Application of the Topological Optimization Technique to the Stents Cells Design for Angioplasty

Transcription

1 Application of th opological Optimization chniqu to th. A. Guimarãs S. A. G. Olivira Emritus Mmbr, ABCM M. A. Duart Snior Mmbr, ABCM Fdral Univrsity of Ubrlândia - UFU Faculty of Mchanical Enginring , Ubrlândia, MG. Brazil Application of th opological Optimization chniqu to th Stnts Clls Dsign for Angioplasty Rstoring th intrnal lumn of artris by mploying an xpandabl msh (stnt) of mtallic or polymric matrial, known as angioplasty, is on of th most common procdur for tratmnt of th obstructiv cardiovascular disass. h stnt for angioplasty hav bn xtnsivly usd in th tratmnt of th cardiovascular disass. hy should b flxibl during th implant procdur and stiff whn implantd into th blood vssl. hs dsign critria dpnd on th matrial, gomtry and th tchnology usd in th stnts manufacturing. h objctiv of this work is to provid th optimizd gomtry of a stainlss stl stnt by mans of th topological optimization tchniqu. h gnral ida of this mthodology is to simulat th implant procss of a stnt using th finit lmnts mthod, xtract th snsitivitis of its flxibility and stiffnss and updat its matrial distribution or topology in an itrativ procdur. h algorithm for th numrical computation of th stnt matrial distribution is basd on th huristics and in th Lagrangian of th optimization problm. h rsults show that th stnt optimal topologis providd by this mthodology hav som advantags whn compard with th traditional gomtris of commrcial stnts. Kywords: biomchanics, mchanical dsign, nonlinar finit lmnts, stnts, topological optimization Introduction h accumulation of substancs in th coronary artris, such as th cholstrol, is on of th causs of dath among adults around th world. In th past, most of cardiovascular disass just wr tratabl through a by-pass surgry (Srruys and Kutryk, 998). Nowadays, a minimally invasiv cathtr-basd procdur has bn xtnsivly usd to unblock th disasd artry. In this procdur, a thin wall mtallic tub known as stnt is also implantd in th rgion of th unblockd artry in ordr to prvnt rstnosis or r-closur of th blood vssl. hr ar svral critria to b considrd in th stnt dsign (Srruys and Kutryk, 998). In this papr, it will b mphasizd only th flxibility and th stiffnss as th stnts mchanical dsign critria. During th implant procdur, th stnt should b abl to track th cathtr into th vssl until th stnosd artry rgion is rachd. hus, th stnt should b flxibl nough to bnd whil attachd to th cathtr. Convrsly, th stnt should also b sufficintly stiff to maintain th artry opnd and to support th prssur applid by th blood vssl wall aftr th implant. Analytical prdiction of th flxibility and stiffnss of a stnt only is possibl in th linar lastic rang (Durig t al., 2000). Indd, th structural analysis of a stnt is highly nonlinar du to th complxity of its gomtry and th larg plastic strain that occur during th implant. In this way, th analysis of th dsign critria abov mntiond has bn studid using nonlinar finit lmnts modls or xprimntal obsrvations of stnts implantd in patints (Srruys and Kutryk, 998; Ahmad and Barrtt, 999; Borgrsn and Sadghi, 2000; Durig t al., 2000; McClan and Eiglr, 2002). On th othr hand, th dsign tchniqus usd in th dfinition of th stnts gomtry ar not usually dscribd in th litratur. It is supposd that th dsignrs us th intuition, xprinc and th rsults of th stnt analysis to improv thir prformanc (Borgrsn and Sadghi, 2000). Howvr, although ths ar valuabl dsign tools, thy do not guarant that th obtaind stnt gomtry is optimum. In this contxt, th structural optimization tools availabl in th litratur provid an automatd mthodology to obtain th Papr accptd March, chnical Editor: Agnor d oldo Flury. optimum gomtry of stnt with maximum flxibility and stiffnss (andrplaats, 999). opological optimization is a particularly usful dsign tool to b usd in th dfinition of th bst matrial distribution of a stnt (Guimarãs, 2005). An advantag of this tchniqu is that problms concrning th maximization of