Flujo pulsante a través de una bifurcación carótida

8/12/2019 Flujo pulsante a travs de una bifurcacin cartida

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P u l s a t i l e F l o w T h r o u g h t h e C a r o t i d B i f u r c a t i o nL D J o u a n d S A B e r g e r

Depar tment o f M echan ica l Engineer ingUniver s i ty o f Ca l i fo rn ia a t Berke ley

B e r k e l e y , C a l i fo r n i a 9 4 7 2 0I n t r o d u c t i o n

I t has long been speculated that f luid dy aa m lcs m igh t play an impor tant role in a theroscleroticdisease s . L o w and high shear s tress, osci l la tory she ar s t ress, and the shear s tress gradient [1-4]hav e a l l been p roposed to r e la te f lu id dyn am ics and a the roscle ros is .

Because of the geomet ry and impor tance o f the ca ro t id b i furca t ion , i t has been a focu s ofresea rch for m ore than a decade. In the carot id , the bifurcat ion resul ts in very high sh ear s t resses ,wh ile the widen ing o f the carot id s inus leads to ve ry low shear st resses . B oth of these features ares i~ if i ca n t in the search to ident ify the s i tes and m ech anis m s o f atheroscleros is . I t appears thatgeom etr ical factors and the pulsati li ty o f the f lo w are both ve ry important to the init ia tion o fa the rosc le ros i s and the growth of the p laque . H o w separa t ion i s an impor tan t a spec t o f f low in thecarot id bifurcat ion. P laque is of ten fou nd in this lo w shea r s tress region. H o w separation a lsocau ses d iagnos tic p rob lems in Magnet ic R eson anc e Ang iograp hy (MR A) [5] , w hich is a w ide lyused mo da l i ty to de te rmine the degree of the s tenos i s. There fore , an unders tanding o f the ex ten t o fthe separation region and its implication fo r the m agn etiza tion saturation in MR A is essential toacc ura te clinical diagnosis.M e t h o d

In the pr esen t s tudy , the pseudo -compress ib i l i ty me thod , in wh ich the cont inu ity equa t ion i swr i t t en as

b~ - p 13 V - u = 0 ,is used fo r s teady f low [6] and for uns teady f low [7] . An implic i t f mite difference formulat ion w asem ploye d . A th i rd-order flux-sp li tt ing procedu re wa s used for the convec t ion te rm, and a seco nd-ord er one fo r the viscou s term [ 7]. Af ter l inear izat ion, the Nav ier -Stokes equat ions are o f thef o r m :

w hic h ma y be rewr it ten asI + 3- 1 Im +~ =:-- / I (D nl m+l - D n+l m)2 J A t \ 0 1 9 /

= - ~nl,m _ Im (1.5Dn+l,m _ 2 D n + 0 .5 Dn -l) ,J At


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w he re I is the t ransformation Jacobia n, At the ph ysic al t ime s tep, A ~ the ar ti ficial time s tep, I tbeiden t i ty matr ix , D - - [p u v w]T, R the res idue , ~ the conv ect ion flux, and f fiiv the visco us f lux. Inthe formu la , n is the phys ica l t ime and m i s the i te r a tion n um ber of the pseudo- time . There fore , i ti s s eco nd-o rder accura te bo th in space and in t ime .

T o inc rease the e f fic iency of the num er ica l s chem e, the pa ramete r 6 , which i s r elated to thepse ud o-w ave propaga t ion speed , needs to be o p t im ized . W hi le a fo rmu la for op t imum 13 has beenes tab l i shed [8] , num er ica l exper iments sho w ed tha t 1< ~ < 5 0 0 .

Th e vesse l wa l l i s a s sumed to be rig id , and the f lu id incompress ib le and New tonian . Lam inarf low i s a l so assumed. At the wa l l , the no - s l ip condi t ion i s impo sed , a s we ll a s ze ro norm alpressu re gradient . Th e veloci ty .prof i les , w hi ch are fu nct io ns o f t ime, are specif ied a t the entranceo f the co m m on ca ro tid (CCA ) and the ex i t o f the ex te rna l ca ro tid (EC A) , and the pr es sure a t theexi t o f the internal carot id CICA) .

