setups thatinvolve different typologies of mechanical loads (whosemain features are summarised in Tab. 1) while performinga parametric study on the values of membrane viscosity µ m . The latter choice is motivated by the large variabil-ity of membrane viscosity values reported in the litera-ture [46, 18, 45, 47, 22, 48, 49, 17, 19, 32].The two kinds of dynamics are symmetrical ( ˜ t = t L /t R → ) in the limit of small load strengths ( σ → ), i.e., inthe limit where the response function of the RBC is dom-inated by the "intrinsic" properties of the membrane; inmarked contrast, we found an asymmetry in the two kindsof dynamics for finite load strengths ( ˜ t = t L /t R = 1 for σ > ), meaning that the loading dynamics is always fasterthan the relaxation one. We found that the asymmetry pro-foundly depends on the kind of mechanical load and wehave demonstrated this non-universality via a quantitativestudy in terms of the applied load strength σ and the valueof membrane viscosity µ m . There are some realistic loadmechanisms, like shear flows, that make the membranerotate during loading, while leaving the membrane relax-ing to the shape at rest without rotation: in this case, thecontribution that the membrane viscosity gives to the char-acteristic times t L and t R differs, and then the ratio ˜ t isa function of both the stress σ and the membrane viscos-ity µ m . From the other side, there are other realistic loadmechanisms, like the stretching with optical tweezers orthe deformation with an elongational flow, in which themembrane deforms without rotating during both processes.In this case, the contribution given by the membrane vis-cosity µ m to the characteristic times is the same duringboth loading and relaxation, and as a consequence, the ra-tio ˜ t is a function of the stress σ only. Finally, even thoughwe showed that both loading and relaxation dynamics arenot universal, we found that for a given value of the stress σ , a linear increase of the characteristic times as functionsof the membrane viscosity µ m is a fair approximation inall cases.We argue our findings offer interesting physical and prac-tical insights on the response function and the unsteadydynamics of RBCs driven by realistic mechanical loads. Conflicts of interest
There are no conflicts to declare. cknowledgements The authors acknowledge L. Biferale and G. Koutsou. Thisproject has received funding from the European UnionHorizon 2020 Research and Innovation Program under theMarie Skłodowska-Curie grant agreement No. 765048.We also acknowledge support from the project “DetailedSimulation of Red blood Cell Dynamics accounting formembRane viscoElastic propertieS” (SorCeReS, CUP No.E84I19002470005) financed by the University of Rome“Tor Vergata” (“Beyond Borders 2019” call).
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Fabio Guglietta
Department of Physics & INFN,
University of Rome “Tor Vergata” ∗ Chair for Computational Analysis of Technical Systems (CATS),
RWTH Aachen University † Computation-based Science and Technology Research Center,
The Cyprus Institute ‡ [email protected] Marek Behr
Chair for Computational Analysis of Technical Systems (CATS),
RWTH Aachen University † Giacomo Falcucci
Department of Enterprise Engineering “Mario Lucertini,”
University of Rome “Tor Vergata" § Mauro Sbragaglia
Department of Physics & INFN,
University of Rome “Tor Vergata” ∗ ∗ Via della Ricerca Scientifica 1, 00133, Rome, Italy † ‡
