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A Discrete-Particle Model of Blood Dynamics in Capillary Vessels

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A Dscrete-Partcle Model of Blood Dynamcs n Capllary Vessels Wtold Dzwnel 1,2, Krzysztof Boryczko and Davd A.Yuen 2 1 AGH Insttute of Computer Scence, Al.Mckewcza 30, , Kraków, Poland 2 Mnnesota Supercomputer
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A Dscrete-Partcle Model of Blood Dynamcs n Capllary Vessels Wtold Dzwnel 1,2, Krzysztof Boryczko and Davd A.Yuen 2 1 AGH Insttute of Computer Scence, Al.Mckewcza 30, , Kraków, Poland 2 Mnnesota Supercomputer Insttute, Unversty of Mnnesota, Mnneapols, Mnnesota 55455, USA Abstract We nvestgate the physcal mechansm of aggregaton of red blood cells (RBC) n capllary vessels, usng a dscrete partcle model. Ths model can accurately capture the scales from 0.001µm to 100µm, far below the scales, whch can be modeled numercally wth classcal computatonal flud dynamcs. We use a dscrete-partcle model n 3D for modelng the flow of plasma and RBCs n a capllary tube. The two stuatons nvolvng neckng and no neckng have been consdered. The flexble vscoelastc red blood cells and the walls of the elastc vessel are made up of sold partcles held together by elastc harmonc forces. The blood plasma s represented by a system of dsspatve flud partcles. We have smulated the flow of cells of dfferent shapes, such as normal and sckle cells. The cells coagulate n spte of the absence of adhesve forces n the model. The total number of flud and sold partcles used ranges from 1 to 3 mllon. We conclude that aggregaton of red blood cells n capllary vessels s stmulated by depleton forces and hydrodynamc nteractons. The cluster of sckle cells formed n the neckng of the condut effcently decelerates the flow, whle normal cells can pass through. These qualtatve results from numercal smulatons accord well wth laboratory fndngs. Keywords: dscrete partcles, vscoelastc blood flow, elastc capllary vessels, flud partcle model submtted to Journal of Collod and Interface Scence, February Introducton As a physologcal flud, blood s a complex suspenson of polydsperse, flexble, chemcally and electrostatcally actve cells, whch are suspended n an electrolytc flud consstng of numerous actve protens and organc substances. The rheology of blood has been studed for many years expermentally, theoretcally and numercally [1-6]. For macroscopc blood vessels of a dameter more than two orders of magntude greater than the sze of a red cell, blood can be consdered as a homogeneous flud. Therefore, macroscopc blood dynamcs can be modelled by solvng classcal hydrodynamc equatons. An overvew of the most recent numercal methods for modelng vascular flow n macroscale can be found n Quarteron, SIAM News, 2001 [3]. Mechancal propertes of blood are usually descrbed by a consttutve equaton, whch s constructed on the bass of expermental vscometrc data. The dervaton of such an equaton applcable for the macroscale s non-trval. If blood s tested dynamcally t would behave lke a nonlnear vscoelastc flud. Moreover, blood s thxotropc, that s, ts vscoelastc characterstcs change wth the level of stran and stran hstory [1,7]. For example, at a vanshng shear rate the blood behaves lke an elastc sold. The consttutve equatons are non-lnear, complex and not well constraned. Takng nto account the complex boundary condtons assocated wth varous geometry of blood vessels and elastc nteractons between vessel wall and blood flow, the smulaton of realstc hydrodynamc behavor of blood n macroscopc vessels s ndeed a very challengng undertakng. Macroscopc vessels represent only a small fracton of crculatory system, although the largest vens contan 50% of blood. The vascular tssue s made of mcroscopc capllary channels. There are about blood vessels whose dameters are comparable wth the dmensons of the red blood cells (RBC).e., 5-10µm [1,6]. Therefore, the majorty of defects n crculatory blood system occurs n capllary vessels where blood flows less vgorously than n larger macroscopc vessels. Fg.1 