Journal of Biomechanical Science and Engineering

Transcription

1 Scence and Engneerng Smulaton Study on Effects of Hematocrt on Blood Flow Propertes Usng Partcle Method * Ken-ch TSUBOTA, ** Shgeo WADA *** and Takam YAMAGUCHI ** **Department of Boengneerng and Robotcs, Graduate School of Engneerng, Tohoku Unversty, Aoba, Senda , Japan E-mal: tsubota@pfsl.mech.tohoku.ac.jp, takam@pfsl.mech.tohoku.ac.jp ***Department of Mechancal Scence and Boengneerng, Graduate School of Engneerng Scence, Osaka Unversty, 1-3 Machkane-yama, Toyonaka , Japan E-mal: shgeo@me.es.osaka-u.ac.jp Abstract The effects of hematocrt (Hct) on blood flow n mcrocrculaton were nvestgated by computer smulaton usng a partcle method. Deformable red blood cells (s) and blood plasma were modeled by assembly of dscrete partcles. It was assumed that an conssted of an elastc membrane and nner vscous flud, and that plasma was vscous flud. The partcles for the membrane were connected wth ther neghborng membrane partcles by stretch/compresson and bendng sprngs. The moton of all the partcles that was subjected to ncompressble vscous flow was solved by the movng partcle sem-mplct (MPS) method based on Naver-Stokes (NS) equatons. The forces nduced by the sprngs to express the elastc membrane were substtuted nto the NS equatons as the external force, whch enabled coupled analyss of elastc moton and plasma flud flow. Two-dmensonal smulatons of blood flow between parallel plates were carred out for varous Hct values. As a result, t was shown that at hgher Hct, s were less deformed nto a parachute shape durng ther downstream moton, ndcatng that mechancal nteracton between s restrcted the deformaton. Mechancal nteracton between s had a sgnfcant nfluence on deformaton and the velocty profle of plasma flow when the Hct value was more than 0.0~0.30. Apparent blood flow resstance ncreased wth Hct, correspondng to prevously reported n vtro expermental results compled to an emprcal formula. Key words: Computatonal Bomechancs, Mcrocrculaton, Red Blood Cells, Deformablty, Plasma Flow, Flud-Sold Coupled Problem, Rheology 1. Introducton *Receved 17 May, 006 (No ) [DOI: /jbse.1.159] Blood s a suspenson of elastc red blood cells (s) n vscous plasma flud (1) (3). In mcrocrculaton, the volumetrc rato of s to whole blood, so called hematocrt (Hct), greatly affects the rheologcal propertes of the blood (1), (), (4). Extensve n vtro expermental studes have been conducted to establsh an emprcal formula that relates blood flow resstance to Hct (1), (), (4). Ths phenomenon ncludes the mechancal factors of (1) the shape and deformaton of an and () mechancal nteracton between s and plasma. Furthermore, these mechancal factors lead to (3) to nteracton va plasma flud. Theoretcal and computatonal approaches have been useful for understandng the role of these mechancal factors n blood flow phenomena (), (5). 159