20 Konstantinou Kavafi Str., 2121 Nicosia, Cyprus § Via del Politecnico 1, 00133, Rome, Italy a r X i v : . [ c ond - m a t . s o f t ] F e b Shear simulation (SHS): characteristic time t cosL In Fig. 1, values of t cosL are reported (see Eq. (13) in main paper) as a function of the shear rate ˙ γ for different values ofmembrane viscosity µ m . While t L measures the time the membrane takes to reach the average deformation D av , thecharacteristic time t cosL measures the period of the oscillations. In Fig. 2 we report the characteristic times t L and t R (see Eqs. (12)-(14) in main paper) as well as the ratio ˜ t = t L /t R asfunction of the average deformation D av . δ parameter In Fig. 3 we report the values of the parameter δ used to interpolate t L , t cosL and t R (see Eqs. (12)-(14) in main paper). Four movies showing the loading-relaxation simulations for all three kinds of mechanical loads simulated are availablein the Ancillary Files:• STS: stretching simulation with F = 50 × − N and µ m = 3 . × − m Pa s (see Sec. 3.1 in mainpaper).• SHS: shear simulation with ˙ γ = 123 s − and µ m = 3 . × − m Pa s (see Sec. 3.2 in main paper).• SHS_mum0: shear simulation with ˙ γ = 123 s − and µ m = 0 . m Pa s (see Sec. 3.2 in main paper).• FRMS: four-roll mill simulation with F roll = 10 s − and µ m = 3 . × − m Pa s (see Sec. 3.3 in mainpaper). 2 Supplementary figures t c o s L [ − s ] ˙ γ [s − ] . . . µ m [ − m P a s ] Figure 1: Values of t cosL as a function of the shear rate ˙ γ (see Eq. (13) in main paper) for different values of membraneviscosity µ m (from lightest to darkest color): µ m = 0 . m Pa s ( ), µ m = 0 . × − m Pa s ( ), µ m = 1 . × − m Pa s ( ), µ m = 3 . × − m Pa s ( ). . . . . . (a) . . . . . (a) (b) . . . . . . . . . . (a) (b) (c) t L [ − s ] D av STSSHSFRMS . . . µ m [ − m Pa s] t R [ − s ] D av ˜ t D av Figure 2: Comparison between the characteristic times t L (panel (a)), t R (panel (b)) and the ratio ˜ t = t L /t R (panel(c)) as functions of the average deformation D av (see Sec. 4 in main paper) for the three simulations performed, i.e.,stretching simulation (STS, ), shear simulation (SHS, ), four-roll mill simulation (FRMS, ), for different values ofmembrane viscosity µ m (from lightest to darkest color): µ m = 0 . m Pa s ( , , ), µ m = 0 . × − m Pa s ( , , ), µ m = 1 . × − m Pa s ( , , ), µ m = 3 . × − m Pa s ( , , ).3 . . . . . . . . (a) . . . . . . . . (a) (b) . . . . . . . . (a) (b)(c) . . . . . . . . (a) (b)(c) (d) . . . . . . . . (a) (b)(c) (d)(e) . . . . . . . . (a) (b)(c) (d)(e) (f) δ - L o a d i n g F [ − N]STSSHSFRMS . . . µ m [ − m Pa s] δ - R e l a x a t i o n F [ − N] δ - L o a d i n g ˙ γ [s − ] δ - R e l a x a t i o n ˙ γ [s − ] δ - L o a d i n g F roll [s − ] δ - R e l a x a t i o n F roll [s − ] Figure 3: Values of the parameter δ used to interpolate t L , t cosL and t R (see Eqs. (12)-(14) in main paper) are reported forthe three simulations performed, i.e., stretching simulation (STS, , panels (a-c), Sec. 3.1), shear simulation (SHS, ,panels (d-f), Sec. 3.2), four-roll mill simulation (FRMS, , panels (g-i), Sec. 3.3), for different values of membraneviscosity µ m (from lightest to darkest color): µ m = 0 . m Pa s ( , , ), µ m = 0 . × − m Pa s ( , , ), µ m = 1 . × − m Pa s ( , , ), µ m = 3 . × − m Pa s ( , , ).4 Supplementary tables
STS SHS FRMSL R L R L R σ = 0 . Pa a . ± . . ± . . ± . . ± . . ± . . ± . b . ± . . ± . . ± . . ± . . ± . . ± . σ = 0 . Pa a . ± . . ± . . ± . . ± . . ± . . ± . b . ± . . ± . . ± . . ± . . ± . . ± . σ = 0 . Pa a . ± . . ± . . ± . . ± . . ± . . ± . b . ± . . ± . . ± . . ± . . ± . . ± . σ = 0 . Pa a . ± . . ± . . ± . . ± . . ± . . ± . b . ± . . ± . . ± . . ± . . ± . . ± . σ = 0 . Pa a . ± . . ± . . ± . . ± . . ± . . ± . b . ± . . ± . . ± . . ± . . ± . . ± . σ = 0 . Pa a . ± . . ± . . ± . . ± . . ± . . ± . b . ± . . ± . . ± . . ± . . ± . . ± . σ = 0 . Pa a . ± . . ± . . ± . . ± . . ± . . ± . b . ± . . ± . . ± . . ± . . ± . . ± . Table 1: Coefficients a L , a R , b L and b R used to fit t L ( µ m ) and t R ( µ m ) for the STS, SHS and FRMS at some fixed valueof the stress σ . We use linear fits, t FIT L = a L + b L µ m and t FIT R = a R + b R µ m , respectively (see Fig. 5 in main paper).Values of a are expressed in [ − s], while values of b are expressed in [ m − Pa −1