Rouleaux aggregates of red blood cells. Ths fgure comes from R de Roeck and M.R.Mackley, Department of Chemcal Engneerng, Unversty of Cambrdge (from [5]) Thromboss s an mportant defect n crculatory system. It s a major cause of most heart attacks and of other severe cardovascular problems, such as schema and angna. The two major components of the thrombotc process are aggregaton and coagulaton. Aggregaton nvolves platelets, a cellular consttuent of the blood. The processes of cell-cell and cell-substrate adheson 2 wthn the movng blood are connected wth a very complex actvaton mechansms nvolvng changes n platelet s surface membrane and chemcal composton of the blood plasma n the neghborhood of njury [1,9]. It s well known that human red blood cells can form aggregates known as rouleaux [1,7] (see Fg.1) whose formaton depends on the presence of the protens fbrnogen and globuln. The slower the blood flow, the smaller s the shear rate and the larger are rouleaux. Neckng of the mcroscopc vessels caused, e.g., by accumulaton of cholesterol plagues n the vessel walls, slows down the flow. Ths stmulates the creaton of larger and larger thromb, resultng n a postve feedback loop, whch can eventually choke the flow. Blood coagulaton s a physologcal mechansm for stoppng leaks n blood vesels. At stes of damaged vascular endothelum, platelets and neutrophls are captured from the flow stream whle a proten, tssue factor, localzed on the vessel wall can bnd a specfc enzyme to ntate a cascade of coagulaton reactons on cellular surfaces, resultng n the converson of prothrombn to thrombn [1,8,9]. Coagulaton nvolves a network of more than 30 enzyme reactons wth numerous forward and feedback loops, presumably for the dual purposes of control and amplfcaton of the ntal perturbaton. Aggregaton and coagulaton are ndependent processes that only nteract n the end, when the fbrn mesh forms on the platelet aggregate. A deep understandng of coagulaton and aggregaton processes s of major medcal mportance [9]. Both coagulaton and aggregaton nvolve the mutual bndng of red blood cells (RBC). Bndng depends on the followng factors: 1) The presence of partcular plasma protens. 2) The mposed flow condtons. 3) The deformaton of RBC. Numerous studes have been performed on nvestgatng the mechansms of red cell aggregaton, resultng n two popular hypotheses: the brdgng mechansm and the depleton layer hypotheses [1]. It s commonplace knowledge that shearng condtons can accelerate the aggregaton process. The chokng of flow by neckng s one such the condton. Geometrcal faults n RBC cells caused by blood dseases, can be the followng factor forcng the aggregaton. Numercal models of blood flow, whch are based on macroscopc contnuum hydrodynamc equatons are not suffcent n the physcs for smulatng the feedback dynamcs between mcroscopc flow and aggregated mcrostructures n capllary vessels. Formaton of RBC clusters cannot be smulated by the classcal computatonal flud dynamcs (CFD). In the capllares RBCs must be treated as ndvdual objects of crcular bconcave shape. Consequently, blood n the mcroscale must be regarded as a two-phase, heterogeneous flud consstng of a lqud plasma phase and a deformable sold phase of blood cells. In the macroscopc models the mcroscopc phenomena such as nteractons between cells resultng from depleton and electrostatc forces, chemcal reactons [8], large densty fluctuatons n plasma soluton, are not present or they are averaged out because of ther coarse-graned nature. In capllary channels t s necessary to consder all of these nteractons, ncludng the tght nteractons between red blood cells and the capllary walls. As shown n [10,11,13], dscretepartcle methods, such as dsspatve partcle dynamcs, flud partcles model and mult-level partcle method, can be used for modelng accurately complex fluds n scales rangng from 10nm to 100µm. In ths paper we present a novel dea based on the use of dscrete-partcles for modellng red blood cells dynamcs n capllary vessels. We nvestgate the mechansm of RBC aggregaton n mcroscopc blood channels employng a flud partcle model. Frst, we brefly descrbe the numercal model, focusng on ts effcent parallel mplementaton. Next we go over the results from smulatons nvolvng 1-3 mllon of partcles n 3-D. Fnally, we dscuss the results and summarze the advantages and dsadvantages of ths new approach. 