2 However, there s lttle nvestgaton on the combned effects of these mechancal factors on blood flow dependng on Hct because establshed numercal methods (usually wth Euler grds) have not enabled the computaton of these factors at the same tme. Recently, the motons of multple s have been nvestgated by computer mechancal smulaton to explan the mechansm of how the Hct determnes the rheologcal propertes of blood n mcrovasculature. Sun and Munn (5) proposed the two-dmensonal lattce Boltzmann method to smulate blood flow consderng ndvdual blood cells suspendng n plasma. They qualtatvely reproduced the motons of multple blood cells observed n experments, such as the axal mgraton of s and appearance of a plasma layer near the vascular wall, and suggested that the behavor determnes the resultng flow resstance that depends on Hct as observed n experments. However, they modeled s as a rgd body and dd not consder ther deformabltes. On the other hand, Boryczko et al. (6) smulated three-dmensonal blood flow n capllary ncludng multple s usng a dscrete partcle model, and suggested that combned effects of Hct and deformaton play an mportant role n apparent blood flow propertes. However, they modeled an as a sold elastc body, and the nner flud of the was gnored. As denoted above, recent advances n computatonal technques combned wth ncreasng computng power allow us to calculate the complcated mechancal nteracton among blood cells suspendng n plasma. In partcular, a partcle method, whch has been used to analyze flud, sold and flud-sold coupled mechancal problems (7), has begun to be appled to blood flow smulaton (6), (8) (1). Ths method does not need mesh generaton such as that used n the fnte element method, and only uses the dscrete partcles as computng ponts that are traced n Lagrangan coordnates. The moton of each partcle s determned by ts mechancal nteracton wth neghborng partcles, whch s modeled to express the analyzng object. The method s advantageous n drectly modelng each blood component (6), (8) (1), such as a blood cell and plasma, usng an assembly of dscrete partcles that are assgned the characterstc propertes of the correspondng blood component. In addton, the complex mechancal/bologcal nteracton between blood components can be expressed n the blood flow smulaton by takng nto account only the nteracton between these partcles. The purpose of ths study s to nvestgate the effects of Hct on blood flow propertes on the blood cellular scale from the vewpont of computatonal mechancs. A partcle method was used to smulate two-dmensonal blood flow between parallel plates, whch s a smplfed flow model of mcrocrculaton. Blood was consdered as a suspenson of multple s, consstng of an elastc surface membrane and nner vscous flud, n vscous flud of blood plasma. Parametrc smulaton studes on Hct clarfed the relatonshps among Hct, deformaton and apparent blood flow resstance.. Methods.1 Partcle method for blood flow smulaton In the prevous study (1), we proposed a partcle method for blood flow smulaton to nvestgate the moton of a sngle deformable nteractng wth vscous plasma flud. In ths study, ths method was extended to nclude multple s. The smulaton procedures (1) (4) are as follows, whch are schematcally llustrated n Fg. 1. The followng equatons are wrtten for a two-dmensonal problem wth unt length n the thckness dmenson. (1) The blood regon was dscretzed by partcles that are assumed to have the characterstcs of s and plasma, as shown n Fg.. It was assumed that an part conssts of a surface membrane and nner flud partcles. Each partcle has physcal quanttes such as the poston r, velocty u, pressure p, and constant densty ρ. () A sprng network model (13) was appled to membrane partcles to express the 160

3 Fg. 1 Algorthm of computer smulaton of blood flow usng partcle method. Fg. Dscretzaton of s and plasma usng partcles. Partcle moton was determned on the bass of NS equatons usng the MPS method. Partcles for the membrane were connected by sprngs to express the membrane's elastc behavor. elastc behavor of deformable s, n whch the membrane partcles were connected wth the neghborng membrane partcles by sprngs for stretch/compresson L and bendng B, as shown n Fg.. The elastc energy stored n the stretch/compresson sprng due to the change n the length l from ts reference l 0 was expressed as N k l li l0 E =. (1) l I = 1 l0 For bendng, the elastc energy can be expressed as a functon of the angle θ between two neghborng stretch/compresson sprng elements. Here, to avod the foldng of sprng elements, we determned the elastc energy for bendng as N kb θi Eb = tan. () I = 1 In Eqs. (1) and (), I s a sprng element, N s the total number of sprng elements, and k l and k b are sprng constants for changes n length and bendng angle, respectvely. Usng total elastc sprng energy of the membrane, E = E l + E b, (3) the elastc sprng force actng on the membrane partcle was obtaned on the bass of the prncple of vrtual work as E F =. (4) r Ths force was calculated from poston r accordng to the results of vector analyss. (3) Assumng ncompressble vscous flow, the moton of all the partcles was determned under a gven boundary condton usng the movng partcle sem-mplct (MPS) method (14), (15). In the MPS method, partcle moton s modeled on the bass of the equaton of contnuty and the Naver-Stokes (NS) equatons wth a sem-mplct tme-marchng algorthm. The gradent vectors and Laplacan of the scalar quantty φ for partcle, whch appear n the NS equatons, are expressed by consderng the nteracton to the neghborng partcles j as ( ) ( ) d φ j φ φ = rj r w rj r (5) 0 n j rj r 161