Po int relaxation is used for the i tera tion w ith 1< 6< 10 0, and ~ increases with lulz . Severalre laxat ion sch eme s have been evaluated. O ur s imu lat ion s indicated that the conjugate gradient w ithincom ple te L U precondi tion ing was qu ite e f f ic ien t f o r s teady f low, b u t no t fo r uns teady f low. Inthe s imulat ion s , the res idues are redu ced th ree o rde rs o f m agnitud e within 40 i tera tions for s tead yf low. There fore , a s imula t ion advances to the ne x t t ime s tep whenev er the max imu m res idue i s l e s sthan 10 4 or af ter 40 i tera tions . Th e calcula t ion f or on e cardiac per iod , consis t ing of 120 t imes t e p s , t a k e s 5 0 h o u r s o n an I B M R I S C 3 4 0 wo r k s t a t i o n . S y s t o l e ta k es u p o n l y 2 5 p e rc e n t o f t h eca rd iac cyc le , and the f low ra te a t the pe ak o f sys to le i s 2 .5 t imes the mean f low r ate. Th eW o m e r s l e y n u m b e r is 7. 3 a n d th e R e y n o l d s n u m b e r i s 3 0 0 .

e s u l t s

T h e g e o m e t r y o f t h e c a r o t i d b i f u r c a t i o n v a r i e s f r o m i n d i v i du a l t o i n d iv i du al . I t c a n b ec h a r a c t e d z ~ b y : t h e a n g l e b e t w e e n t he I C A a n d C C A , t h e a n g l e b e t w e e n t h e E C A a n d C C A , t h ed i a m e t e r s o f t h e C C A , I C A a n d E C A a n d t h e d i a m e t e r o f t he c a r o t i d i n u s . T h e f l o w d e p e n d s o nt h e R e y n o l d s n u m b e r , t h e f l ow - ra t e a t i o t h e r a t i o o f t h e f l o w r a m i n t h e I C A t o t h a t n t h e C C A )a n d t h e p ul sa ti le w a v e f o r m . T h e g e o m e t r y u s e d i n t h i s s t u d y ( F i g . I ) w a s o b t a i n e d f r o ma v e r a g i n g 6 7 a n g l o g r a m s [ 9 ] . T h e b i f u r c a t i o n m a g i c b e t w e e n t h e I C A a n d E C A i s 5 0 . T h ed i a m e t e r o f t h e I C A a n d E C A a m 0 . 7 a n d 0 . 6 o f t h e d i a m c t e r o f t h e C C A , r e s p e ct i v el y . T h ew i d e n i n g o f t h e I C A i s 1 . 0 6 f t h e d i a m e t e r o f t h e C C A . T h e c o d e w a s v a l i d a t e d y c o m p a r i n g t h es i m u l a t i o n r e su l ts i t h t h e s t e a d y a n d u n s t e a d y f l o w s t h r o u g h a 9 0 c u r v e d d u c t [ I0 ].

H i g h w a l l s h e a r s c a n b e f o u n d o n t h e d i v i d e r w a l l a n d n e a r th e b if ur ca ti on , a n d t h e f l o ws e p ar a t e s f r o m t h e m a i n f l o w c l o s e t o t h e n o n - d i v i d e r w a i l s . T h e m a x i m u m w a l l s h e a r f o r p ul sa ti lef l o w i s 1 0 N / m 2 i n t h e 3 D s i m u l a t i o n n d 8 . 9 N / m 2 i n t h e 2 D s i mu l at i on ; n c on tr as t, n s t e a d y f l o wt h e m a x i m u m w a l l s h e a r s 4 . 5 N / m 2 i n t h e 3 1 ) s i m u l a t i o n a n d 3 . 5 N / m 2 i n t h e 2 1 ) s i m u l a t io n , a l lf o r a R e y n o l d s n u m b e r o f 3 0 0 a n d f l o w - r a m r a t i o . 6 . C o m p a r e d w i t h 1 .5 N / m 2 [ l I ] a s t h e v a l u ea b o v e w h i c h t h e b l o o d v e s se l i la te s n d b e l o w w h i c h t h e v e s s e l u m e n d e c r e a se s , o u r s i m u l a t i o n si n di c at e t h a t t h e w a l l s h e a r i s b e l o w t h i s l e v e l i n m o s t o f t h e f l o w r e g i o n e x c e p t n e a r t h eb i f u r c a t i o n .


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Se vera l carotid bifurcations with different bifu rca tion angles w ere studied. Although their f lowpatte rns vary slightly (Fig. 3), the distributions o f the wa ll shea r cha nge dramatically (Fig. 2). F orex am ple , the w all shea r decre ases at the bifurcation-internal wall w hen the angle of the IC Adec rease s. Th is clearly demonstrates the eff ec t o f the centrifugal force. In pulsatile flow , the f lowpattern and shear stress distribution in the carotid bifurcation are affected more by inertia than bygeo m etry (Fig . 4-5) . In addit ion , f low separat ion appears in both the ICA and ECA . Once theflow decelerates , the secondary f low bec om es s t ro nge r immediately. Figure 6 show s the veloci tycon tours and the secondary f low a t the widen ing o f the ICA .

i s c u s s i o nTh e carot id b ifurcation generates very s trong s eco nda ry f low s in the ICA and E CA , and these

seco nda ry f low s remain s t rong unti l they exi t the carot id s inus. The secondary f low is w ea kwithin the separation region, which m ay cau se saturat ion of magnetization in M RA [5]. Theinstantaneous streamlines are m ore com plicate d fo r pulsatile flow than for steady flo w w ith thesam e f low ra te and Reyno lds num ber (F igs . 3 ,5 ).