flxibility and stiffnss of structurs subjctd to th larg strains ar wll stablishd in th litratur (Yin and Anathasursh, 200; Nishiwaki t al., 200; Mayr t al., 996; Maut t al., 998). hn, th sam formulation usd in th algorithms of topology optimization of stiff and flxibl structurs can also b applid and adaptd to th stnts dsign problm. hrfor, th objctiv of this work is to mploy topology optimization as a mchanical dsign tool of stainlss stl stnts by simulating th implant procdur. In th following, it will b dscribd th structural modls usd in th study of stnts, th formulation of th optimization problm and th mthodology usd in th numrical computation of th stnt optimal topology. Finally, th stnt optimal matrial distribution will b shown and compard qualitativly with th gomtry of commrcial stnts. Nomnclatur A = factor usd in th updating of th flxibl topology, dimnsionlss B w = factor usd in th updating of th hardnd topology, dimnsionlss C 2 U = mutual strain nrgy dnsity, J/m d = diffrntial lmnt of volum, m E = Young s modulus,n/m 2 E p = plastic modulus, N/m 2 E t = tangnt modulus, N/m 2 f = volum fraction of th optimal topology, dimnsionlss F = body forc, N F s = surfac (normal) forc, N I = indx usd in th summation of th finit lmnts, dimnsionlss K = global stiffnss matrix, N/m m = mov limit, dimnsionlss N = numbr of finit lmnts from th dsign spac, dimnsionlss p = powr of pnality of th SIMP modl, dimnsionlss J. of th Braz. Soc. of Mch. Sci. & Eng. Copyright 2008 by ABCM July-Sptmbr 2008, ol. XXX, No. / 26

2 . A. Guimarãs t al P(x) = objtiv function usd in th maximization of th stnt cll flxibility, dimnsionlss U = displacmnt fild of th structur, m = volum of th dsign spac, m (x) = volum of th stnt cll optimal topology, m W = lastic-plastic strain nrgy dnsity, J/m x = rlativ dnsity, dimnsionlss Grk Symbols u = prscribd displacmnt applid to th dsign spac of th structur, m ε = total strain, dimnsionlss ε = lastic strain, dimnsionlss ε p = plastic strain, dimnsionlss η = numrical damping cofficint, dimnsionlss Ζ = load factor, dimnsionlss σ r = yild strss, N/m 2 Γ = rgion of surfac, m 2 Subscripts rlativ to lastic rang min rlativ to th minimum valu o rlativ to proprtis from th solid matrial p rlativ to plastic rang r rlativ to yild strss t rlativ to tangnt rlativ to forc applid by th vssl wall 2 rlativ to dirction of th flxibility rlativ to raction forc applid by th vssl wall Suprscripts rlativ to finit lmnt j rlativ to itration indx rlativ to transpos Dfinition, Charactristics and Modls of Stnts A stnt can b dfind as any dvic with circular sction usd to rinforc th wall of a vssl (Srruys and Kutryk, 998). As can b sn in Fig., th stnt structur is cylindrical and is formd by a rptitiv gomtrical pattrn, known as clls. In applications for angioplasty, th stnt diamtr bfor th implant varis of.5mm to th 2.5 mm. In this phas, th stnt should only track th cathtr into th vssl. o opn artris whr dposits rstrict th blood flow, cardiologists us an inflatd balloon to push th obstruction asid. Whn th prssurizd balloon inflats, th stnt xpands, that is, it incrass its diamtr until it contacts th innr surfac of th artry wall (som stnts ar auto-xpansiv). Aftr th implant procss, th final diamtr of th stnt implantd in th artry may rach until fiv tims its original diamtr (Srruys and Kutryk, 998; Borgrsn and Sadghi, 2000). h importanc of this contact btwn stnt and blood vssl is to prvnt any diamtr rduction of th artry wall aftr th implant and also to prvnt thrombus formation. Figur. Modl of finit lmnts of a thr-dimnsional stnt (Guimarãs t al., 2006). Figur 2 shows th plan modl of a stnt with svral clls bfor th xpansion procss (Guimarãs t al., 2006). h xpandd stnt has som rgions whr th strss lvl xcds th yild strss. h hardning of ths rgions and th stiffnss of th clls maintain th artry wall and th stnt with th xpandd diamtr. In th longitudinal dirction of th cll, it can b obsrvd th prsnc of sinusoidal structurs. h rol of ths structural lmnts is to improv th flxibility of th stnt during th implant, in