Slides: Lecture notes: Course: N BMT N BMT 4786 (3T160 ...vosse/3T160/slides.pdfintra-vascular...
124
Cardiovascular Fluid Mechanics Course: BMT (8W090) N (3T160) Lecturers: BMT F.N. van de Vosse N M.E.H. van Dongen Lecture notes: 4786 Cardiovascular Fluid Mechanics Slides: www.mate.tue.nl/ vosse/8W090 Eindhoven University of Technology faculty of BioMedical Engineering (BMTE) faculty of Applied Physics (NT) Cardiovascular Fluid Mechanics 1 TU/e
Transcript of Slides: Lecture notes: Course: N BMT N BMT 4786 (3T160 ...vosse/3T160/slides.pdfintra-vascular...
Card
iovascular
Flu
idM
echan
ics
Co
urse:
BM
T(8W
090)
N(3T
160)
Lectu
rers:B
MT
F.N.van
de
Vosse
NM
.E.H
.vanD
on
gen
Lectu
ren
otes:
4786C
ardiovascu
larF
luid
Mech
anics
Slid
es:w
ww
.mate.tu
e.nl/vo
sse/8W090
Ein
dh
ovenU
niversity
of
Techn
olo
gy
faculty
of
Bio
Med
icalEn
gin
eering
(BM
TE
)facu
ltyo
fA
pp
liedP
hysics(N
T)
Card
iovascular
Flu
idM
echan
ics1
TU
/e
Card
iovascular
flu
idm
echan
ics:b
ackgro
ud
fun
ction
:tran
spo
rt
of
en
of
nu
trients
and
waste
pro
du
cts
of
heat
imm
un
esystem
mech
anism
:
heart
asa
pu
mp
stron
gly
bifu
rcating
netw
ork
blo
od
pressu
rereg
ulatio
n
cloth
ing
path
olo
gy:
heart
failure
heart
valved
isease
athero
sclerosis
majo
rcau
seo
fd
eath(U
S/E
U)
pulmonary
circulation
aorta
apulmonaris
RA
LA
vcava
RV
LV
vpulmonaris
aorticvalve
mitralvalve
tricuspidvalve
pulmonaryvalve
lungs
tissuesandorgans
Card
iovascular
Flu
idM
echan
ics2
TU
/e
Card
iovascular
Flu
idM
echan
ics:b
ackgro
un
d
cardiovascu
larresearch
:
cardiovascu
larp
hysiolo
gy
cardiovascu
larp
atho
log
y
diag
no
stics-
blo
od
pressu
rean
dfl
owm
easurem
ents
-M
RI,C
Tim
agin
g-
ultraso
un
dan
dM
RIvelo
city-
ultraso
un
dw
allmo
tion
measu
remen
ts
treatmen
t-
med
ication
(pressu
rereg
ulatio
n,g
ene
therapy)
-vascu
larp
rosth
esis(d
esign
)-
heart
valvep
rosth
esis(d
esign
)-
treatmen
ten
surg
icalplan
nin
g(im
agin
g+
mo
delin
g)
extracorp
orealsystem
s-
cardio
-pu
lmo
nary
bypass
surg
ery-
dialysis
-card
iacassist
devices
Card
iovascular
Flu
idM
echan
ics3
TU
/e
Ath
erosclero
sis:p
atho
log
y
narrow
ing
of
arteriallum
en-
coro
nary
arteries(h
eartattack)
-cereb
ral/carotid
arteries(stro
ke)-
periferalarteries
(claud
icatio)
wid
enin
go
farteriallu
men
-ao
rtican
eurysm
s-
cerebralan
eurysm
s
heart
valves-
valvesten
osis
-valve
insu
fficien
ce
flu
idm
echan
ics+
wallm
echan
ics
No
on
et.al(1977)
Card
iovascular
Flu
idM
echan
ics4
TU
/e
Ath
erosclero
sis:p
atho
log
y
siteso
fp
reference:
Zarin
set.
