Numerical analysis of pulsatile blood flow in realistic coronary bypass models

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NUMERICAL ANALYSIS OF PULSATILE BLOOD FLOW IN
REALISTIC CORONARY BYPASS MODELS
JAN VIMMR∗, ALENA JON´
Aˇ
SOV ´
A∗AND ONDˇ
REJ BUBL´
IK∗
∗Uni e si y o Wes Bohemia, Depa men o Mechanics
Uni e zi ni 22, CZ-306 14 Pilsen, Czech Republic
e-mail: j imm @kme.zcu.cz
Key wo ds: Ao o-co ona y Bypass, End- o-side Anas omosis, Side- o-side Anas omo-
sis, Hemodynamics, Pulsa ile Flow, FVM
Abs ac . The pape ’s objec i e lies in nume ical modelling o pulsa ile blood low in
comple e ao o-co ona y bypass models econs uc ed om CT da a, especially in models
wi h indi idual and sequen ial bypass g a s. Uns eady blood low is desc ibed by he non-
linea sys em o he incomp essible Na ie -S okes equa ions in 3D, which is nume ically
sol ed using de eloped compu a ional algo i hm based on he ully implici p ojec ion
me hod and on he cell-cen ed ini e olume me hod o hyb id uns uc u ed e ahed al
g ids. Ob ained nume ical esul s a e discussed wi h ega d o dis ibu ion o eloci y,
wall shea s ess and oscilla o y shea index a p oximal and dis al anas omoses, i.e., in
a eas p one o de elopmen o in imal hype plasia.
1 INTRODUCTION
Acco ding o he mos ecen Eu opean ca dio ascula disease s a is ics[1] almos hal
o a ec ed people had ei he b ain s oke o hea in a c ion. Conside ing also he in-
c easing numbe o pa ien s and su gical ea men s connec ed wi h s enosed o occluded
a e ies, he unde s anding o ca dio ascula disease o igin and de elopmen is c ucial
o i s u u e p e en ion and ea men . In he case o ischemic hea disease, one o
possible su gical in e en ions is he implan a ion o enous bypass g a s, o en c ea ing
a de ou be ween ao a and he damaged co ona y a e y b anch. The ailu e a e o
implan ed bypass g a s is mos ly ela ed o he de elopmen o in imal hype plasia, an
abno mal healing p ocess in he anas omosis egion. I is ypical o he hickening o u-
nica in ima leading o dec ease in he a e ial lumen and consequen ly o g a ailu e[2].
Nowadays a he oscle osis and in imal hype plasia alike a e hypo hesized o be igge ed
by non-uni o m hemodynamics leading o mo phological and me abolic changes wi hin
he essel wall[3]. Reci cula ion zones and low and oscilla ing wall shea s ess a e some o
he supposed igge ing ac o s ha a e esponsible o endo helium ac i a ion[4]. Beside
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XI In e na ional Con e ence on Compu a ional Plas ici y. Fundamen als and Applica ions
COMPLAS XI
E. Oña e, D.R.J. Owen, D. Pe ic and B. Suá ez (Eds)
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Jan Vimm , Alena Jon´aˇso ´a and Ondˇ ej Bubl´ık
clinical esea ch, nume ical in es iga ion o bypass hemodynamics ep esen s a aluable
insigh in o he p oblem and p o ide so be e unde s anding o blood low in luence on
ca dio ascula disease occu ence and de elopmen .
The objec i e o his pape is he modelling o pulsa ile blood low in pa ien -speci ic