3 Numercal model Blood n capllary vessels can be vewed physcally as a two-phase, heterogeneous flud consstng of a lqud plasma phase and an elastcally deformable sold phase flled wth blood cells. The plasma, rch n many other blood components such as lpds, enzymes and platelets, cannot be studed n the mcroscale as a contnuous medum but rather as an electrolytc suspenson wth many mcrostructural components. As shown n [10-13], the mesoscopc scales present n complex flud can be accurately modelled usng the dscrete partcle approxmaton. In the model two types of partcles are defned accordngly: Flud partcles (FP), whch represent a porton of plasma flud lumps of flud suspenson from Fg.2a. The nteractons between these partcles are defned by the collson operator from flud partcle model (FPM) [14]. FPM s an extenson of the dsspatve partcle dynamcs (DPD) [15] method. Sold partcles (SP), the peces of matter defnng nodes of the elastc grd. They represent the sold components of the vascular system,.e., the vessel walls and red blood cells shown n Fg.2b,c,d. Interactons among the partcles Flud partcle model (FPM), descrbng the plasma flud, dffers from the dsspatve partcle dynamcs (DPD) frst descrbed by by Hoogerbrugge and Koelman [15]. The flud partcles can rotate n space and can be understood as relatvely large mass packets, although they are stll partcles n the statstcal mechancs sense. The nteractons between the partcles are represented by the collson operator of a fnte range (unlke n smoothed partcle dynamcs (SPH) [16]) and a broad class of partcle methods [17] where they are derved n a canoncal manner from the force laws of contnuum mechancs and are drectly based on a regularzed stress tensor. In contrast to classcal dsspatve partcle dynamcs, the nteracton range for flud partcle model can be shorter due to more realstc nteracton potental between droplets n FPM. It reduces some of the defcences n DPD model [14], whch are usually compensated by a larger cut-off radus n the DPD nteractons. The flud partcles possess several attrbutes ncludng mass m, poston r, translatonal v and angular v veloctes and type (see Fg2a). The droplets nteract wth each other va a collson operator Ω standng for the two-body, short-ranged force [14]. Ths type of nteracton s a sum of the conservatve F C, two dsspatve components F T and F R (translatonal and rotatonal) and a Brownan forcef ~. Summarzng: Ω = V 1 ~ S ~ 1 ~ A ( rj ) e j m( A( rj ) 1+ B( rj ) ej ) o v + rj ( ϖ + ϖ j ) e j + σ A( rj ) dw j + B( rj ) tr[ dw ] 1 + C( rj ) dwj oe j γ (1) 2 where: r j = r r j s a vector pontng from partcle to partcle j and e j =r j /r j, D s the model dmenson, dt s the tmestep, γ - scalng factor for dsspaton forces, dw S, dw A, tr[dw]1 - are respectvely the symmetrc, antsymmerc and trace dagonal random matrces of ndependent ~ ~ ~ Wener ncrements and A(r), B(r), C(r), A( r), B( r), C( r),v (r) are normalzed weght functons dependent on the separaton dstance r=r j. We have assumed that f a dstance between partcles and j, r j = r j, s greater than a cut-off radus R cut, the value of Ω=0. As shown n Español [14], the sngle-component FPM system yelds the Gbbs dstrbuton as the steady-state soluton to the Fokker-Planck equaton. Consequently, t obeys the fluctuaton dsspaton theorem, whch defnes the relatonshp between the normalzed weght functons. D 4 Fg.2 a) Flud partcle model, b) RBC model made of flexble partcles, c) RBC model after volume renderng, (RBC surface come from Data Explorer renderng) d) real mage of deformed RBC extracted from Snce the rheology of the plasma s approxmately Newtonan [1], there s lttle doubt that the non-newtonan features of human blood come from the red blood cells (RBC). The nteracton of the blood cells wth the blood flow becomes the key ssue. Normal human RBC assumes a