Comparative Study of Netonian Sinusoidal Blood Flow through Normal and Stenosed Carotid Artery

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1 IOSR Jornal of Mahemacs (IOSR-JM) e-issn: , p-issn: X. Volme 13, Isse 6 Ver. III (Nov. - Dec. 2017), PP Comparave Sdy of Neonan Snsodal Blood Flow hrogh Normal and Senosed Carod Arery Mohammad Mar Rahman 1,a),Md. Anwar Hossan 1,b), Kharzzaman Mamn 1,c), Mos. Nasrn Aher 1,d). 1 (Deparmen of Mahemacs, Dhaa Unversy of Engneerng and Technology, Gazpr-1700, Bangladesh) a)correspondng ahor: marrahman1975@gmal.com b) anwarmah23@gmal.com,c) presdenmamn@gmal.com,d) nasrn @yahoo.com Absrac: A nmercal smlaon s performed o nvesgae Newonan snsodal flows behavor on hree dmensonal dealzed carod arery () and sngle senosed (75% by area) carod arery(s). The wall vessel s se as rgd drng smlaon. Bfrcaed blood vessel are smlaed by sng hree-dmensonal flow analyss. Snsodal and parabolc velocy profles are se o o f he condons of nle bondares of arery. The nvesgaon has a Reynolds nmber range of 90 o Low Reynolds nmber model has been sed as governng eqaon. The nvesgaon has been carred o o characerze he flow behavor of blood n wo geomery, namely, () Normal carod arery () and () Senosed carod arery (S). The Newonan model has been sed o sdy he physcs of fld. The fndngs of he wo models are horoghly compared n order o observe here behavoral seqence of flows. The nmercal resls were presened n erms of velocy, pressre, wall shear sress dsrbons and cross seconal veloces as well as he sreamlnes conor. Senoss dsrbs he normal paern of blood flow hrogh he arery as redced area. A senoss regon velocy and pea Reynolds nmber rapdly ncrease and Reynolds nmber reach ransonal and rblen regon. These flow flcaon and rblence have bad effec o he blood vessel whch maes o accelerae he progress of senoss. Keywords : Aheroscleross, Carod arery, Snsodal blood flow, Senoss, Vscosy Dae of Sbmsson: Dae of accepance: I. Inrodcon Sroe s a prevalen and severe medcal condon ha mposes an mmense loss of healh and heavy economc brden worldwde. Globally, sroe s he second leadng case of deah and he leadng case of long-erm dsably [1, 2]. The adverse effecs are boh eensve and long-lasng, as one hrd of new sroes resl n deah and mos sroe srvvors are afflced by a nerologcal dsably ha reqres rehablaon or long-erm care [3]. Moreover, once a sroe has occrred, he rs of sroe recrrence ncreases sbsanally [4, 5]. By far, he mos common form of sroe s de o schema a resrcon or shorage n cerebral blood spply whch accons for 78% of sroes [6]. Ischema s mos ofen cased by a hrombs or clo ha has emblazed and ravelled o he bran casng areral occlson of a narrow cerebral vessel [7, 10] The hrombs ypcally orgnaes from a promal locaon n he body, sch as he hear (cardogenc embolsm) or from 2 aheroscleroc plaqe n a large promal arery (arery-o-arery embolsm), commonly he aora or nernal carod arery (I). Aheroscleross localzed n he carod areres s no only prevalen, b also represens a sgnfcan rs for clncal cerebrovasclar evens. The carod arery s he mos freqenly nvolved se n large arery aherohromboc sroe. Carod arery aherohromboss s esmaed o be responsble for 10 30% of all sroes [4, 6-8]. he geomery of he flow vessel s a ey deermnan of s flow characerscs. I has been well esablshed ha senoses, bfrcaons, and srface roghness are assocaed wh flow dsrbances