ao o-co ona y bypass models wi h emphasis placed on he hemodynamics s udy in in-
di idual and sequen ial ypes. We deno e he indi idual ype as he bypass wi h one
p oximal and one dis al (end- o-side) anas omosis and he sequen ial ype as he one wi h
one p oximal and wo dis al (side- o-side and end- o-side) anas omoses. As is appa en
om he p esence o mul iple anas omoses, he sequen ial bypass echnique is o en em-
ployed o connec se e al co ona y a e ies by one g a . Howe e , a majo unknown o
such bypasses lies in he esul ing blood supply o each connec ed co ona y a e y and he
possibili y o in imal hype plasia o ma ion a one o bo h es ablished anas omoses. In
his ega d, we pe o m a nume ical analysis o pulsa ile blood low in wo ao o-co ona y
bypass models econs uc ed om CT da a p o ided by he Uni e si y Hospi al Pilsen,
Czech Republic. Fo esul s discussion, special emphasis is placed on he e alua ion and
analysis o main hemodynamic ac o s such as dis ibu ion o eloci y, wall shea s ess
(WSS) and oscilla o y shea index (OSI) in a eas ha may be p one o de elopmen o
in imal hype plasia.
2 PROBLEM FORMULATION AND BYPASS MODELS
In compa ison o o he published pape s, which mos ly deal wi h dis al bypass pa s,
especially he dis al anas omosis[5], he p esen s udy conside s comple e ao o-co ona y
bypass models wi h ealis ic geome y, i.e., bo h p oximal and dis al anas omoses a e mod-
elled. In his way, he p oblem o bounda y condi ions may be adequa ely app oached. In
human essels, blood beha es as an incomp essible non-New onian luid. As is shown in
o he s udies[6], blood’s shea - hinning beha iou may be neglec ed in selec ed cases such
Figu e 1: CT scans o indi idual ao o-co ona y bypass model – posi ion o he p oximal end- o-side
anas omosis (le ) and he dis al end- o-side anas omosis ( igh )
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Jan Vimm , Alena Jon´aˇso ´a and Ondˇ ej Bubl´ık
Figu e 2: Indi idual ao o-co ona y bypass model – ( om le o igh ) p ima y econs uc ion om
CT da a, model a e smoo hing, e ahed al compu a ional mesh wi h deno ed compu a ional domain
bounda ies
as in human ao a. Fu he mo e, p e ious nume ical simula ions o s eady non-New onian
blood low pe o med by he au ho s o his pape in an idealized co ona y bypass model[7]
showed negligible non-New onian e ec s. The e o e, in he p esen s udy, blood’s com-
plex heological p ope ies a e neglec ed and blood is assumed o be an New onian luid.
Fu he , all nume ical simula ions o pulsa ile blood low a e ca ied ou o bypass models
wi h igid and impe meable walls. The au ho s a e awa e ha his assump ion, especially
in connec ion wi h he elas ic ao a, ep esen s a ele an limi a ion o he cu en ma h-
ema ical model. Howe e , an imp o emen in ela ion o essel compliance is planned in
one o hei u u e s udies.
Fo he pu pose o he p esen s udy, wo se s o CT da a we e p o ided by he Uni-
e si y Hospi al Pilsen, Czech Republic. The i s da a se con ained he CT scans o an