bconcave dscod shape wth an averaged dameter of 8µm, thckness of 2µm [1,7]. Red blood cells are known to change shape n response to the local flow condtons. Deformaton affects the physologcal functon of the red blood cell and ts hydrodynamc propertes. At a concentraton of 50%, a suspenson of rgd spheres cannot flow, whereas blood s flud even at 98% concentraton n volume [1]. The cells n our model consst of rectangular grd of sold partcles (SP) (see Fg.2). The partcles nteract wth partcles n ther Moore neghborhood [23] by conservatve elastc forces F C =F H where: F H = χ ( r ) e (2) j a j Besdes conservatve force, the collson operator for SPs ncludes the addtonal dsspatve component smlar to F T for flud partcles. Ths artfcal vscosty prevents RBC from breakngup due to collsons wth the fast partcles. In the model we assume that the value of a j {1, 2, 3}. As shown n Fg.2a, a j depends on the poston of neghborng partcle n the Moore neghborhood. The elastcty of the object made of sold partcles s not only a functon of χ but also depends on the type of grd assumed and ts resoluton. At a fner resoluton the radus j 5 of nteracton should be extended to the neghborng layers for matchng to the requred elastcty. The lack of a self-consstent procedure for matchng the real materal parameters (e.g., elastcty) s a drawback of ths model. Couplng the mmersed boundary method and neo-hookean membrane model [6] wth a partcle approach gves an approxmate formula for matchng nterpartcle forces parameters to realstc propertes of blood cells. We have assumed that there are no attractve forces between cells,.e., the partcles from dfferent cells (and channel wall) rebound due to conservatve repulsve forces smulated here by the repulsve part of the Lennard-Jones force. In realstc blood the presence of partcular plasma protens, notably fbrnogen and mmuno-globulns, plays an mportant role n the aggregaton of blood cells. Many laboratory studes on blood flow suffer from nablty to elmnate thrombn, whch actvates platelets, as well as preventng fbrn formaton, whch stablzes platelet deposts [8]. Neglectng ths factor allows us to examne other elements responsble for aggregaton. They are drectly caused by the capllary flow, RBC elastcty and the exstence of depleton forces. Depleton forces are volumetrc forces wth an entropc character, resultng from the granularty of the plasma suspenson represented n the model by flud partcles. The nteractons between flud partcles and cells are mmcked numercally by nteractons between flud partcles and grds of sold partcles representng cells. We assume also that the forces between flud partcles and sold partcles are smlar to those gven n Eq.(1) One can consder more complcated systems n whch flud partcles nteract wth surfaces coverng the SP grd, rather than wth the separate SP partcles. Along the blood vessel wall there s a layer of endothelal cells. These cells cannot move, but they can deform. They respond to the shear exerted on the vessel wall by the flowng blood. The cells form a contnuous layer through whch any exchange of matter between the tssue and the blood takes place. In the model the blood channel s made of massve partcles hollowed out computatonally (see Fg.2b). The blood vessel conssts of several layers of partcles to prevent the plasma partcles from leakng outsde of the channel. The wall partcles nteract wth one another wth forces smlar to sold partcles n RBC cells. In contrast to RBC, the Brownan force for wall partcles s non-zero n order to avod excessve energy dsspaton from the system and to smulate the random deformaton of wall. Interactons between the wall and both plasma and sold partcles are repulsve n character and are gven by Eq.1. Tmesteppng and numercal mplementaton The temporal evoluton of the partcle ensemble obeys the Newtonan laws of moton: 1 v& = Ω j( r, v, ϖ ) r & = v (3) m ; 1 j r j r cut ( r, v, ϖ ) & = Nj N j rj F I j j; r j r cut ϖ = 2 1 The equatons of moton represent stochastc dfferental equatons (SDE) due to stochastc nature of the Brownan force component. The leap-frog numercal scheme