Th e f low always separates in the carot id s inus exce pt for ve ry low Reynolds num bers (< 100);the extent of the separation is affected m ore b y the f low -rate rat io than by the Reynolds nu m ber.O ur simulations indicate that the sepa ration reg ion is not close d even for steady flow , a nd thisgre atly affec ts both the particle residence t im e in the carotid sinus and the saturation o fm agn etization. Th e separation region gro w s dur ing deceleration and shrinks during acceleration.

Several d i fferent pulse-w ave form s hav e bee n s tudied . The shear s t ress in the separat ionreg ion rem ains low throughout the pulse cyc le desp ite cha nge s in the flow rate. In addition, them ea n wa ll shear stress is determined princ ipally by the m ea n flow rather than by the pulsatil ity~Re f e r e n c e s

[1] C. G. Caro, J . M. Fitzgerald, R. C. Schroter: . Proc. Royal Soc. London Ser. B, Vol. 177,109 (1971)

[2] D. L. Fry: Circ. Res. Vol. 24 , 93 (1969)[3] D . N. K u, D. P. Giddens: Atherosclerosis Vol. 3, 31 (1983)[4] M. L i , C. Kleins t reuer , G. A. Trusk ey: J . Biomech. Eng. Vo l. 117 , 350 (1995)[5] R. van Tyen , D. Saloner, L. D. Jou, S. A. Be rger: Magn. Reson. Med. Vol. 31, 184 (1994)[6] A. J. Chorin: J . Comp. Phys. Vol. 2 , 12 (1967)[7] S. E. Rogers , D. Kwak, C. Kids: A/AA J . Vol . 29 , 603 (1991)[8] C. L. M erkle , Y . H. Choi: Int . J . Numer. Methods Eng. Vol. 25, 293 (1988)[9] B. K . Bharadvaj : J . Biomech. Eng. Vol . 15 ,349 (1982)

[10] J . A. C . H ump hrey , A. M. K. Tay lo r , J . H. W hi te law: J . F l u i d M ech . Vol . 83 ,509 (1979)[11] D. P. Giddens: Appl. Mech. Rev. V ol. 43 , 98 (1990)


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Figure 1 The grid representation o f the carotid bifurcation

z11

u n i i [

- - I t ) ' / ,10" -. . . . . 2 5 V 2 5 ". . . . . . . 4 0 " / 1 0 "

i i i i0 2 4 6

C o m m n n - i n t c r n a l w a l l- 2

:-: - - 1 0" /40 " [

; : 40"/ lO"

-

O 64 8Bi t u r c a t i ~ l n - in t , z m a i w a l l

Figure 2 The wall shear in the IC A fo r different bifurcation angles


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Figure 3. The streamlines in the carotid bifurcation for different bifurcation angles (ICA/ECA)(a) 10 /40 , (b) 25*/25*, (c) 40 /10 at Re = 3 00 and Q~/Qc = 0.65.

10

6E8 4

2

-2

i i i i

~ _ t = (I.OWI - t = 0.2.2T.. .. t = 0.4YTt t = 0.59T

i i i i-2 0 2 4 6

C o m m o n i n t e r n a l w a l l

10

2

0

I - \ / -, =0 22T/ - - .. .. t = 0.45T . . .. .. .. t = 0.59T. : ' - . . .r -% / . ' : . ' - - 2 ' . .

I J L I-2 0 2 4 6

B i f u t c a t i o n i n a t m a l wail

Figure 4. The wall shear in the ICA under physiological wave form


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Figure 5 . The s t r eaml ines i n t he caro t id b i fu rca t io n und er t he physio logica l wave fo rm(a) t = 0.09 I , Co) t = 0.22 T (e nd sy sto le) , (c) t = 0.4 5T, (d) t = 0.59 T (diastole)

a) b) ~)

d) (e) r ) l )Figure 6 . The ve loc i ty con tou rs an d v ec to r p lo t i n the ICA

Steady flow : (a) 10 /40 , (b) 25 /25 , (c) 40/10 Pulsat i le f low: (d) t = 0. 09 T, (e) t = 0.2 2T , (f) t = 0.45T, (g) t = 0.59T

Flujo pulsante a través de una bifurcación carótida

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