and comple flow [9-11].The nflence of vessel geomery on flow s eemplfed n he areral vasclare. Even n he normal, healhy vasclare, vessel bends and bfrcaons, as well as he plsale flow of he cardac cycle, nrodce secondary flows and oscllang shear sresses. Aheroscleross adms ye more complcaons, parclarly where areral dsease s severe enogh o case a vessel senoss [15]. Whereas flow n normal areres s comple b ypcally lamnar, senoss are assocaed wh shear layers, eddy formaon, dsal flow dsrbances [15], and velocy flcaons [16]. Toros, rreglar vessel srfaces are lely o frher conrbe o flow dsrbances [9, 10]. A he senosed carod arery, flow frs enconers an asymmerc bfrcang ncon, hen mmedaely goes hrogh a senoc flow consrcon, followed downsream by a sdden epanson no he carod blb. The bfrcaon mposes a flow dvson ha affecs he bondary layers along boh he nsde and osde bends, whch s dependen on he eac branch geomery. DOI: Page

2 The comparson of senosed flow behavor wh he normal one can provde he proper ndersandng of nderlyng mechansm behnd he developmen of aheroscleross. The flow rns o be abnormal n he redced cross seconal area of he arery saed by Kader and Shenory [7]. The flow behavor n he senosed arery s qe dfferen n comparson o he normal one. Sress and ressance o flow s mch hgher n senosed arery. Cha and Shread [9] fond ha he flow hrogh he consrced be s characerzed by hgh velocy e generaed a consrced regon. Kader and Shenory [7] fond he resls from nmercal smlaon whch demonsraed ha velocy and senoc e lengh ncreases n ncreasng he severy of senoss. Ther resls also demonsrae ha he 3D senoc CFD model s capable of predcng he changes n flow behavor for ncreased severy of senoss. Yong e al. [17] sded he wall shear sress and pressre graden n he senoss and evalaed he case of plaqe rpre. The ahors sded plsale blood flow hrogh he senoss wh elasc wall o observe he lmen movemen. Accordng o her sdy he pea WSS occrred s before mnmm lmen poson. Pno e al. [18] performed nmercal smlaon assmng a physologcal plsale flow hrogh dfferen models of senoss. In case of sbec-specfc anaomcally realsc senosed carod bfrcaon sbeced o plsale nle condon, he smlaon resls demonsraed he rapd flcaon of velocy and pressre n pos-senoc regon by S. Lee and S. Lee [16]. Ahmed and Gddens [19] sded boh seady and plsale flow hrogh 25%, 50% and 75% consrcon of a rgd be where Reynolds nmber ranged from 500 o In anoher sdy, he senoss havng dfferen geomerc profles le rapezm, sem-ellpse and rangle were analyzed consderng he Non-Newonan behavor by Lorenzn and Casalena [20]. Mos of he researcher consdered senosed carod arery or normal one. B n hs sdy we se normal and senosed carod arery ogeher o ndersand he flow behavor correcly. The nvesgaon has a Reynolds nmber range of 90 o The nmercal resls are presened n erms of velocy, pressre, wall shear sress dsrbons.ths paper s organzed as follows. In secon Model descrpon we dscss he geomery and blood properes. In secon Governng eqaon and Nmercal mehod we dscss he governng eqaon, bondary condon, grd ndependence chec, valdaon and nmercal scheme. In secon Resls and dscsson we dscss and smmares he wss, velocy and pressre dsrbons hrogh he carod arery. II. Model descrpon 2.1 Geomery Three dmensonal carod arery () and 75% senosed(by area) carod arery (S) are sed as geomery for hs sdy shown n Fg. 1 (a). Bfrcaon angles I and E 18.5 ; he rads of he blood vessel C(4mm), mddle of senoss (2mm), end of I(2.5mm), end of E(2mm); lengh of he arery from VI() are a poson () 72mm, () 80mm, (l) 88mm, (m) and (n) 103mm, (o)and (p) 118mm and (q) and (r) 142mm; percen of senoss (75% by area). Fg. 1(b) shows he mesh n cross-seconal plane of a carod arery. (a) (b) Fgre 1: (a) Model of a carod arery ; (b) mesh n cross seconal plane of a carod arery. 