ordr to achiv bst mobility. Figur 2. Plan modl of th stnt with svral clls (Guimarãs t al., 2006). opology Optimization as a Stnt Dsign ool opology optimization can b dfind as a structural dsign tool that gnrats automatically th bst layout of th lmnts or componnts of a structur (Rozvany, 997). his procdur must locat th structur lmnts that will b ithr solid or void. Solid lmnts hav rlativ dnsity qual to (on) and void lmnts hav rlativ dnsity qual to 0 (zro). For xampl, Fig. shows th optimal topology of a cantilvr bam with minimal strain nrgy. In this cas, th volum of th optimal topology is 40% of th original dsign spac volum (rctangular bam). Figur. Optimal topology of a cantilvr bam. In this papr, th dsign problms of th stnt cll subjctd to th plastic strain providd by th xpansion of th balloon and th lastic bnding du to th curvatur imposd by th cathtr will b tratd sparatly using topology optimization. First off all, th topology of th stiff rgion from th plastically dformd stnt cll, shown in th Fig. 2, will b optimizd by maximizing th hardning. Subsquntly, th lastic strain nrgy of th flxibl longitudinal structur from th stnt cll will b maximizd and its optimal topology computd. All topologis shown in this papr hav a volum quals to 20% of th total dsign spac volum. his valu is rcommndd and usd by th manufacturrs in ordr to avoid th rjction of th stnts implantd in th vssl (Srruys and Kutryk, 999). In fact, th lss th mtallic surfac ara of a stnt contacts th artry wall, th lss is th probability of rjction. 262 / ol. XXX, No., July-Sptmbr 2008 ABCM

3 Application of th opological Optimization chniqu to th Stnt Cll with Maximum Hardning h stnt cll should b subjctd to plastic strain only during th balloon xpansion procss. h plastic strain nrgy absorbd by th stnt during th implantation incrass th yild strngth of th stainlss stl. hus, aftr th hardning, it will b ncssary to incras th strss lvl in ordr to xpand or comprss th diamtr of th xpandd stnt placd in th artry. h purpos of this is to nsur that th stnt diamtr producd by th angioplasty is maintaind. It is dsird to maximiz th hardning of th stnt to improv th prformanc during th balloon xpansion. On way of maximizing th hardning is to maximiz th toughnss or total strain nrgy in th lastic and plastic rangs (Mayr t al., 996; Maut t al., 998): W = { ε }[ E p ]{ ε} d () 2 whr d is th volum diffrntial lmnt of a continuous solid body, {ε} rprsnts th total strain dfind by th sum of th lastic and plastic strains, {ε } and {ε p }, givn by (Karasudhi, 99): { p ε } = { ε } + { ε } (2) and [E p ] is th plastic modulus, that is a linar approximation of th strss-strain curv for th lastic and plastic rangs of th stainlss stl (Karasudhi, 99; Mayr t al., 996). his matrial modl, known as Bilinar Isotropic Hardning (BISO) modl, is usually usd for larg strain analysis of ductil matrials, such as a stainlss stl stnt (Karasudhi, 99). Anothr advantag of th BISO modl is th fast computation of th strss componnts of a structur subjctd to plastic strain, which is particularly usful in th finit lmnt analysis (Mayr t al., 996; Ahmad and Barrtt, 999). h plastic modulus, E p, is dfind by (Karasudhi, 99; Mayr t al., 996): Et E E p = () E - E t whr E is th Young s modulus and E t is th linar approximation of th plastic rang, calld tangnt modulus. It should b notd that in th Eq. (), th variabl E p dnots th plastic modulus matrix that corrlats th quivalnt strss tnsor with th plastic strain tnsor (Mayr t al., 996). On th othr hand, th Eq. () is a scalar rlation that rprsnts ach componnt of th plastic modulus tnsor as a function of th componnts of th tangnt and Young s modulus tnsors. In th BISO modl, th dfinition of th plastic strain incrmnts, dε p, is basd on th flow rul (Karasudhi, 99; Maut t al., 998). Gomtrically, dε p is a normal vctor to th yild surfac rprsnting th strss stat of th structur. For larg strain of ductil matrials, th on Miss lliptical yild surfac provids th distortion nrgy causd by any strss combination. Physically, th plastic