4 and d = n λ φ 0 [ ( φ j φ ) w( rj r )] j, (6) respectvely (15). In these equatons, w( r j r ) s a kernel functon that monotoncally decreases wth the dstance r j r between two partcles and j, d s the number of space dmensons, n 0 s the objectve value of the number densty of partcles, and λ s the constant that expresses the ncrease n the statstcal dffuson of the dstrbuton of physcal quanttes (15). Wth respect to membrane partcles, the elastc sprng force descrbed n Eq. (4) was substtuted to the NS equatons as the external force term and explctly solved, whch enabled us to perform a coupled analyss of vscous flud (plasma and nner flud of s) and an elastc membrane ( membrane). Thus, for a membrane partcle a dscrete form of the NS equatons can be wrtten as D u 1 p u 1 = + + F, (7) Dt ρ m where m s the representatve mass of partcle and can be regarded as constant ρd 0 wth d 0 beng the mean dstance between two neghborng partcles. Equaton (7) assumes that membrane partcle can be modeled as an elastc thn membrane surrounded wth the flud n a volume d 0. (4) The procedures () (3) were repeated to obtan the change n blood flow wth tme. There are several methods of expressng the elastcty of s n blood flow smulaton wth a partcle method. Boryczko et al. (6) connected all the partcles of the part, ncludng both the membrane and nner flud. Therefore, the nner flud of an was modeled as an elastc sold but not as a flud. In another approach, an elastc membrane can be expressed by at least a few layers of elastc partcles. However, ths method would need an extensve number of partcles to dscretze the entre smulaton regon. In ths study, the partcles of an membrane were modeled by an assembly of partcles n one layer and they were explctly connected by stretch/compresson and bendng sprngs. Ths gves a practcal way to smulate the large deformaton of s consderng the mechancal propertes of both the elastc membrane and nner flud of an.. Two-dmensonal model of blood flow between parallel plates A two-dmensonal smulaton model was constructed for blood flow between parallel plates, as shown n Fg. 3. The model conssted of s, plasma and rgd plates. The length of the flow channel L and the dstance between the plates D were 90.0 µm and 9.0 µm, respectvely. Bconcave s were arranged at equal dstances d apart, and ther long axes were set perpendcular to the flow drecton. The bconcave shape was obtaned as the fnal state of the shape change smulaton based on the sprng network model (13), the detals of whch are descrbed n the Appendx. The blood flow models were constructed for varous Hct values from 0.10 to 0.49 by adjustng the dstance d between s from.0 µm to 4.5µm. As a boundary condton, a constant and unform velocty u 0 = m/s was Fg. 3 Two-dmensonal model of blood flow between parallel plates. Bconcave s were placed wth varous dstances d to adjust Hct values from 0.10 to

5 appled to the nlet, where the Reynolds number wth respect to the plate separaton dstance D was Inflow of s was ntermttent, and the nterval of nflow was determned from the Hct value. A zero-pressure condton was appled to the outlet. A nonslp condton was assumed between the plasma and the, and between the plasma and the plates. The vscosty and densty were Pa s and kg/m 3, respectvely, set to be the same as those of water. The mean dstance between two neghborng dscrete partcles was set to 0.5 µm, and thus the smulaton area contaned approxmately partcles. The sprng constants of the membrane were set as k l = N m n Eq. (1) for stretch/compresson, and k b = N m n Eq. () for bendng (see the Appendx for sprng constant values). The reference length of the stretch sprng was l 0 = 0.5 µm, the same as the mean dstance between two neghborng partcles. 3. Results and Dscusson 3.1 deformaton and movement The dynamcal behavor of elastc s n flowng blood between parallel plates was nvestgated by computer smulaton. Fgure 4 shows the characterstc movement and deformaton of s n blood flow n the case of Hct = The smulaton tme t s normalzed by the tme T 0 = L/u 0. The s moved downstream n the flowng plasma at a constant velocty. At the onset of the flow (t/t 0 = 0 and 0.10), the s mantaned ther concave shape at the upstream and were deformed nto a convex shape at the downstream, smlar to the parachute shape observed n experments (). From t/t 0 = 0.30 to 0.60, the s were kept deformed nto a parachute shape, and the blood flow reached a steady state. Fgure 5 shows the mechancal behavor of s n blood flow at tme t/t 0 = 1.0 for varous Hct values from 0.10 to Ths parametrc study on Hct demonstrated that at hgher Hct, s were less deformed. The degree of deformaton of s was quantfed by a deformaton ndex ε = (h h 0 )/h 0, where h and h 0 are the projecton length of an aganst the cross secton of the flow channel and ts ntal value, respectvely, as shown n Fg. 6. Ths fgure shows the tme course of the change n the mean value of the deformaton ndex, ε M, n the mddle part of the flow channel wth a length of L/3 ndcated by the dotted box n Fg. 5. Deformaton ndex ε M monotoncally ncreased for all the Fg. 4 Smulaton results of tme course of change n behavor n blood flow n the case of Hct = The open-crcle marker nsde an ndcates the same at a dfferent tme. 163

