MULTIOBJECTIVE NONLINEAR SHAPE OPTIMIZATION OF STENT BASED ON EVOLUTION PRINCIPLES Wllam Annccharco Insttuto de Materales y Modelos Estructurales Facultad de Ingenería. Unversdad Central de Venezuela Caracas, Venezuela. anncchw@ucv.ve George S. Dulkravch Department of Mechancal and Materals Engneerng Florda Internatonal Unversty Mam, Florda, U.S.A. dulkrav@fu.edu ABSTRACT The treatment of atherosclerotc stenoses s accomplshed by a well-establshed nterventonal method called Percutaneous Translumnal Angoplasty (PTA). Generally, ths procedure s accompaned wth the deployment of small tube shape structure to support the wall of the njured artery and mprove the lmtatons of balloon angoplasty, such as restenoss and abrupt closure. Numerous computatonal studes have been carred out to nvestgate the expanson and mechancal behavor of dfferent stent desgns. They have lmted ther analyss to study and compare dfferent commercal and theoretcal stent desgns usng some medcal and mechancal crtera, n conjuncton wth parameters obtaned from laboratory test. However, none of them have treated ts desgn as a mult-objectve optmzaton task wth dfferent optmzaton crtera. In ths sense, the objectve of ths work s to present and dscuss an evolutonary multobjectve optmzaton algorthm based on evoluton prncples and Pareto s domnance crtera to select, generate and evaluate compromse solutons that generaton after generaton lead to the optmum (or quasoptmum) external shape of ths knd of prosthess. The man advantage of ths approach s that all the dfferent desgn crtera can be ncluded n one sngle run. NOMENCLATURE C: stran energy of the structure. D : stent s dameter at the fnal deployment. D un : stent s dameter after balloon deflaton (un). F: set of n>1 objectve functons. K : stffness matrx of the fnte element of the model of the structure. L: ntal stent s length. L : stent s longtudnal length at the fnal deployment. L un : stent s longtudnal length after balloon deflaton (un). P: ng vector of the structure. X: Vector of p desgns varables. f(x): objectve functon. g(x): equalty constrant functon. h(x): nequalty constrant functon. l: Number of nequalty constrants. m: number of equalty constrants. n: number of objectve functons. p: number of desgns varables. u: dsplacement nodal vector of the structure. u : dsplecement vector of the fnte element of the model of the structure. : Von-Mses stress of fnte element e. e max : maxmum Von-Mses stress of the structure. INTRODUCTION Cardovascular dseases and manly atherosclerotc stenoses are of the greater causes of mortalty n the western countres (e.g. 38% of all deaths n Amerca last year). In ths knd of dseases, arteres get blocked by accumulaton of cholesterol plaques over the vessel walls, narrowng ts calber and dmnshng the amount of blood flood to the rrgated tssues, causng pan and sometmes patent dead. The treatment of atherosclerotc stenoses s accomplshed by a well-establshed nterventonal method called Percutaneous Translumnal Angoplasty (PTA). It
represents a mechancal soluton for a clncal problem and s the most frequent therapeutcal nterventon world wde [1]. The study, mprovement and development of PTA has deserved a great and steadly growng medcal and and scentfc nterest [2] due to ts socoeconomcal mpact n the soluton of ths dsease. In ths procedure, a balloon s nflated n the blocked zone of the artery to reopen t and restore blood flow perfuson to the downstream tssues. The development and use of ths technque was an effectve soluton to ths dsease by several years, however n most cases, the only dlataton of the arteral walls dd not lead to rehabltaton of the vessel for a long tme. The dea of ntravascular stent was ncally ntdroduced by Dotter [3] n 1969 to deal wth the arteral narrowng, and t was not untl begnnng of 1990 years when these ntravascular prosthess were used to solve the lmtatons of PTA [4]. The stents are small tube-lke structure (stent, see fg. 1) desgn to support the wall of the njured artery, reopen arteral lumen to allow blood flow, correct any deformaton (curvature) of the arteral wall and resst the forces exerted by arteral compresson [5]. Dfferent types of stents can be found n the market. In terms of the manufacture process, they are classfed n slotted tubular stents and col stents. The frsts are obtaned by cuttng slots n small stanless steel tubes wth lasser devces and the second are manufactured by weavng tny steel rods. Based on the technque used to ther deployment, they are classfed n expanded by ballon (plastc deformaton) or selfexpandng (elastc or superelastc deformaton). Most of them are made of stanless steel (316L) or shape memory alloys (Ntnol), among others materals. Fg. 1. Detal of an ntravascular stent. The develepment of ths methodology has extended ts use not only to treatment of coronary dseases, t has been successfully appled to reopen and dealng wth urnary dseases and other major arteres such as lac, renal, etc, [6] The techncal lterature on ths subject s extensve wth numerous works related wth the evaluaton of mechancal propertes of ths devce by means of fnte elements methods, among the most relevants can be menton: Dumouln and Cocheln [7] characterzed and evaluated some mechanal propertes of ballon expandble stents (P308 Palmaz stent) usng lnear and non-lnear