al(1983)C
ardiovascu
larF
luid
Mech
anics
5T
U/e
Ath
erosclero
sis:d
iagn
ostics
detectio
nm
etho
ds:
ang
iog
raphy
(geo
metry)
intra-vascu
laru
ltrasou
nd
(geo
metry)
ultraso
un
dD
op
pler
(velocity
,wallm
otio
n)
MR
I(velocity
,wallm
otio
n)Bran
ds
(1996)
Card
iovascular
Flu
idM
echan
ics6
TU
/e
Card
iovascular
Flu
idM
echan
ics:b
ackgro
un
d
cardiovascu
larflu
idm
echan
ics:
un
derstan
din
gth
ecircu
lation
of
blo
od
evaluatio
no
fen
dovascu
lartreatm
ent
develo
pm
ent
and
evaluatio
no
fd
iagn
ostic
techn
iqu
es
develo
pm
ent
and
testing
of
vascular
and
heart
valvep
rosth
esis
develo
pm
ent
and
testing
of
extracorp
orealsystem
s
this
cou
rsefo
cuses
on
:
fluid
mech
anics
flowp
atterns
instraig
ht,cu
rvedan
db
ifurcatin
gtu
bes
fluid
-solid
interactio
nw
avep
rop
agatio
n,atten
uatio
nan
dreflectio
n
hem
o-rh
eolo
gy
susp
ensio
nrh
eolo
gy,
shear
thin
nin
go
fb
loo
d,
flowin
the
micro
-circu
lation
Card
iovascular
Flu
idM
echan
ics7
TU
/e
Card
iovascular
Flu
idM
echan
ics:co
nten
ts
gen
eralintro
du
ction
:
overviewcircu
latory
system
win
dkesselm
od
el
basic
equ
ation
s:
con
stitutive
equ
ation
s
equ
ation
so
fm
otio
n:
,
fluid
mech
anics
of
the
heart:
un
der
con
structio
n
New
ton
ianflow
inb
loo
dvessels:
steadyan
du
nsteady
flowin
tub
es:
steadyan
du
nsteady
flowin
curved
tub
es:
Card
iovascular
Flu
idM
echan
ics8
TU
/e
Card
iovascular
Flu
idM
echan
ics:co
nten
ts
mech
anics
of
the
vesselwall:
mo
rph
olo
gy
con
stitutive
equ
ation
s
wave
ph
eno
men
ain
blo
od
vessels:
wave
pro
pag
ation
,attenu
ation
and
reflection
vascular
imp
edan
ce
hem
o-rh
eolo
gy:
mo
rph
olo
gy
shear
thin
nin
gan
dvisco
elasticm
od
els:
,
susp
ensio
nrh
eolo
gy
no
n-N
ewto
nian
flowin
blo
od
vessels:
in-elastic
and
visco-elastic
flowin
tub
es
flowo
fb
loo
dcells
insm
allvessels
Gen
eralintro
du
ction
9T
U/e
Gen
eralintro
du
ction
:co
nten
ts
the
cardiovascu
larsystem
:
the
heart
the
systemic
circulatio
n
pressu
rean
dflow
:
arteries
micro
-circulatio
n
veno
us
system
asim
ple
mo
delo
fcard
iovascular
system:
perio
dic
defo
rmatio
nan
dflow
the
win
dkesselm
od
el
vascular
imp
edan
ce
Gen
eralintro
du
ction
10T
U/e
Th
ecard
iovascular
system:
the
heart
circulato
rysystem
:
pu
lmo
nary
circulatio
n
Æ
righ
tatriu
m(1
kPa)
Æ
righ
tven
tricle(4
kPa)
Æ
a.p
ulm
on
aris
Æ
lun
g
Æ
v.pu
lmo
naris
systemic
circulatio
n
Æ
leftatriu
m(1
kPa)
Æ
leftven
tricle(13
kPa)
Æ
aorta
Æ
periferaltissu
e
Æ
v.cava
pulmonary
circulationsystem
ic
circulation
aorta
a.pulmonaris
RA
LA
v.cava
RV
LV
v.pulmonaris
aorticvalve
mitralvalve
tricuspidvalve
pulmonary
valve
Gen
eralintro
du
ction
11T
U/e
Th
ecard
iovascular
system:
the
heart
isovolumetric
contraction:valves:
mc,ac
up
,
con
stant
zero,
zero
ejectionphase:
valves:m
c,ao
up
/dow
n,
also
zero,
up
/dow
n
isovolumetric
relaxation:valves:
mc,ac
dow
n,
con
stant
zero,