6 Inlet (u = u 0 ) 0 150[Pa] Outlet Flow Hct = 0.10 Hct = 0.0 Hct = 0.30 Hct = 0.40 Hct = 0.49 Fg. 5 behavor n blood flow at normalzed tme t/t 0 = 1.0 for varous Hct values from 0.10 to The mddle part of the flow channel wth a length of 3/L ndcated by a dotted box s used to evaluate apparent flow propertes n Fgs. 6, 7, 11 and 1. Fg. 6 Change n deformaton ndex of s n mddle part of flow channel for varous Hct values. Fg. 7 Mean value of deformaton ndex over tme from tme t/t 0 = 1.0 to 3.0 as functon of Hct. values of Hct, and remaned constant from tme t/t 0 = 1.0 to 3.0, although the value was fluctuated due to the numercal nstablty of the partcle method and the ntermttent nflow condton of s. Fgure 7 shows the mean value of the deformaton ndex over the tme from t/t 0 = 1.0 to 3.0, ε M, as a functon of Hct. Average deformaton ndex ε M remaned constant when Hct was smaller than 0.0, and monotoncally decreased wth ncreasng Hct. The deformaton ndex ε M for Hct = 0.49 was smaller by 37% than Hct = The result agrees wth the suppressng effects of Hct on deformaton observed n n vtro experments (16). Prevous expermental work suggests that hgh Hct n tubes greater than 7 µm can cause transton of flow from sngle-fle wth parachute shapes to mult-fle wth slpper shapes (). At hgh Hct n our smulatons mult-fle flow was not predcted. Ths could be because the ntal arrangement n our smulaton was set to straght sngle-fle. The mult-fle flow observed expermentally could be explaned by an ntal random 164

7 arrangement. Thus future work s necessary to test the effects of ntal arrangement and the causes of sngle- to mult-flow transtons. 3. Mechancal nteracton among s and plasma Fgure 8 shows the flud force actng on an membrane n the cases of Hct = 0.10, 0.30 and Arrows on s shown n Fg. 8 llustrate the flud force vectors per the unt length on the membrane, f Flud [N/m], that were calculated from the vscous and pressure force terms n NS equatons. The force f Flud s normalzed by the stretch/compresson sprng constant k l and the reference length of the membrane element, l 0, ntroduced nto the elastc model as f Fludl 0 /( kl / l0) = ffludl0 / kl. The force vectors were tangental to the membrane n the vcnty of the parallel walls, as shown by closed crcles n Fg. 8. Ths ndcates that the vscous drag force was generated by the dfference n velocty between the fxed parallel walls and the. Approachng the central regon of the flow channel, force vectors became normal to the membrane, as shown by closed squares n Fg. 8. The normal force vectors ndcate that the pressure force s domnant n the flud force on the membrane. Fgure 9 shows the magntude of the flud force, F l / k, actng on the membrane as a functon of Hct. Flud 0 l Fg. 8 Flud force f l Flud 0 actng on membrane for Hct values of 0.10, 0.30 and The force s normalzed by stretch/compresson sprng constant k l and the reference length, l 0, of the stretch sprng as f l / k. The length of an arrow Flud 0 ndcates the magntude of the normalzed force. l Fg. 9 Magntude of flud force, F Flud l0 / kl, actng on membrane as functon of Hct. The defnton of F Flud s explaned n Sec. 3.. The magntude of the force F Flud l0 / k l s llustrated for pressure force, vscous force, and total flud force (sum of pressure and vscous forces). 165