strain of a ductil matrial is producd by its distortion nrgy (Norton, 996). Following th topology optimization nomnclatur, th mathmatical dfinition of th stnt cll dsign problm with maximum hardning is: Maximiz: W = 2 { ε }[ E p ]{ ε } d Subjctd to: ( x ) f = (4) 2 p { Fs } d Γ { ε}[ E ]{ ε} d = ζ { u} 0 x min x (6) such that f is th prcntag of th optimal topology volum, (x), with rspct to th total volum,, of th initial dsign domain. Equation (5) dscribs th quilibrium of th topology to b satisfid during th optimization. h lft sid of th Eq. (5) is th total strain nrgy accumulatd by th stnt cll. h right on rprsnts th xtrnal work of th prscribd displacmnt, { u}, applid in som rgion of th surfac, Γ, in th dsign spac. In th surfac intgral of th Eq. (5), {F s } is an unit and fictitious normal forc applid to th dsign spac and ζ, calld load factor, rprsnts th magnitud of {F s } that has th sam ffct of th prscribd displacmnt, { u}. Equation (6) ar th minimum and maximum valus (sid constraints) that th rlativ dnsity or dsign variabls, x, can assum. h maximum valu, x=, is th rlativ dnsity of th solid matrial and th minimum valu, x min = 0.00, of th rlativ dnsity clos to zro is usd in ordr to avoid a singularity of th global stiffnss matrix of finit lmnts. Stnt Cll with Maximum Flxibility h stnt cll should b flxibl as wll, sinc it will b subjctd to larg dflctions in th lastic rang whn th cathtr bnds in th blood vssl bfor th balloon xpansion (Srruys and Kutryk, 998; Borgrsn and Sadghi, 2000). A masur commonly usd in th topology optimization fild of th structural flxibility, is th mutual strain nrgy, C 2 U, dfind as follows (Nishiwaki t al, 200; Yin and Ananthasursh, 200): 2U = U KU 2 C (7) whr U dnots th displacmnts of th structur causd by th forc F, U 2 is th displacmnt fild du to th forc F 2 and K rprsnts th global stiffnss matrix of th lastic solid body illustratd in th Fig. 4. Indd, Eq. (7) masurs th displacmnt or flxibility of th structur in th dirction of th unit dummy load, F 2, whn th forc F is applid. hus, th largr is th mutual strain nrgy, th largr is th flxibility of th structur. Figur 4. Concpt of flxibility of an lastic body. By considring th flxibl stnt cll structural modl, th forc F may b intrprtd as th bnding load applid by th blood vssl wall at th sam plan of th stnt cll. h location of th Γ (5) J. of th Braz. Soc. of Mch. Sci. & Eng. Copyright 2008 by ABCM July-Sptmbr 2008, ol. XXX, No. / 26

4 . A. Guimarãs t al load F 2 rprsnts th point of contact of th bnt stnt cll with th vssl wall. Usually, th stnt matrial is stiffr than th vssl wall matrial (Srruys and Kutryk 998). hrfor, it is xpctd that th contact strain is prfrntially locatd in th vssl wall. Mathmatically, th stnt cll with maximum flxibility is obtaind from th maximization of th Eq. (7). Howvr, th simpl maximization of Eq. (7) dos not guarant that th structur will not dform indfinitly. In this cas, th problm is ill conditiond and th Eq. (7) will tnd to th infinit. h solution for this inconvnint is to limit th mutual strain nrgy to a finit valu during th maximization of th Eq. (7). Most paprs daling with th flxibl structurs topology optimization introducs a lumpd stiffnss at th point of application of th forcs F or F 2 to rstrict th flxibility of th structur in th dsird dirction, (.g. Nishiwaki t al. 200; Yin and Ananthasursh 200). In this work, it will b adoptd th traditional approach usd in th formulation of objctiv function, P(x), of a flxibl topology. It is dfind by th ratio btwn th mutual strain nrgy and th complianc of th structur (Yin and Ananthasursh 200): U KU P(x) = - 2 (8) U K U whr th numrator and th dnominator rprsnt th structural flxibility and stiffnss masurs rspctivly. In this formulation, th usr dos not nd know th valus of th wighting cofficints of th flxibility and stiffnss of th structur. Whn th Eq. (8) is maximizd, th mutual strain nrgy is maximizd and th complianc is minimizd simultanously. h global stiffnss matrix, K, in th dnominator of Eq. (8) is obtaind by rstricting th dgr