as n [15] represents only a crude approxmaton of the stochastc ntegrator. It generates serous artfacts, leadng to unphyscal correlatons and monotoncally ncreasng (or decreasng) temperature drft. Due to the large nstabltes observed for the leap-frog scheme, we have used a hgher-order temporal O( t 4 ) scheme for ω [10-13]: n+ 1/ 2 = 2 n 1/ 2 n 2 / 3 n n ( N N 1 ) ϖ ϖ ϖ + (5) (4) 6 whle the values of v n+1 and w n+1 are predcted by usng the O( t 2 ) Adams-Bashforth procedure. As shown n [11], both the hydrodynamc temperature and hydrodynamc pressures do not exhbt a notceable drft for one mllon tmesteps. For smulatons requrng more accurate conservaton of thermodynamc quanttes, another ntegrator, whch uses a thermostat, should be employed. We consder here an sothermal three-dmensonal system, whch conssts of M partcles confned n a long cylnder wth perodc boundary condtons n the z drecton. The flow s drected from top to the bottom. The partcle system s accelerated by an external force correspondng to a gven pressure gradent. For the mult-component system of flud and sold partcles wth dfferent nteracton ranges we have assumed that cut-off radus R cut ~max k (R cut,k ). The partcular value of k ndcates the type of nteracton. In these smulatons we used the same R cut for all the nteractons excludng nteractons between wall partcle-wall partcle and RBC partcle-rbc partcle (for the same cell), whch uses nvarable neghbor lsts. The forces are computed by usng O(M) order lnk-lst scheme [18]. The force on a gven partcle ncludes contrbuton from both sold and flud partcles that are closer than R cut and whch are located wthn the cell contanng the gven partcle or wthn the adjacent cell. The procedures for computng the collson operator and ntegratng Newton s equatons are dsplayed n Fg.3a. Match harmonc forces to nput data Match FPM forces to nput data communcaton overlay Generate ntal condtons P3 P4 Compute W j Central forces Ω = V j r j T ( ) e + F V(rj) harmonc potental j IF Fj Sold partcle - flud partcle FPM forces C T R ~ Ω = F + F + F + F j j j and F R non-central j j F T P1 Sold partcle r & = v v & 1 = m j F leap- frog+ Adams- Bashford j Move partcles a f FPM partcle r & = v v 1 & = m j F j same as DPD 1? & = I j N hgher order scheme j Fg.3 A dagram of partcle model mplementaton, b) geometrc decomposton of the partcle system nto dfferent processors P (plasma s not shown). As shown n Fg.3b, the parallel algorthm s facltated by geometrc decomposton of the tube onto P domans and by mappng them onto P processors. By usng SPMD paradgm (sngle program multple data), commonly used for parallelzaton of MD code, each processor follows an dentcal predetermned sequence of operaton to calculate the forces actng on the partcles wthn an assgned doman. Ths code s wrtten n FORTRAN 95 wth a MPI envronment. The detals of the parallel FPM algorthm and the speedups obtaned on up to 32 processors of IBM SP and SGI/Orgn 3800 systems at the Mnnesota Supercomputng Insttute are descrbed n [12]. P0 b P2 7 Results The FPM method can predct the transport propertes of the flud, thus allowng one to adjust the model parameters such as: densty, temperature, nternal pressure and vscosty, by usng the lmt of the contnuum equatons [14]. The plasma represents a suspenson of protens, enzymes and other cells (e.g., platelets, leukocytes etc.). We have assumed that ts densty and vscosty s 2% hgher than water. Structurally, red blood cells are bconcave dscs of dmensons 8µm dameter, 2.5 µm thck at the edge and 1 µm thck at the center. They can be envsaged as soft bags contanng hemoglobn. The RBC s a hghly deformable entty, as demonstrated when cells pass through capllares wthn the body, the dameters of whch are of the order 2-4 mµ. The capllary dameter s about 1,5-3 tmes larger than the dameter of the cell. The cells were constructed by usng partcles on strngs model descrbed n the prevous secton (see Fg.2). Lkewse we have fabrcated deformed sckle cells and crescent-roll cells representng some blood dseases. We have
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