zero
fillingphase:
valves:m
o,ac
con
stant,
dow
n
2
up
/dow
n,
zero
00.2
0.40.6
0.81
0 5 10 15 20
time [s]
pressure [kPa]
mc
aoac
mo
atrial
aortic
ventricular
00.2
0.40.6
0.81
−200 0
200
400
600
time [s]
flow [ml/s]
mc
acm
oao
aortic
mitral
Gen
eralintro
du
ction
12T
U/e
Th
ecard
iovascular
system:
systemic
circulatio
n
the
heart:
pu
mp
ing
of
blo
od
into
aorta
main
tainrelatively
hig
harterialp
ressure
almo
sto
nly
inertialfo
rces(in
equ
ilibriu
mw
ithp
ressure)
arterialsystem:
transp
ort
of
blo
od
totissu
es
pu
lsating
flowto
mo
reo
rless
steadyflow
main
tainrelatively
hig
harterialp
ressure
regu
lation
bysm
oo
thm
uscle
cells
bo
thin
ertialand
viscou
sfo
rces
Gen
eralintro
du
ction
13T
U/e
Th
ecard
iovascular
system:
systemic
circulatio
n
capillary
systemsystem
:
exchan
ge
of
nu
trients
and
gases
with
tissue
large
volu
me,low
velocity
inh
om
og
eneo
us
fluid
main
lyvisco
us
forces
veno
us
system:
collectio
nan
dtran
spo
rtto
the
heart
large
storag
evo
lum
e
regu
lation
bysm
oo
thm
uscle
cells
main
lystatio
nary
inertialan
dvisco
us
forces
Gen
eralintro
du
ction
14T
U/e
Th
ecard
iovascular
system:
systemic
circulatio
n
dim
ensio
ns
20kgD
og
(Miln
or,1989):
1:left
atrium
,2:left
ventricle,
3:ao
rta,4-7:
arteries,8:
arterioles,
9:cap
illaries,
10:ven
ules,
11-14:vein
s,15:
venae
cavae,
16:rig
ht
atrium
,17:rig
ht
ventricle,
18:a.
pu
lmo
nalis,19-20:
pu
lmo
nary
arteries,21:
arterioles,
22:cap
illar-
ies,
23:ven
ules,24-25:
pu
lmo
nary
veins,
26:vv.
pu
lmo
nalis
05
1015
2025
300
100
200
300
level [−]
volume [ml]
05
1015
2025
300
100
200
300
400
length [mm]
05
1015
2025
300 2 4 6 8 10
log number [−]
05
1015
2025
300 20 40 60
diameter [mm]
systemic circulation
pulmonary circulation
Gen
eralintro
du
ction
15T
U/e
Pressu
rean
dflo
w:
arteriesao
rticp
ressure
pu
lse:
chan
ges
alon
gao
rta
Æ
peak
delays
wavesp
eed
Æ
increase
inam
plitu
de
Æ
steepen
ing
of
the
fron
t
Æ
mo
derate
fallof
mean
result
of
aortic
com
plian
ce
Æ
defin
edas:
Æ
dep
end
so
nm
aterial
Æ
dep
end
so
ng
eom
etry
Æ
dep
end
so
ntran
smu
ralp
ressure:
00.2
0.40.6
0.81
10 12 14 16 18
time [s]
pressure [kPa]
abdominal
ascending
Gen
eralintro
du
ction
16T
U/e
Pressu
rean
dflo
w:
arteriesC
om
plian
ce/disten
sibility:
disten
sibility:
thin
walled
tub
eslin
earelastic:
variesstro
ng
lyalo
ng
the
arterialtree
arteriesare
no
n-lin
earan
isotro
pic
visco-elastic
0.81
1.21.4
1.61.8
0.9 1
1.1
1.2
relative pressure [−]
relative cross−section [−]
Gen
eralintro
du
ction
17T
U/e
Pressu
rean
dflo
w:
arteries
arterialflow:
driven
byp
ressure
grad
i-en
t
pu
lsating
linear
theo
ry:
6to
10h
armo
nics
OK
00.5
110 12 14 16 18
pressure [kPa]
00.5
1−
100 0
200
400
flow [ml/s]
00.5
1−
100 0
200
400
flow [ml/s]
00.5
1−
100 0
200
400
flow [ml/s]
00.5
1−
100 0
200
400
flow [ml/s]
00.5
1−
100 0
200
400
flow [ml/s]
00.5
1−
100 0