of frdom of th point of application of th forcs F or F 2, shown th in Fig. 4. h displacmnt fild, U, is drivd from this condition. Using th objctiv function modl dfind abov, th topology optimization problm of th flxibl stnt cll will b: Minimiz: P(x) = U KU2 - U K U Subjctd to: ( x ) f = (4) KU = F (9) Algorithm for th Gnration of th Optimal opologis h stnt volum fraction is th only activ constraint of both formulations abov dscribd. Furthrmor, th maximization of th hardning and flxibility has a larg numbr of dsign variabls. hs dsign variabls ar th rlativ dnsity of ach finit lmnt from dsign spac. For problms of this natur, th optimality critria mthod is th optimizr mor fficint to obtain th optimal solution (Mayr t al., 996; Maut t al., 998; Sigmund, 200; Yin and Ananthasursh, 200). hrfor, it will b usd in this work to provid th stnt cll optimal topologis. h optimality critria mthod is basd on th Kuhn-uckr conditions applid to th Lagrangian of th objctiv function and constraints. It can b dmonstratd that th stationary condition of th Lagrangian of th lastic-plastic strain nrgy dfind by th Eq. () subjctd to th constraints (4), (5) and (6) of th hardning maximization problm of th stnt cll is givn by (Guimarãs, 2005): Bw = { } [ E p ] { ε ε x } N { } [ E p ] { ε ε } i= λ x N ( x ) x j+ η ( x ) = ( B ) ( x ) j w (2) () whr λ is th Lagrangian multiplir of th volum constraint, N is th numbr of finit lmnts usd in th discrtization of th dsign spac, η is a numrical damping cofficint to dcras th convrgnc rat in th computation of th optimal topology, j dnots th itration indx usd in th updating of th topology and i is an indx which rprsnts th summation of all finit lmnts from th dsign spac. Equation (2) dpnds on th snsitivity of th plastic modulus, which is th drivativ of th Eq. () with rspct to th x. In this work, it will b mployd th SIMP (solid isotropic matrial with pnalization) approach as an intrpolation modl of rlativ dnsity of th stnt cll. In this cas, th optimal topology is totally dfind and fr of porous (Sigmund, 200) and th snsitivity of th plastic modulus in th Eq. (2) is asily computd. In th cas of th flxibility maximization problm of th stnt cll, th stationary condition of th Lagrangian applid to th Eq. (8), (4), (9), (0), () and (2) is dfind by (Guimarãs, 2005): KU 2 = F 2 (0) K = () U F 0 x min x (6) K A = U 2 U - U KU x U KU K 2 U U + 2 (U KU ) x λ ( x ) x (4) From th stnt dsign point of viw, F is th raction forc of th vssl wall on th stnt cll du to th contact that occurs whn th cathtr bnds. In fact, F is qual to F 2 but in opposit dirction. In this situation, th dgrs of frdom of th point of application of F ar rstrictd for th computation of th U from th Eq. (). j+ j j ( x ) = ( x ) - A (5) whr th rlativ dnsity in th itration j+ was updatd basd on th gradint of th Lagrangian multiplid by -, which rprsnts th stpst dscnt dirction of this optimization problm. Onc again, th SIMP modl dfind for th flxibl stnt cll will b usd in th updating of th topology. Unfortunatly, Eq. () and Eq. (5) dos not guarant a stabl convrgnc sinc abrupt changs may occur in th formation of th 264 / ol. XXX, No., July-Sptmbr 2008 ABCM

5 Application of th opological Optimization chniqu to th topology. In ordr to stabiliz th formation of th topology, th following procdur was usd in this work (Maut t al., 998; Sigmund, 200; Yin and Ananthasursh, 200): j j+ max[ 0. 00,( x ) - m] ( x ) min[,( x ) + m] (6) whr m is th mov limit that rprsnts a maximum prmissibl valu of th chang of rlativ dnsity in ach itration. h rol of this paramtr is to avoid th formation of discontinuitis in th topology during th optimization. h choic of th valu of this paramtr dpnds on th bhavior of objctiv function to b optimizd. In th prsnt papr, th valu of m for ach problm will b chosn obsrving th convrgnc procss of th optimal topology. If th objctiv function bgins to oscillat, th valu of m is rducd so that th convrgnc procss bcam stabl. Figur 5 illustrats th stps of th topology optimization algorithm to b usd in both