2.2 Blood properes The densy of he blood s 1050gm 3. In a Newonan model for he blood vscosy, he vale of μ s 3 reaed as a consan sally se o pa. s. DOI: Page

3 DOI: Page III. Governng Eqaon And Nmercal Mehod 3.1 Governng eqaon De o consrcon, blood passes hrogh he hroa and pos senoc regon wh hgh velocy. Flow velocy a pos senoc regon ncreases b pressre of ha regon decreases. So, moderae and severe areral senoss resrcs he flow severely wh creang rblen flow characerscs. For hs regon, rblen flow s epeced a pos senoc regon especally for 75% senoc arery. Neher lamnar flow modelng nor sandard wo-eqaon models are sable for hs nd of blood flow. So, Wlco low-re rblence model s more accepable for flow analyss fond by Varghese and Franel [22]. Therefore, low Re rblen model s aen for calclaon. Now, he Naver-Soes eqaon can be gven by- 0 (1) p 2 1 (2) Snce each erm of hs eqaon s me averaged, he eqaon s referred o as a Reynolds averaged Naver-Soes (RANS) eqaon. Drng hs procedre, several addonal nnown parameers appear whch reqre addonal eqaons o be nrodced as rblence models. The se of RANS eqaons are- 0 (3) p (4) In hs eqaon s an addonal erm nown as he Reynolds s sress ensor, whch can be appromaed by sng Bossnesq s hypohess- 3 2 (5) Eddy vscosy can be modeled as a fncon of he rblence nec energy and specfc dsspaon rae ; herefore s referred o as he wo-eqaon rblen model. The rblence nec energy and specfc dsspaon rae of sandard model are deermned by followng wo eqaons: The eqaon: S Y G (6) The eqaon: S Y G (7) In hese eqaons, G represens he generaon of rblence nec energy de o mean velocy gradens. G represens he generaon of ω. and represen he effecve dffsvy of and ω, respecvely. Y and Y represen he dsspaon of and ω de o rblence. S and S are ser-defned sorce erms. A low Reynolds nmber correcon facor conrols he nflence on he overall srcre of he flow feld, dependng pon local condons, and s gven as-

4 rr Re R 0 1 Re R (8) Where, Re, R 8, 0, 0.072, 1. 4 Closre Coeffcen for he Transonal Model are- 1, 0.52, , 0.09, 0.072, 8and R 8 Dealed descrpons of seen harmonc coeffcens are shown n TABLE 1. Here, Reynolds nmber vares from 90 o Snce cardac plse cycle s 0.82sec, s plsaon, fond from he calclaon rad 0.82 sec R. Fg. 2(a) and 2(b) show oscllaory physologcal waveform and parabolc nle velocy profle respecvely. In fgre 2(a), a, b, and c represen he posons of early sysole (0.0615sec), pea sysole ( sec), and dasole (0.533 sec) respecvely. Fgre 2: (a) oscllaory snsodal waveform, (b) parabolc nle velocy profle, and (c) velocy dsrbon n 75% senoc arery from dfferen mesh szes. 3.2 Grd ndependence chec An eensve es s carred o wh dfferen szes of mesh sch as mesh 1 ( elemen), mesh2 ( elemen) and mesh3 ( elemen) respecvely. Fg 2(c) shows he velocy dsrbons for 75% senosed carod arery wh menoned mesh szes. In all cases, he velocy dsrbons are same. I mples ha he solon s grd ndependence. 