indi idual g a , Fig. 1, whose p oximal end was a ached o he ao a and he dis al one
was sewn o an occluded b anch o co ona y a e ies. The co esponding p ima y model
a e econs uc ion in so wa e Ami a is shown in Fig. 2 oge he wi h he inal smoo hed
model and he uns uc u ed e ahed al compu a ional mesh, which was gene a ed in he
sys em Al ai Hype mesh. Fig. 3 gi es a de ailed iew a he mesh in he dis al anas-
omosis egion and in he p e-anas omosis co ona y bi u ca ion. Fo his bypass ype,
ollowing inle and ou le bounda y condi ions a e p esc ibed, see bounda ies deno ed in
Fig. 2,
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Jan Vimm , Alena Jon´aˇso ´a and Ondˇ ej Bubl´ık
Figu e 3: Indi idual ao o-co ona y bypass model – iew a he dis al end- o-side anas omosis egion
wi h co ona y bi u ca ion
•ao a inle ∂Ω(A)
Iand ao a ou le ∂Ω(A)
O– ime-dependen low a e Q( ) and p es-
su e p( ), espec i ely, a e applied. The alues o low a e and p essu e a e aken
om li e a u e[8], Fig. 4;
•co ona y a e y ou le ∂Ω(CA)
O– cons an p essu e equal o a e age a e ial p essu e
12 000 Pa is gi en;
•occluded co ona y a e y is ea ed as a igid wall bounda y ∂ΩW.
In o de o pe o m nume ical compu a ions wi h non-dimensional p imi i e a iables,
e e ence alues ha e o be se . Fo he indi idual g a , he e e ence eloci y U(1)
e is
Figu e 4: Indi idual ao o-co ona y bypass model – ime-dependen low a e Q( ) p esc ibed a he ao a
inle ∂Ω(A)
I(le ) and ime-dependen p essu e p( ) p esc ibed a he ao a ou le ∂Ω(A)
O( igh )
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Jan Vimm , Alena Jon´aˇso ´a and Ondˇ ej Bubl´ık
chosen o be equal o 0.1592 m ·s−1, co esponding o a e age ao a inle low a e, Fig. 4
(le ), and he e e ence diame e is D(1)
e =0.036 m, co esponding o a e age ao a
diame e . Acco ding o p esc ibed bounda y condi ions[8], blood’s densi y and iscosi y
a e conside ed o be cons an η(1) =0.0049 Pa ·s and (1) = 1055 kg ·m−3, espec i ely.
The second se o p o ided CT scans con ained da a o a sequen ial ao o-co ona y
bypass wi h one side- o-side anas omosis and one end- o-side anas omosis, Fig. 5. The
igu e shows he econs uc ed bypass model and he uns uc u ed e ahed al compu a-
ional mesh as well. De ailed iew a bo h he side- o-side and end- o-side anas omoses is
gi en in Fig. 6. Since he au ho s o his pape ha e access o physiological da a measu ed
wi hin a eal side- o-side anas omosis[9], i s nume ical simula ions o pulsa ile blood low
in he sequen ial ao o-co ona y bypass model a e done o he side- o-side anas omosis
Figu e 5: Sequen ial ao o-co ona y bypass model – ( om le o igh ) CT scan wi h deno ed posi ion o
he side- o-side anas omosis, p ima y econs uc ion om CT da a, model a e smoo hing, e ahed al
compu a ional mesh
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Jan Vimm , Alena Jon´aˇso ´a and Ondˇ ej Bubl´ık
Figu e 6: Sequen ial ao o-co ona y bypass model – iew a he dis al side- o-side anas omosis (le ) and
a he dis al end- o-side anas omosis ( igh )