fatgue, collapse and post-bucklng analyss. Etave est al. [8] compare the most representatve types of stents ( slotted-tubular and col stents) wth regard ther mechancal characterstcs and gve some parameters for ther desgn. Mglavacca et al. [9] present a smlar approach, but only dealng wth slotted- tubular stent. In addton to study some mechancal propertes of ths knd of endoprosthess, They gve some recommendatons to mnmze the dogbonnng effect caused by unequal expanson of the stent. In Graca et al. [10] s smulated by fnte elements the mechncal behavour of stanless steel stents by consderng global flexblty and crtcal bucklng pressure. Other authors, Prendergast et al. [11] and Lally et al. [12], besde studyng the mechancal propertes of these devces, have also takng nto account the nteracton between the stent and the artery walls n order to analyse how ts desgn generates reestenoss mechansm, whch s the cause of the posteror vessel obstructon. For nstance, n [11] s presented the study of the nteracton of dfferent commercal stents and ther nteracton wth the arteral tsue, reportng prolapse of the tssue and the arteral wall stresses. On the other hand, n [12] two commercal stents are compared n terms of the level of the stresses produced n the vascular tssue of the artery, whch s drectly related wth the rate of restenoss observed n clncal studes. Recently, New drug-elutng stents have been developed to reduce the restenoss of the stent wth promssng future. However, It s wdely accepted that a better development n the desgn of these devces, ncludng the recent drug coatng stent, s requred to solve or reduce n an optmum way the reestenoss problem presented n the balloon angoplasty wth stent [13]. Much of the above researches have tested and proposed stent desgns concerned wth some of the followng ssues (a) suffcent rgdty to resst the compressve forces exerted by the vessel wall, (b) suffcent flexblty to navgate tortuous vessels, (c) scaffoldng propertes: stents must be able to hold open the vessel and scaffold the stenotc materal plaque aganst the vessel wall,
(d) mnmal longtudnal contracton when expanded, (e) mnmal shearng between the stent and tssue durng expanson because ths denudes the vessel of ts endothelal cell lnng. Those research papers have lmted ther analyss to the study and comparson of dfferent commercal and theoretcal stent desgns usng some of the above crtera n conjuncton wth parameters obtaned from laboratory test. However, none of them have treated the stent s shape desgn as mult-objectve optmzaton task wth dfferent optmzaton crtera. In ths sense, the objectve of ths work s to present and dscuss a mult-objectve evolutonary optmzaton methodology that n an comprehensve way takes all of the dfferent desgn crtera n order to obtan an optmzed external shape of ths knd of prosthess. A specalzed evolutonary mult-objectve algorthm based on structural evoluton prncples [14] and Pareto s domnance methodology [15, 16, 17] are used to select, generate and evaluate compromse solutons that generaton after generaton lead to the quas-optmum or optmum soluton of the optmzaton problem under the dfferent crtera selected n ther desgn. Ths paper s structured as follows: the optmzaton problem; the structural evolutonary prncples used to deal wth the problem of generate comprse desgns, after that, a numercal example s presented and dscussed to show the versatlty and the external shapes found by the mult-objectve optmzaton approach; and fnally the conclusons and future research. STRUCTURAL EVOLUTIONARY MULTIOBJECTIVE OPTIMIZATION The optmzaton problem conssts n optmze at the same tme a set of functon objectves constraned by a set of equalty and nequalty functons. The formulaton of ths problem s stated by: = f1( ) fn( ) ( ) = = ( ) = Mnmze F X,, X gj X 0 j 1,,m subjected to h k X 0 k 1,,l (1) The dea s to fnd a desgn vector of varables, X *, such that: * ( X ) = mn f ( X) = 1,, n f (2) However, ths s not the common stuaton and the objectve functons behave among then n the opposte sense. One way to solve ths problem s to fnd the greater number of solutons whch fulflled domnaton crtera of Pareto optmzaton for mult-objectve problems. In Pareto optmzaton, a soluton vector, X (1), s sad to domnate soluton vector X (2), f the followng condtons are satsfed at the same tme: 1. Evaluaton of soluton X (1) s less or equal than evaluaton of soluton X (2) for all objectves: f () 1 ( 2) ( ) f ( ) X X = 1,,n (3) 2. Evaluaton of soluton X (1) s strctly better than soluton X (2) n at least one objectve functon: { } () 1 ( 2) n( ) n ( ) n 1,2,,n : f X < f X (4) Generally, there s not a common mnmum (or maxmum) for all objectve functons. In the strct sense of the word there s not a mnmzaton at all, so that, the task of the desgner s to dentfy the greater number of possble Pareto mnmum and n terms of them select the most sutable soluton that n a compromse way solve the set of objectve functons. In general, structural evolutonary optmzaton s based on the prncple of slowly removng the neffectve materal of the structure, n a way that the performance of the structure s mproved n terms of the optmzaton crtera. In order to generate the ntal desgn populaton, modfy them n accord the optmalty crtera, evaluate the dfferent mert functons and obtan the Pareto front, an optmzaton algorthm s developed usng the suggestons gven n [15], whch s based on the dea of rankng the populaton by means of non-domnated solutons [16], and they are dstrbuted usng the sharng technques ntroduced by Goldberg [17]. An optmalty crteron based on stress performance of the structure can be used to control the strength of the desgn. In ths sense, the potental falure of a structure s gven by stress or deformaton levels hgher than the maxmum allowed stress or deformaton level. On