3.3 Valdaon Before sarng of presen nvesgaon he nmercal smlaon s needed o be valdaed. Valdaon of he presen nmercal compaon s done by plong he seady velocy profle a downsream from he senoss hroa and comparng wh he velocy profle of Varghese and Franel [22]. For hs case, a parabolc velocy profle s assmed as nle bondary condon. The mean nle velocy corresponds o Reynolds nmber 500 and he flow s assmed o be seady. The resls are shown n he Fgre 3, where a good agreemen can be fond wh Varghese and Franel [22] Low Re -ω (presen) Varghese and Franel Fgre 3: Valdaon wh he velocy profle of Varghese and Franel [22]. DOI: Page

5 3.4 Nmercal scheme The nmercal smlaons are performed by well-nown sofware ANSYS Flen A pressre based algorhm s chosen as he solver ype. Ths solver s generally seleced for an ncompressble fld. As here s no hea ransfer n he blood flow process, energy eqaon s no solved. Snce rblen s epeced n 75% senoc carod arery a pos senoc and bfrcaon regon, a low Reynolds nmber rblen model s sed hrogho he wor. In solon mehods, he SIMPLE algorhm s seleced for pressre-velocy coplng. Frs Order Upwnd scheme s employed as a nmercal scheme for dscrezaon of he momenm eqaon. The me sep s se o sec wh 2000 nmber of oal me seps. Mamm 10 eraons are performed per each me sep. IV. Resls And Dscsson Of Snsodal Blood Flow The compaonal resls of snsodal blood flow are condced o sdy he nflence of senoss on he flow behavor. The flow parameers le velocy, pressre, WSS are observed from ransverse and longdnal conors a specfc nsans of plse cycle for comparng he flow varaon. The dscsson s caegorzed wh he observaons of flow varaon sarng from early sysole, pea sysole and dasole, respecvely Wall Shear Sress(WSS) dsrbon WSS s an mporan facor o deermne he effec of severy of areral senoss. I depends on he vscosy of he fld and velocy graden and, s defned as d, where s he vscosy and d s he velocy dr dr graden. Fgres, 4(a), 4(b), 4(c) show he conors of WSS n normal carod arery and 75% (by area) senosed carod arery and Fgs.5(a), 5(b), 5(c) show he comparsons of WSS n boh senoc arery a earlysysole, pea sysole and dasole respecvely. In boh cases, WSS s nformly dsrbed a C regons. The resls reveal ha he flow paern changes abrply n he regon of senoss hroa and bfrcaon onwards n he downsream whereas no sgnfcan change s observed n he psream and s sgnfcanly observed ha he changes of WSS s de o senoss and bfrcaon. Srprsngly a sharp rase of WSS and mmedaely declne are observed n he senoc hroa. From he comparsons, or fndng s ha mamm WSS s observed a pea sysole de o senoss. Afer bfrcaon WSS becomes hgh de o snoss a all cases and hen WSS becomes more low a pea sysole and dasole. Fgre 4:Comparson beween Carod arery and Senosed carod arery WSS conor a (a) early sysole (b) pea sysole and (c) dasole. DOI: Page

6 Fgre 5:Comparson beween Carod arery and Senosed carod arery WSS dsrbon a (a) early sysole (b) pea sysole (c) dasole. 