problms, maximization of th flxibility and hardning. Although thy ar diffrnt problms tratd sparatly, it will b usd th sam mthodology in th drivation of th optimal topologis basd on th optimality critria mthod. Initially, all finit lmnts from dsign spac hav rlativ dnsity qual to volum fraction chosn by th usr. For th maximization problm of th flxibility, th nodal displacmnt filds U, U 2 and U ar obtaind by solving th quilibrium quations (9), (0) and (). hs displacmnts will b xtractd from a subroutin implmntd in th Matlab (Sigmund, 200). For th hardning problm, it will b usd th finit lmnt softwar ANSYS in ordr to xtract th lastic and plastic strain fild from th stnt dsign spac. Subsquntly, th objctiv function snsibility is calculatd. Finally, th Lagrangian multiplir, λ, from volum constraint is dtrmind using a bi-sctioning algorithm and th topology of stnt cll is updatd. h subroutins of updating of th stnt cll topology wr implmntd in Matlab (Sigmund, 200). j dnsity filtring of th finit lmnts of structur (Sigmund, 200). In this approach, th rlativ dnsity of ach finit lmnt will b computd according to th wighting avrag of th strain nrgy snsibility of th nighboring lmnts. hus, bfor th updating of th topology using th Eq. (6), th snsitivity of th strain nrgy of ach finit lmnt is computd, as illustratd in th flowchart (Fig. 5). h objctiv of this procdur is to obtain a mor uniform matrial distribution and without th chckrboard pattrns. Structural Modl for th Stnt Cll Optimization Figur 6 shows th dsign spac and th boundary conditions that simulat th balloon xpansion for th hardning maximization problm (Ahmad and Barrtt, 999). Du to symmtry of th stnt cll, it will only b considrd th half of its dsign spac. h traction displacmnts, u, of sam magnitud applid to th uppr and lowr cornrs from th right dg simulat th stnt xpansion procss causd by th balloon. Nonlinar finit lmnt analysis from softwar ANSYS simulats this xpansion procss svral tims during th optimization. u Figur 6. Structural modl for th maximization of th hardning of th stnt cll. u abl. Paramtrs of th stnt cll structural modl subjctd to th plastic strain. Paramtrs of th stnt modl Magnitud Young s modulus 90x0 9 N/m 2 angnt modulus 00x0 6 N/m 2 Yild strss 250x0 6 N/m 2 Poisson s ratio 0. Stnt cll lngth.5x0 - m Stnt cll hight.25x0 - m raction displacmnt 0.22x0 - m and 0.44x0 - m Figur 5. Flowchart of th stnt cll optimization topology algorithm. In topology optimization, thr still xists a natural tndncy to th formation of rgions lik chckrboards (Sigmund, 200). In ths rgions, th optimal matrial distribution is not continuous and oscillats from solid to void. h mor fficint approach for th tratmnt of this problm, to b usd in this papr, is th rlativ abl shows th gomtrical paramtrs as wll as th matrial proprtis to b usd in th hardning maximization of th stnt cll (Norton, 996; Srruys and Kutryk, 998). In this situation, it will b gnratd two hardnd stnt cll topologis corrsponding to two diffrnt valus of u dscribd in th ab.. h smallr displacmnt, u=0.22mm, modls th xpansion of a stnt to b implantd in an artry with a small diamtr. Similarly, th displacmnt, u=0.44mm, simulats th xpansion of th stnt in a vssl with a largr diamtr. h vrtical load F simulats th forc applid by th vssl wall on th stnt whn th cathtr bnds. h dummy load F 2 is th dsirabl dirction of th stnt flxibility. It is applid at th middl of th bottom dg of th dsign spacs shown in th Fig. 7 and Fig. 8. Morovr, th flxibl stnt cll should maintain its shap whn J. of th Braz. Soc. of Mch. Sci. & Eng. Copyright 2008 by ABCM July-Sptmbr 2008, ol. XXX, No. / 265