200
400
flow [ml/s]
flow
3 2 1 045
00.5
110 12 14 16 18
pressure [kPa]
1
00.5
110 12 14 16 18
pressure [kPa]
2
00.5
110 12 14 16 18
time [s]
pressure [kPa]
5
00.5
110 12 14 16 18
pressure [kPa]
pressure
0
00.5
110 12 14 16 18
pressure [kPa]
4
3
Gen
eralintro
du
ction
18T
U/e
Pressu
rean
dflo
w:
capillary/ven
ou
ssystem
capillarysystem
:
bifu
rcating
netw
ork
flowth
rou
gh
tissue
flowth
rou
gh
arterioven
ou
san
astom
osis
(bypass)
stron
greg
ulatio
n
(no
n-)lin
ear(tim
e-dep
end
ent)
resistance
veno
us
system:
wallth
inn
erth
anarteries
mu
chlow
erp
ressure
than
inarteries
po
ssibility
toco
llapse
presen
ceo
fvalves
(almo
st)co
nstan
tp
ressure
Gen
eralintro
du
ction
19T
U/e
Asim
ple
mo
del:
perio
dic
defo
rmatio
nan
dflo
w
perio
dic
fun
ction
s:
Fo
urier
transfo
rm:
with
:
the
com
plex
Fo
urier
coefficien
ts
the
ang
ular
frequ
ency
of
the
basic
harm
on
ic
no
te:
linear
theo
ry:su
perp
ositio
no
fso
lutio
ns
per
harm
on
ic
Gen
eralintro
du
ction
20T
U/e
Asim
ple
mo
del:
the
win
dkesselm
od
el
win
dkesselm
od
el:
simp
leco
mp
liance:
con
stant
periferalresistan
ce:
then
:
or
afterF
ou
riertran
sform
a-tio
n:
often
used
,stron
glim
itation
s
aortic flow
00.2
0.40.6
0.81
−100 0
100
200
300
400
500
flow [ml/s]
aortic pressure w
indkessel model
00.2
0.40.6
0.81
11 12 13 14 15 16 17
time [s]
pressure [kPa]qa
pa
C
Rp
Gen
eralintro
du
ction
21T
U/e
Asim
ple
mo
del:
vascular
imp
edan
ce
lon
gitu
din
alimp
edan
ce:
defin
ition
:
expresses
flowin
du
cedby
alo
calpressu
reg
radien
t
pro
perty
of
asm
allarterialsegm
ent
inp
ut
imp
edan
ce:
defin
ition
:
expresses
flowas
aresu
lto
fa
given
(inp
ut)
pressu
re
pro
perty
of
specific
siteo
fth
earterialtree
transverse
imp
edan
ce:
defin
ition
:
expresses
the
flowd
rop
du
eto
the
storag
eo
fth
evessel
pro
perty
of
asm
allarterialsegm
ent
Basic
equ
ation
s22
TU
/e
Basic
equ
ation
s:in
tro
the
stateo
fd
eform
ation
:
stress
disp
lacemen
tan
dstrain
velocity
and
rateo
fd
eform
ation
con
stitutive
equ
ation
s
equ
ation
so
fm
otio
n:
Reyn
old
s’transp
ort
theo
rem
con
tinu
ityeq
uatio
n
mo
men
tum
eeq
uatio
ns
initialan
db
ou
nd
aryco
nd
ition
s
Basic
equ
ation
s23
TU
/e
State
of
defo
rmatio
n:
stress
mech
anicaleq
uilib
rium
:
sum
of
allforces
equ
alszero
bo
dyw
illneith
eraccelerate
no
rd
eform
cuttin
gp
lane:
plan
e
with
no
rmal
surface
force
top
revent
de-
form
ation
and
acceleration
of
the
two
parts
stressvecto
r
:
ineach
po
int
represen
tation
:
Ω
Γ
n
s
Cau
chystress
tenso
r
:
defin
esth
esu
rfacefo
rceo
rsu
rfacestress
canb
ed
ecom
po
sedin
an
orm
aland
shear
stress
Basic
equ
ation
s24
TU
/e
State
of
defo
rmatio
n:
disp
lacemen
tan
dd
eform
ation
disp
lacemen
tvecto
r
:
defin
ition
:
infin
itesimalm
aterialvector
:
willstretch
and
rotate
to
defo
rmatio
ng
radien
tten
sor
:
defin
ition
:
cartesianco
ord
inates
(2D):
t=0
Ω0 dx01x0