4.2.Velocy dsrbon In fgre 6, nle velocy n boh geomeres begns wh same magnde for all case. Veloces a he hroa and pos senoc regon for all cases of S are relavely hgher han ha of accordng arery. Recrclaon regon s observed n S a pos senoc regon for all cases. In he oher hand, velocy s lamnar n C regon for all cases. The hghes velocy s observed n S a pea sysole and he lowes velocy s observed n a dasole. Fgre 6: Comparson beween Carod arery and Senosed carod arery a dfferen plse cycle. Comparson beween Carod arery and Senosed carod arery velocy conors a cross-secon (a) early sysole (b) pea sysole and (c) dasole a dsance from velocy-nle () 0.0mm (VI),() 0.072mm, () 0.08mm (mddle of senoss), (l) 0.088mm (bfrcaon pon) and (m) 0.103mm are shown n fgre In all cases we see no change n velocy dsrbon before senoc regon. Fgre 4.4 shows de o bfrcaon velocy decreases n all cases and velocy s hgher n nner sde han he oer sde of he arery. The presence of senoss ncreases he phenomena,so he possbly of formaon plaqe n oer wall s hgh. Carod Arery Senosed Carod Arery DOI: Page

7 pressre(pa) pressre(pa) pressre(pa) Fgre 7:Comparson beween Carod arery and Senosed carod arery velocy conor a cross-secon (a) early sysole (b) pea sysole and (c) dasole a dsance from velocy-nle () 0.0mm (VI),() 0.072mm, () 0.08mm (mddle of senoss), (l) 0.088mm (bfrcaon pon) and (m) 0.103mm Pressre dsrbons Pressre conor n boh areres a early-sysole, pea-sysole, and dasole are shown n Fgre 8 and cross-seconal pressre dsrbon along dameer are shown n fgre 9. Fgre 8(a) shows he conor of pressre n and S a early-sysole. In he pressre s nform and s varyng wh a small varaon hrogho he geomery. Hgh pressre a pre-senoc regon and sdden declne of he pressre a he hroa are characerzed n S. Agan, fgre able 1 show ha he pressre a mddle pon of cross-secon (), () (l), (m) and (n) for and S are as follows: () mmhg () mmhg (l) mmhg (m) mmhg (n) mmhg Early sysole S Pea sysole S Dasole S Table 1: Pressre a mddle pon of dfferen cross-secon The resls clearly noced ha S can case faal deah by blasng he pre senoc areral regon of he paen of hgh blood pressre. From fgre 8(c), we observe ha, pressre s nform and changng wh a small varaon before he hroa. From able 1, s observed ha, he pressre a before hroa and hroa for are mmhg and mmhg and for S are mmhg and mmhg respecvely. I s well nown ha pressre dfference may provde some nd of era force ha may drve blood hrogh he senoss wh hgh velocy. Snce The pressre dfference n S s hgher han ha of for all case. So a every case S creaes more force han a her pre-senoc regon. Fgre 8:Comparson beween Carod arery and Senosed carod arery pressre conor a (a) early sysole (b) pea sysole and (c) dasole. ( ) S S S rr rr r R DOI: Page

8 pressre(pa) pressre(pa) pressre(pa) pressre(pa) pressre(pa) pressre(pa) pressre(pa) pressre(pa) pressre(pa) pressre(pa) pressre(pa) pressre(pa) ( ) (l ) ( m ) ( n ) rr S S rr S rr S rr (a) (b) (c) Fgre 9: Comparson beween Carod arery and Senosed carod arery pressre dsrbon a cross-secon along dameer (a) early sysole (b) pea sysole and (c) dasole a dsance from velocy-nle () 0.072mm, () 0.08mm (mddle of senoss), (l) 0.088mm (bfrcaon pon), (m) 0.103mm(I) and (n) 0.103(E). 4.4Sreamlne conors Varaons of sreamlnes conors have been nvesgaed for and S o sdy he flow paern of blood hrogh arery. Sreamlne dsrbon for he models a early-sysole, pea-sysole, and dasole s shown n fgre10. A early sysole here s no vore n and sreamlnes rn smoohly hrogho he geomeres. B n S a very small vore s noced afer he hroa and bfrcaon regon, ecep hs mos of he sreamlnes rn smoohly hrogho he geomery. A pea all areres creae vore. Two large vorees a he pos senoc regon near he hroa are seen n S and a lle rblence s seen a I n. B here are dfferences n vore sze and shape. A dasole all areres creae sgnfcan sze of vore. Lengh of vore s ncreased n S. Sgnfcan sreamlne dfferences are observed among he areres. S creae large dfference n sreamlne from. S rr S rr S rr S rr S rr S rr rr S S rr DOI: Page