8 F Flud [N/m] was defned as l a fflud dl F =, (8) Flud la where l a [m] s the total perpheral length of the membrane. The magntude of the total flud force (sum of pressure and vscous forces) ncreased wth decreasng Hct. Ths was due to an ncrease n pressure force wth decreasng Hct, whereas vscous force dd not change wth Hct, as shown n Fg. 9. The total flud force remaned constant when Hct was smaller than around 0.0. The dependence of the flud force actng on the membrane on Hct was assocated wth flud flow. Fgure 10 shows the velocty vectors of the flow of plasma and the nner flud of an (left) n the cases of Hct = 0.10, 0.30 and 0.49, and ther components on the axal drecton of the flow channel as a functon of y coordnate, whch s set perpendcular to the axal drecton (rght). Ths graph demonstrates that at lower Hct, the axal velocty of plasma was spatally more dstrbuted n the central regon of the flow channel. Comparng the dstrbuton of plasma flow around an (Fg. 10) and the flud force actng on the (Fg. 8), t was suggested that the velocty dstrbuton of plasma flow caused the force dstrbuton actng on the membrane, and that ths flud force dstrbuton enhanced deformaton nto a parachute shape. The velocty profle of the nner flud of the was flat, demonstratng that the velocty of the nner flud was not spatally dstrbuted and was the same as that of the membrane because the nner flud flow was restrcted by ts surroundng membrane. In the hgh-hct cases, the velocty profle of plasma was flat because the plasma flud was packed n between the neghborng s and ts flow was restrcted, smlar to the mechansm of the restrcton of the nner flud flow by the membrane. On the other hand, the flat velocty profle dd not generate the dstrbuton of the flud force actng on the membrane, resultng n restrcton of deformaton. Ths ndcates that both the effects of plasma on s and vce versa worked to restrct deformaton n hgh-hct cases. Both the degree of deformaton (Fg. 7) and the magntude of the flud force actng on membrane (Fg. 9) as a functon of Hct dd not change when Hct values were less than 0.0~0.30, n whch case the dstance between s was more than 7.0~ When the Hct values were less than 0.0~0.30, the plasma flow profles obtaned by smulaton were the same as the flow profles of plasma wthout s when the dstance between plasma and s ranged from /D (= 4.5 µm) to D (= 9.0 µm). These results demonstrate that mechancal nteracton between s has a sgnfcant nfluence on deformaton and plasma flow propertes when the Hct values are more than 0.0~0.30, n whch the dstance between s s wthn the sze of an. 3.3 Blood flow resstance Fgure 11 shows the tme course of the change n apparent pressure drop p n the mddle part of the flow channel. In ths graph, pressure drop p s normalzed by the pressure drop ( p) 0 of plasma flow wthout s. Pressure drop p ( p) 0 ncreased at the onset of flow, and gradually decreased untl tme t/t 0 = 1.0. After that, pressure drop p ( p) 0 converged to a constant value. The sold lne n Fg. 1 shows the mean value of the pressure drop over the tme from t/t 0 = 1.0 to 3.0, p ( p) 0, as a functon of Hct. Average pressure drop p ( p) 0 monotoncally ncreased from 1.0 to 1.36 wth ncreasng Hct from 0.10 to Ths result s consstent wth prevously reported results of n vtro experments compled to an emprcal formula (1), (4), as shown by a dotted lne, n whch the dfference n pressure drop p ( p) 0 between the smulaton and n vtro experments was less than.%. The blood flow resstance obtaned by smulaton as a functon of Hct qualtatvely 166

9 Fg. 10 Velocty vectors of plasma around (left) and ther components n axal drecton of flow channel (rght) for Hct values of 0.10, 0.30 and The magntude of velocty s normalzed by the nlet velocty value u 0 on the rght. The focus of nterest s the square regon wth D = 9.0 µm on each sde n the mddle part of the flow channel. agrees wth n vtro expermental results usng a mcro flow channel (4). The dfference of.% between the smulaton and experments mght ndcate the quanttatve correspondence between smulaton and experments. However, the smulaton result s two dmensonal, whereas the expermental results are three dmensonal. In addton, model parameters ntroduced n s, such as membrane sprng constants and the vscosty of nner flud, were determned to express both the deformaton nto parachute shape and the apparent flow resstance expressed by an emprcal formula. Therefore, t s an mportant future work to compare the results of a three-dmensonal smulaton wth expermental results, 167