the contrary, lower levels of stress and deformaton can be nterpreted as neffcent use of materal. Based on ths concept, a local stress optmalty crteron can be stated n order to slowly remove neffcent materal of those areas of the structure wth lower stress levels, due to n these areas the materal s not used effectvely. In upcomng generatons of desgns the stress dstrbuton levels wll be more unform. Ths crteron s mplemented by usng the rate between the Von-Mses elemental stress ( and the maxmum Von-Mses stress ( e ) ) of max the structure, f ths ndex s less than the remoton rate (RR ) the fnte element s takng off the desgn. Ths procedure contnue untl no more fnte element can be removed under ths condton. In the next cycle, the remoton rate s ncremented by the evoluton rate (ER). The evoluton process s mantaned untl there s no materal to remove or s stopped when stress levels of the structure are lower than a certan percentage of the maxmum allowed stress (e.g.: 25%). e max < RR, +1 RR = RR + ER (5) The followng optmalty crteron s based on the stran energy deformaton of the structure (C). Ths concept s used as an ndrect measure of the average stffness of the structure, and s defned by: 1 T C = { P} { u} (6) 2 In ths crteron, the materal s taken off based on the dea of mnmzng the average stffness of the structure, whch generates the greatest possble structure s flexblty. In ths sense, the ndex (α ) s used to evaluate how the remoton of a partcular fnte element of the stent s model affect the stran deformaton energy of the total structure of the stent [14]. Ths ndex s defned as: T { u } [ K ]{ u } 1 α = (7) 2 In ths case, the structural evolutonary algorthm removes teraton by teraton a number of fnte elements, correspondng to one or two percent of the total numbers of fnte elements of the model, wth the greater values of the ndex (α ) f the objectve s to mnmze the average stffness of the structure, but f the objectve s to obtan the lght structure wth the hghest possble stffness, the elements to be removed are those wth the lower values of α. STENT DESIGN CRITERIA The mechancal propertes used most to evaluate the behavor of ths prosthess are [7-10]: () the nternal pressure requred to deploy and deform the stent to ts fnal placement; () the elastc ntrnsc retracton of the stent due to stent s materal elastc deformaton (elastc dameter recol). Ths parameter s measure n terms of the relatve reducton n stent s dameter after balloon deflaton: dameter recol = D - D D un (8) () the measurement of the relatve reducton n stent s length after balloon deflaton (elastc longtudnal recol): longtudnal recol = L - L L un (9) (v) the strong of the stent to compresson forces exerted by the atherosclerotc tssue of the artery. Ths resstance s evaluated as the requred pressure to reduce the stent s dameter ten percent after balloon deflaton and elastc recol; (v) the longtudnal shortenng of the stent: longtudnal shortenng = un L - L L (10) (v) the area covered by the stent, defned as the percentage rate between the metal lateral surface n contact wth the arteral cylndrcal wall surface; (v) the stent flexblty; and (v) the stress and stran levels of the devce n ts deformed state. The goal of ths paper s to present and dscuss a mult-objectve methodology approach whch allows the evaluaton of stent desgns takng n
consderaton the above dscussed crtera, the numercal example shows compromse solutons obtaned by mnmzaton of stress levels and the maxmzaton of the devce s flexblty, whch s requred to delver the stent to the deployment zone n the artery. NUMERICAL EXAMPLE The ntal 3D Fnte element model s depcted n fgure 2. A typcal stent of 4 mm of dameter and 30 mm of length was modeled wth 3700 parallelepped elements and 4550 nodes. The model s constraned to deform n the radal drecton. The materal used to smulate the expanson of the stent s 316L stanless steel. The nelastc consttutve response s descrbed through a Von Mses-Hll plastcty model wth sotropc hardenng. The Young modulus s 190 GPa, the Posson rato 0.3, the yeld stress 205 MPa [18]. Fg. 4: Rounded geometrcal patter scheme. The fnal confguraton mesh of the rounded geometrcal pattern s presented n fgure 5. Fg. 2. Intal 3D fnte element mesh. Intally, the model was ed by an nternal unform radal pressure up to ts reachng the double of ts ntal dameter. The followng fgures show some of the successful cuttng schemes obtaned by the algorthm Fg. 5: Lateral vew of the fnal mesh of the rounded geometrcal pattern scheme. the followng fgure, the Von-Mses stresses of the last pattern show the maxmum stresses are located n some of the lnks of the pattern. Fg. 3: Damond shaped scheme Fg. 6: Von Mses stresses of the rounded geometrcal pattern Fnally, Fgures 7 dsplay a cad reconstructon of the last desgn.