9 Fgre 10: Comparson beween Carod arery and Senosed carod arery velocy sreamlne a (a) early sysole (b) pea sysole and (c) dasole. V. Conclson Nmercal smlaons are performed on plsale blood flow hrogh carod arery and 75% severy senosed carod arery. Snsodal velocy profles are mposed a nle for he smlaons. Wall shear sress dsrbon shows ha as severy ncreases, shearng of flow also ncreases for all cases. And mamm sress s eered n hroa regon a pea sysole. Pressre dsrbon demonsrae ha a every case senosed carod arery creaes more force han normal carod arery a her pre-senoc regon and 75% areral senoss can case faal deah by blasng he pre senoc areral regon of he paen of hgh blood pressre. De o senoc consrcon, he flow behavor changes abrply n he downsream of he senoss. Drng pea sysole he flow ncreases a he hroa regon formng a e and laer dsrps sddenly formng eddes de o pressre drop n downsream sde. Analyss of sreamlne shows ha a recrclaon regon s vsble a he pos senoc regon a 75% senoc carod arery for all cases and recrclaon regon ncreases wh ncreasng mnmm velocy a he nle flow. Fnally, sgnfcan flow changes are noced n senosed carod arery. So senoss s very harmfl condon o any paen. References [1]. Lopez AD, Mahers CD, Ezza M, Jamson DT and Mrray CJ., Global and regonal brden of dsease and rs facors, 2001:sysemac analyss of poplaon healh daa, Lance 367: , (2006). [2]. Lloyd-Jones D, Adams RJ, Brown TM, Carnehon M, Da S, e al., Hear dsease and sroe sascs-2010 pdae: a repor from he Amercan Hear Assocaon Crclaon 121:e46-e215, (2010). [3]. Haney GJ, Jamroz K, Broadhrs RJ, Forbes S and Anderson CS., Long-erm dsably afer frs-ever sroe and relaed prognosc facors n he Perh Commny Sroe Sdy, Sroe 33: , (2002). [4]. Mohan KM, Crchon SL, Greve AP, Rdd AG, Wolfe CD, e al., Freqency and predcors for he rs of sroe recrrence p o 10 years afer sroe, he Soh London Sroe Regser. J Nerol Nerosrg, Psychary 80: (2009). [5]. Paala J, Haapanem E, Meso AJ, Meso TM, Aro V, e al., Recrren schemc evens n yong adls afer frs-ever schemc sroe., Ann Nerol 68: , (2010). [6]. O'Donnell MJ, Xaver D, L L, Zhang H, Chn SL, e al., Rsfacors for schaemc and nracerebral haemorrhagc sroe n 22 conres(he INTERSTROKE sdy): a case-conrol sdy., Lance 376: , (2010). [7]. S. Khader and B. Shenoy., Effec of ncreased severy n paen specfc senoss,world ornal of Modellng and Smlaon, Vol. 7 No. 2,pp ,(2011). [8]. M.X. L, J.J. Beech-Brand, L.R. John,P.R.Hosns and W.J. Easson., Nmercal analyss of plsale blood flow and vessel wall mechancs n dfferen degrees of senoss,j. Bomech, 40, ,(2007). [9]. C, Cha and G. Shread., Changes n flow and wall sresses hrogh areral consrcon offse from he cenre of he vessel, The Anzam Jornal, 50: C744-C759,(2009). [10]. Ags, A.D., B. Arff, S. A. G. McG. Thom, X.Y. X and A. D. Hghes., Analyss of comple flow and relaonshp beween blood pressre, wall shear sress and nma-meda hcness n he hmen carod arery,am. J. physol. Hear Cr. Physol.293: ,(2007). [11]. Bale-Glcman, J.K. Selby, D. Saloner and O. Savas., Epermenal flow sdes n eac-replca phanoms of aheroscleroc carod bfrcaonsnder seady np condons, J. Bomech. Eng. 125(1):38-48,(2003). DOI: Page

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