wi h ollowing inle and ou le bounda y condi ions, Fig. 6 (le ),
•g a inle ∂Ω(G)
Iand co ona y a e y inle ∂Ω(CA)
I– ime-dependen low a es
Q( )(G)and Q( )(CA), espec i ely, a e applied acco ding o li e a u e[9], Fig. 7;
•g a ou le ∂Ω(G)
Oand co ona y a e y ou le ∂Ω(CA)
O– cons an p essu e equal o
a e age a e ial p essu e 12 000 Pa is gi en.
Simila ly o he case o indi idual g a , e e ence alues ha e o be chosen. The e e ence
eloci y U(2)
e =0.1193 m ·s−1is de e mined om a e age g a inle low a e, Fig. 7, and
he e e ence diame e D(2)
e =0.0045 m se equal o a e age g a diame e . Acco ding
o p esc ibed alues o bounda y condi ions[9], blood’s densi y and iscosi y a e η(2) =
0.0037 Pa ·s and (2) = 1060 kg ·m−3, espec i ely.
3 MATHEMATICAL MODELLING
3.1 Ma hema ical model
Le us conside a ime in e al (0,T),T>0 and a bounded 3D compu a ional domain
Ω⊂R3wi h bounda y ∂Ω. Acco ding o assump ions es ablished in he p e ious sec ion,
co ona y blood low in his compu a ional domain may be modelled as uns eady lamina
iso he mal low o incomp essible New onian luid ha in he space- ime cylinde ΩT=
Ω×(0,T) is ma hema ically desc ibed by he non-linea sys em o incomp essible Na ie -
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Jan Vimm , Alena Jon´aˇso ´a and Ondˇ ej Bubl´ık
S okes (NS) equa ions w i en in he non-dimensional o m as
∂ j
∂xj
=0,(1)
∂ i
∂ +∂
∂xj
( i j)+ ∂p
∂xi
=1
Re(s)
∂2 i
∂xj∂xj
i, j =1,2,3,(2)
whe e is he ime, iis he i- h componen o he eloci y ec o =[ 1,
2,
3]Tco e-
sponding o he Ca esian componen xio he space a iables ec o x=[x1,x
2,x
3]T,pis
he p essu e and Re(s)=U(s)
e D(s)
e (s)/η(s),s=1,2 is he e e ence Reynolds numbe . Fo
he indi idual ao o-co ona y bypass model and o he side- o-side anas omosis model,
we ge Re(1) = 1 234.3 and Re(2) = 153.8, espec i ely.
3.2 Nume ical me hod
The nume ical solu ion o he non-linea ime-dependen sys em o he incomp essible
NS equa ions (1) – (2) is based on he ac ional s ep me hod and he cell-cen ed ini e
olume me hod o mula ed o hyb id uns uc u ed e ahed al g ids, whose con ol ol-
ume Ωkis shown in Fig. 8. The p inciple o hyb id g id sys ems was p oposed by Kim e
al.[10] Time disc e iza ion o he incomp essible NS equa ions (2) is pe o med using he
implici second o de C ank-Nicolson scheme wi h linea iza ion o he con ec i e e m.
Le us in oduce a a iable δˆ i=ˆ i− n
i, whe e ˆ iis he in e media e eloci y and n
iis
he eloci y compu ed a he ime le el n. Le us u he in oduce a p essu e co ec ion
unc ion Ψ de ined as Ψ = (pn+1 −pn)∆ , whe e pn+1 and pna e he p essu e alues
compu ed a he ime le els nand (n+ 1), espec i ely. A e he disc e iza ion, we ge
Figu e 7: Time-dependen low a es p esc ibed
a he g a inle Q( )(G)and a he co ona y
a e y inle Q( )(CA)o he side- o-side anas omo-
sis model
Figu e 8: A e ahed al con ol olume Ωk=
A1A2A3A4wi h bounda y ∂Ωk=4
m=1 Γm
k
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Jan Vimm , Alena Jon´aˇso ´a and Ondˇ ej Bubl´ık
ollowing sys em o equa ions
(δˆ i)k+∆
2|Ωk|
4