6 . A. Guimarãs t al subjctd to raction forc applid by th blood vssl wall, F. hn, th stnt cll stiffnss should also b maximizd du to th raction forc, F. F F 2 F Figur 7. Structural modl for th maximization of th flxibility of th stnt cll with th fixd lft dg. F F 2 F 7. Howvr, both modls will b usd to provid th stnt flxibl topologis. Rsults and Discussion h dformd shap of th stnt cll matrial distributions, with maximum hardning corrsponding to th traction displacmnts of 0.22x0 - m and 0.44x0 - m, ar illustratd in th Fig. 9 and Fig. 0, rspctivly. As th traction displacmnt of magnitud u is applid on th uppr and lowr right cornrs from th dsign spac (Fig. 6), th total displacmnt of stnt cll in th vrtical dirction aftr th xpansion is qual to 0.44mm and 0.88mm (0.22x0 - m and 0.44x0 - m multiplid by two), rspctivly. hrfor, th prcntual vrtical displacmnts of th stnt cll with rspct to th dsign spac hight ar 5% and 70% (0.44mm/.25mm and 0.88mm/.25mm). For ths situations, th stnt diamtr will also incras 5% and 70% sinc this is proportional to th circunfrntial lngth of th cross sction. h structurs shown in th Fig. 9 and Fig. 0 wr providd from th contours of th optimal topologis gnratd by using th algorithm shown in th Fig. 5 (Guimarãs, 2005). In both cass, th topologis wr computd from a msh with 0x25 finit lmnts. Sinc ach finit lmnts analysis usd in simulation of xpansion of th stnt cll is nonlinar, it was ncssary to divid th total solution intrval in 200 stps in ordr to solv th structural quilibrium quations st during th updating of th topologis. his valu is rcommndd by th program ANSYS to avoid th divrgnc of th nonlinar solution procss (ANSYS Inc., 2002). Morovr, th program ANSYS should slct th bst solvr to b usd in th solution of th nonlinar quilibrium quations. Figur 8. Structural modl for th maximization of th flxibility of th stnt cll with th lft dg fr to mov in th horizontal dirction. abl 2. Paramtrs of th flxibl stnt cll structural modl subjctd to th larg lastic strain. Paramtrs of th stnt modl Magnitud Young s modulus Poisson s ratio 0. Stnt cll lngth Stnt cll hight 0.8 F F 2 F - Figur 9. Stnt cll optimal matrial distribution subjctd to th xpansion displacmnt of 0.22x0-m. h physical modls for th maximization of th flxibility of th stnt cll ar shown in th Fig. 7 and Fig. 8. h matrial proprtis, dimnsions of th dsign spac and th magnitud of th forcs applid to both structural modls ar dfind in th ab. 2. In this situation, th paramtrs of th stnt modl wr normalizd. h diffrnc btwn both modls is th boundary condition applid in th lft dg. h cantilvr bam illustratd in th Fig. 7 do not tak account th movmnts in th intrfac of th flxibl longitudinal structur with th stnt cll dsign spac subjctd to th plastic strain. On th othr hand, in th modl shown in th Fig. 8, all nods from th lft dg ar fr to mov in th horizontal dirction. In practic, th intrfac btwn th hardnd and flxibl structurs of th stnt cll is fr to mov as in th vrtical dirction as in th horizontal on. hus, th structural modl shown in th Fig. 8 is mor ralistic whn compard to th modl from Fig. Figur 0. Stnt cll optimal matrial distribution subjctd to th xpansion displacmnt of 0.44x0-m. 266 / ol. XXX, No., July-Sptmbr 2008 ABCM

7 Application of th opological Optimization chniqu to th hr ar significant diffrncs btwn th stnt cll matrial distributions shown in th Fig. 9 and 0. In th topology with maximum hardning subjctd to th xpansion displacmnt of 0.22x0 - m (Fig. 9), th matrial is distributd in th vrtical dirction by conncting th uppr and lowr cornrs of th right dg from th dsign spac. Whn this stnt cll xpands, th plastic strain distribution concntrats in this rgion (Guimarãs, 2005). For th stnt cll topology illustratd in th Fig. 0, it is not mor sn th prsnc of this matrial distribution clos to right dg. In addition, it can b also obsrvd that th dtails of this topology ar shiftd to th lft from th dsign spac whn compard to matrial distribution shown in th Fig. 9. In this cas, th plastic strain fild concntrats in a rgion clos to th hols of th stnt cll matrial distribution shown in th Fig. 0 (Guimarãs, 2005). Figurs and 2 illustrat th dformd shaps of th flxibl stnt cll topologis that wr optimizd from th structural modls shown in th Fig. 7 and Fig. 8, rspctivly. In th ranalysis procss, th forc F was rplacd by a displacmnt of normalizd magnitud quals to 0.08 applid on th flxibl stnt optimal topologis. For ths situations, it can