Naconal de Cenca y Tecnología (FONACIT) of Venezuela for the grants provded to support ths research. Fgure 7, Reconstructon of one of the fnal desgns usng a CAD tool and smoothng ts edges. CONCLUSIONS In ths paper were dscussed dfferent crtera used n the evaluaton of mechancal propertes of ntravascular prosthess, called stents, and how they can be used to propose a mult-objectve methodology, based on the prncples of evolutonary structural optmzaton and Pareto technques, to generate compromse external shape confguratons whch satsfy conflctve objectves. The use of optmzaton technques n addton to fnte element smulaton has become an mportant tool to study, analyze and understand the mechancal behavor of ths knd the devces. Its development allows proposng desgns of stents n concordance wth ts functon, decreasng the amount of expermental and clnc test whch are more expensve and complcated than numercal smulaton. The authors realze that the complete desgn of ths knd of devce requre a multdscplnary approach, whch nvolve bologcal, mechancal and manufacture aspects and the evolutonary optmzaton approach s one of the best way to deal wth ths problem. In ths work have been consder the devce s mechancal behavor but the robustness of the optmzaton tool allow the future ncorporaton of the others mechancal and medcal consderatons. ACKNOWLEGMENTS Dr. Annccharco wshes to acknowledge the support provded by the Consejo de Desarrollo Centífco Humanístco of the Unversdad Central de Venezuela (CDCH) and by the Fondo REFERENCES [1] Flesch, M., and Meer, B., Management and Outcome of Stents n 1998: Long-Term Outcome, Cardology Revew, 7, 215 218 (1999). [2] Amercan Heart Assocaton, Heart and Stroke Statstcal Update, Amercan Heart Assocaton, Dallas, Texas (1999, 2000). [3] Dotter C. T., Translumnally-placed col sprng end arteral tube grafts, long-term patency n canne poplteal artery. Investgatve Radology 4, 329-401, 1969. [4] Fschmann D. L., Leon M. B., Bam D. S., et al., A randomzed comparson of coronary stent placement and balloon angoplasty n the treatment of coronary artery dsease. New England Journal of Medcne 331, 96-501, 1994. [5] Sgwart U., Stents: a mechancal soluton for a bologcal problem?, European Heart Journal 18, 1068-1072, 1997. [6] European Markets for stents n perpheral vascular and non vascular applcatons. HIS Health Group RPAF1226, 2002. [7] Dumouln C., Cocheln B., Mechancal behavour modelng of balloon-expandable stents, Journal of Bomechancs 33, 1461-1470, 2000 [8] Etave F., Fnet G., Bovn M., Boyer J., Roufol G., Thollet G., Mechancal propertes of coronary stents determned by usng fnte element analyss, Journal of Bomechancs 34, 1065-1075, 2001. [9] Mglavacca F-, Petrn L., Colombo M., Aurccho F., Petrabssa R., Mechancal behavor of coronary stents nvestgated through the fnte element method, Journal of Bomechancs 35, 803-811, 2002. [10] Graca L., Puèrtolas S., Domngo S., Puèrtolas J. A., Fnte element smulaton of bendng and bucklng for a stanless steel stent n Internatonal Congress on Computatonal Boengneerng, Doblare, Cerrolaza, Herrera (Eds.), 676-682, 2003. [11] Prendergast P. J., Lally C., Daly S., Red A. J., Lee T. C., Qunn D., Dolan F., Analyss of Prolapse n Cardovascular Stents: A consttutve Equaton for Vascular Tssue and Fnte Element Modellng, ASME journal of Bomechancs Engneerng 125, 692-699, 2003. [12] Lally C., Dolan F., Prendergast P. J., Cardovascular stent desgn and vessel stresses: a
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