10 Fg. 11 Change n apparent pressure drop n mddle part of flow channel for varous Hct values. Pressure drop p n the longtudnal axs s normalzed by pressure drop ( p) 0 n the case of plasma flow wthout s. Fg. 1 Mean value of pressure drop over tme from t/t 0 = 1.0 to 3.0 as functon of Hct. Sold and dotted lnes show the results of the smulaton and n vtro experments compled to an emprcal formula (1), (4), respectvely. whch wll provde us a quanttatve evaluaton of the relatonshps of mechancal behavors n mcrocrculaton between blood components and the resultng rheology of blood. 4. Conclusons A two-dmensonal computer smulaton of blood flow was carred out usng a partcle method to consder the mechancal behavors of deformable multple s. Smulaton results demonstrated a suppressng effect of Hct on deformaton nto a parachute shape. Ths was assocated wth the flud force actng on an, whch was determned by the mechancal nteracton among s and plasma. The mechancal nteracton determned deformaton and plasma flow propertes, determnng blood flow resstance as a functon of Hct. Acknowledgments The authors thank Prof. Sech Koshzuka of the Unversty of Tokyo for provdng the source code of the basc part of the MPS method. They are also grateful to Prof. Ryo Kobayash of Hroshma Unversty for helpful comments on the elastc membrane model. Ths work was supported by the followng grants; Grants n Ad for Scentfc Research by the Mnstry of Educaton, Culture, Sports, Scence and Technology (MEXT) of Japan (Scentfc Research (A) ( ) and Scentfc Research n Prorty Areas (768)). "Revolutonary Smulaton Sofrware (RSS1)" project of next generaton of IT program of MEXT. The frst author was fnancally supported n part by MEXT Overseas Advanced Educatonal Research Practce Support Program, Support Projects n the New Felds of Interdscplnary Research Creaton. Appendx The shape change of a swollen was smulated on the bass of the sprng network model (13), as shown n Fg. 13. As an ntal state, the was assumed to have a crcular shape wth a dameter of Φ = 6.0 µm. The crcular was dscretzed nto N = 76 membrane partcles that were connected by stretch/compresson and bendng sprngs wth the neghborng partcles, as denoted n Sec..1. The membrane partcles moved so that the total elastc energy became mnmum by solvng a set of moton equatons for each partcle, m r + γ r = F,. (9) by the fnte dfference method (FDM). Here, a dot ( ) denotes the tme dervatve, and m and γ are the representatve mass and vscosty of the, respectvely. Force F due 168

11 Fg. 13 Shape change smulaton of swollen due to volumetrc reducton usng sprng network model based on mnmum energy prncple. An ntal crcular changed to have a bconcave shape at the fnal state of the smulaton. to total elastc sprng energy E s expressed as ( E + Γ s ) F =, (10) r nstead of Eq. (4) n order to ntroduce an areal constrant wth penalty functon Γ s. Ths penalty functon was defned as ks s s0 Γ s = s, (11) 0 where s and s 0 are the area and ts reference value, respectvely. The sprng constants n Eqs. (1) and () were set to have the same values as those used n Sec..3. The other parameters were set as follows; the mass m = µg and vscosty γ = N s /m n Eq. (9), and the penalty coeffcent k s = N m n Eq. (11). As a result of shape change smulaton n the case of volumetrc reducton to 70% of the ntal crcular shape (that s, reference volume s 0 was set to 70% of the ntal shape), a bconcave was obtaned at the fnal state. The sze of the bconcave was h 0 = 8.0 µm n axal length and w 0 =.6 µm n the thckness of the concave part. Ths bconcave shape corresponds to that observed n a normal. Sprng constants k l and k b n Eqs. (1) and () determne the elastc propertes of membrane n the blood flow smulaton. The bendng constant, k b, can be reduced to a bendng stffness coeffcent B of the membrane by comparng bendng elastc energy E b n Eq. () wth the analytcal soluton E b calculated by coeffcent B and membrane curvature C (13). Assumng a two-dmensonal crcular shape of membrane as shown n the left sde of Fg. 13, the bendng energy E b s calculated by lne ntegrals along the crcumference L = πφ wth the constant curvature C = 1/(φ /), πφ ' Bπ Eb = B C dl = B dl =. (1) L 0 φ φ Based on prevous reported constant B values (B = 0.4~ N m ) (17), k b = 0.6~ N m. In a prelmnary calculaton, however, these values of k b resulted n excessve deformaton of n a two-dmensonal blood flow smulaton. Ths s due to the fact that a membrane structure n two-dmensonal problem s more easly deformed than that wth three-dmensonal shape. Therefore, the constant k b was set approxmately a hundredfold n ths study to compensate for the decreased stffness of the membrane n two-dmensonal problem and to express realstc deformaton. In our two-dmensonal smulatons ths adjustment resulted n deformaton and pressure drop comparable to real three-dmensonal cases. Stretch/compresson constant k l n two-dmensonal problem represents both planar shear deformaton and ncompressblty of membrane n three-dmensonal one. The value of the constant k l n ths study was chosen to ensure that s n no flow state have a bconcave shape whch s determned by the rato of k l to k b. 169

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