b obsrvd th prsnc of a matrial suddn variation in som rgions of th topologis illustratd in th Fig. and 2. h major rol of ths discontinuous rgions placd in th stnt topologis is to absorb th lastic dflction nrgy and to improv thir flxibility. Figur. Flxibl stnt cll optimal matrial distribution with th lft dg fixd. Figur 2. Flxibl stnt cll optimal matrial distribution with th lft dg fr to mov in th horizontal dirction. h topologis with maximum hardning shown in this papr hav som diffrncs whn compard with th gomtry of th commrcial stnts illustratd in th Fig. 2. h stnt cll matrial distribution subjctd to th xpansion of 0.22x0 - m (Fig. 9) is similar to th topology of th complianc minimization problm (Sigmund, 200). Whn th xpansion displacmnt was incrasd to 0.44x0 - m, th rlativ dnsity of th local matrial distribution in vrtical dirction clos to th right dg was dcrasd. From this obsrvation, it can b concludd that th stnt cll topology subjctd to th xpansion of 0.44x0 - m is lss stiff and mor asily xpandd whn compard to th othr xampl. Anothr diffrnc is th siz of th hols nar th middl of th lft dg in th two topologis. hs dtails suggst that th stnt cll dsign is basd on th diamtr of th artry whr th stnt will b implantd. On th othr hand, in commrcial stnts, th cll gomtry dos not dpnd on th diamtr of th blood vssl whr th stnt will b implantd. h flxibl stnt cll topologis providd in this papr hav two faturs that ar not found in commrcial stnts. h suddn variation of matrial distribution of th flxibl topologis shown in this papr is not usually found in commrcial stnts. As shown in th Fig. 2, commrcial stnts us continuous structural lmnts of curvd gomtry in th longitudinal dirction in ordr to improv th flxibility of th cll. Morovr, commrcial stnts ar dsignd only to absorb th lastic strain nrgy during th implant procdur. In th stnt clls topologis illustratd in this work, th stiffnss of point of application of th forcs on th dsign spac was also maximizd. In commrcial stnts, only th matrial stiffnss guarants thir structural intgrity in larg dflctions. h figur 2 illustrats th plan modl of th stnt aftr th union of th flxibl and stiff topologis shown in th Fig. 0 and Fig.. Subsquntly, this plan modl was wrappd into a cylindrical shap to gnrat th thr-dimnsional stnt modl. h diamtr of this stnt modl bfor th angioplasty is mm (Guimarãs t al., 2006). If th plan modl shown in th Fig. 2 is subjctd to th prcntag displacmnt quals to 70%, th diamtr of thr-dimnsional stnt will incras 2.mm. hus, th final diamtr of th xpandd stnt will b 5.mm. In a typical angioplasty, this stnt modl could b usd, for xampl, in a disasd vssl diamtr into a rang of 4mm to 6mm. Conclusions In th prsnt papr, it was proposd a mthodology for th optimal topology dsign of th plan modl of stainlss stl stnt clls by considring sparatly thir flxibility and hardning. h stnt cll topologis wr computd using an algorithm basd on th huristics and in th optimality critria mthod. It was dmonstratd that is possibl to dsign stnts cll using th topology optimization tchniqu. Although thr ar significant diffrncs btwn th topologis providd in th prsnt work and in th commrcial stnts gomtry, all xampls mt th flxibility and hardning critria. h stnt clls subjctd to th plastic strain providd in this papr shown that th matrial distribution dpnds on th siz of th artry diamtr whr thy will b implantd. In commrcial stnts, th lastic dflction nrgy tnds to distribut along th flxibl structural lmnt with curvd gomtry. In th flxibl optimal topologis shown in th prsnt papr, th lastic nrgy is concntratd in th rgions with suddn variation of matrial distribution. Furthrmor, th structural intgrity of th flxibl optimal topologis is prsrvd sinc th stiffnss of th point of contact btwn th stnt cll and th vssl wall is maximizd. Acknowldgmnt h authors would lik to acknowldg th financial support providd by th CNPq. J. of th Braz. Soc. of Mch. Sci. & Eng. Copyright 2008 by ABCM July-Sptmbr 2008, ol. XXX, No. / 267

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