m=1 δˆ imVn
m+ n
im ·jnm
kδˆ jm +2 n
imVn
m|Γm
k|
+∆
2|Ωk|
4

m=1
pn
m·inm
k|Γm
k|=∆
2Re(s)|Ωk|
4

m=1
∂
∂nm
k
(δˆ im +2 n
im)|Γm
k|,(3)
(ˆ i)k=( n
i)k+(δˆ i)k,(4)
4

m=1
∂Ψ
∂nm
k|Γm
k|=
4

m=1
ˆ im ·inm
k|Γm
k|≡
4

m=1
ˆ
Vm|Γm
k|,(5)
( n+1
i)k= (ˆ i)k−1
|Ωk|
4

m=1
Ψm·inm
k|Γm
k|,(6)
(pn+1)k=(pn)k+1
∆ (Ψ)k,(7)
Vn+1
m=ˆ
Vm−∂Ψ
∂nm
k
,(8)
whe e ∆ is he ime s ep, |Ωk|is he olume o he con ol olume Ωk,k=1,...,N
CV ,
Fig. 8, (Φ)k=1
|Ωk|ΩkΦdΩ is he in eg al a e age o an a bi a y low quan i y Φ o e
con ol olume Ωk,|Γm
k|,m=1,...,4 is he a ea o he m- h ace Γm
ko he con ol
olume Ωk,inm
kis he i- h componen o he ou wa d uni ec o nm
k=[
1nm
k,2nm
k,3nm
k]T
no mal o he ace Γm
kand ˆ
Vm=ˆ im ·i
nm
kdeno es he in e media e ace-no mal eloci y a
he aceΓ
m
k. No e ha he alues o ace-no mal eloci y Vn+1
mcompu ed wi h he help o
Eq. (8) a e used as alues o ace-no mal eloci y Vn
min Eq. (3) a he nex ime le el. Fo
he de e mina ion o alues δˆ im, n
im,pn
m,Ψ
mand de i a i es ∂Ψ
∂nm
k,∂
∂nm
k(δˆ im +2 n
im) a
he m- h ace Γm
ko he con ol olume Ωk, he applica ion o an in e pola ion me hod is
used. Fo mo e de ails on he desc ibed nume ical me hod, see Vimm e al.[11] In ela ion
o pape ’s objec i es o model pulsa ile blood low, bounda y condi ions a e p esc ibed
as
•inle Γm
k⊂∂ΩI: im = iI,∂ im
∂nm
kΓm
k
=0,∂Ψ
∂nm
kΓm
k
= 0, whe e he inle eloci y
ec o componen s iI a e gi en in non-dimensional o m o co esponding inle
bounda ies o bo h bypass models, see Sec. 2;
•ou le Γm
k⊂∂ΩO:pmnm
k−1
Re(s)
∂ m
∂nm
k=pOnm
k,Ψm= 0, whe e pOis he p esc ibed
non-dimensional alue o ou le p essu e, see Sec. 2;
• igid wall Γm
k⊂∂ΩW: im =0,∂Ψ
∂nm
k

Γm
k
=0.
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Jan Vimm , Alena Jon´aˇso ´a and Ondˇ ej Bubl´ık
The solu ion o Eq. (3) leads o a sys em o linea algeb aic equa ions ANS ·xNS =bNS
o 3 ·NCV unknown alues (δˆ i)k,i=1,2,3, k=1,2,...N
CV , whe e NCV is he numbe
o con ol olumes wi hin he hyb id uns uc u ed compu a ional mesh. Simila ly, he
solu ion o Poisson equa ion (5) o he p essu e co ec ion unc ion leads o a sys em
o linea algeb aic equa ions AP oi ·xP oi =bP oi o NCV unknown alues (Ψ)k,k=
1,2,...N
CV . Since bo h sys ems o equa ions con ain la ge spa se ma ices ANS and
AP oi, i is a ou able o use an i e a i e solu ion. In ou case, BICGSTAB me hod wi h
incomple e LU p econdi ione is applied. This kind o me hods is a s anda d pa o he
MATLAB so wa e. Fo he incomple e LU decomposi ion, he UMFPACK lib a y is
used.
4 NUMERICAL RESULTS, DISCUSSION AND CONCLUSIONS
Le us no e ha in his sec ion all nume ical esul s a e p esen ed in hei dimensional
o m. Fo he case o indi idual ao o-co ona y bypass, Fig. 9–10 show eloci y ec o s
a bo h he p oximal and dis al anas omoses a wo ime ins an s 1=0.055 s and 2=
0.095 s, espec i ely. The posi ion o bo h p e-sys olic ime ins an s wi hin he he ca diac
cycle a he ao a is deno ed in Fig. 4 o he alues o inle low a e and ou le p essu e. In
he case o p oximal anas omosis, he p esence o a ’bulge’ illed wi h a weak eci cula ion
zone seems o signi ican ly a ec eloci y dis ibu ion and consequen ly he blood supply
o he dis al anas omosis. Mo eo e , he p esen shape o he p oximal junc ion egion and
i s close p oximi y o high- eloci y ao a may enhance he isk o h ombus o ma ion due
o he possibili y o blood cells accumula ion and pla ele ac i a ion. Rega ding he dis al
anas omosis o he indi idual bypass model, he low dis ibu ion be ween he p e- and
Figu e 9: Veloci y dis ibu ion wi hin he indi idual ao o-co ona y bypass model and eloci y ec o s
a he p oximal end- o-side and dis al end- o-side anas omoses a he ime 1=0.055 s
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