dx02
F
Ω u
x
dx1
θ t=t
dx2
O
Basic
equ
ation
s25
TU
/e
State
of
defo
rmatio
n:
disp
lacemen
tan
dd
eform
ation
stretch:
shear:
toco
nstru
ctco
nstitu
tiverelatio
ns:
Cau
chy-Green
defo
rmatio
nten
sor:
Fin
ger
tenso
r:
t=0
Ω0 dx01x0
dx02
F
Ω u
x
dx1
θ t=t
dx2
O
Basic
equ
ation
s26
TU
/e
State
of
defo
rmatio
n:
disp
lacemen
tan
dd
eform
ation
po
lard
ecom
po
sition
:
defo
rmatio
ng
radien
tten
sor
isstretch
tenso
r,
isro
tation
tenso
r
then
:
since:
soF
ing
erten
sor
d
oes
no
tco
ntain
rotatio
n
same
ho
lds
for
Cau
chy-Green
tenso
r
Basic
equ
ation
s27
TU
/e
State
of
defo
rmatio
n:
velocity
and
rateo
fd
eform
ation
velocity
of
materialp
oin
ts
:
defin
ition
:
velocitygradient
tensor
:
rateo
fch
ang
eo
fa
infin
itesimalm
aterialvector
:
but
also:
and
for
afixed
instan
to
ftim
e(
):
Basic
equ
ation
s28
TU
/e
State
of
defo
rmatio
n:
velocity
and
rateo
fd
eform
ation
rateo
fd
eform
ation
tenso
r
:
relation
velocity
grad
ient
and
defo
rmatio
ng
radien
t
:
show
ing
that:
deco
mp
ositio
n:
gives:
with
rateo
fd
eform
ation
tenso
r
and
vorticity
tenso
r
:
Basic
equ
ation
s29
TU
/e
Co
nstitu
tiveeq
uatio
ns:
intro
Fin
ger
tenso
r:
relation
an
d
:
gives:
extrastress
tenso
r:
defin
ition
:
viscou
sflu
ids:
elasticso
lids:
viscoelastic
fluid
san
dso
lids:
Basic
equ
ation
s30
TU
/e
Eq
uatio
ns
of
mo
tion
:R
eyno
lds’tran
spo
rteq
uatio
n
Reyn
old
s’transp
ort
theo
rem:
scalar,vector
or
tenso
r
intim
ed
epen
den
tvolu
me
w
ithb
ou
nd
-ary
movin
gw
ithvelo
city
rateo
fch
ang
e:
Gau
ss-Ostro
grad
skii:
t
t
n
x
tv
x
t
Basic
equ
ation
s31
TU
/e
Eq
uatio
ns
of
mo
tion
:co
ntin
uity
equ
ation
integ
ralform
:
Gau
ss:
differen
tialform
:
t
t
n
x
tv
x
t
Basic
equ
ation
s32
TU
/e
Eq
uatio
ns
of
mo
tion
:m
om
entu
meq
uatio
n
integ
ralform
:
Gau
ssan
d
:
differen
tialform
:
t
t
n
x
tv
x
t
Basic
equ
ation
s33
TU
/e
Basic
equ
ation
s:in
itialand
bo
un
dary
con
ditio
ns
inn
orm
aldirectio
n:
the
Dirich
letco
nd
ition
:
or
Neu
man
nco
nd
ition
:
intan
gen
tialdirectio
n:
the
Dirich
letco
nd
ition
:
or
Neu
man
nco
nd
ition
:
initialco
nd
ition
s:
velocity
and
stress(p
ressure
and
velocity)
New
ton
ianfl
ow
inb
loo
dvessels
34T
U/e
New
ton
ianflo
win
blo
od
vessels:in
trod
uctio
n
inco
mp
ressible
New
ton
ianflow
:
viscou
sflow
in-viscid
flow
bo
un
dary
layerflow
flowin
straigh
ttu
bes:
fully
develo
ped
flow
entran
ceflow
flowin
curved
tub
es:
steadyflow
un
steadyflow
flowin
bran
ched
tub
es:
remark
on
wallsh
earstress
New
ton
ianfl
ow
inb
loo
dvessels
35T
U/e
Inco
mp
ressible
viscou
sflo
w:
govern
ing
equ
ation
s
inco
mp
ressible:
den
sity
isco
nstan
t
con
tinu
ityeq
uatio
n
New
ton
ianco
nstitu
tivem
od
el:
Cau
chystress
extrastress
with
shear
rate
New
ton
ianfl
ow
inb
loo
dvessels
36T
U/e
Inco
mp
ressible
viscou
sflo
w:
govern
ing
equ
ation
s
Navier-S
tokes
equ
ation
s:
sub
stitutio
no
fN
ewto
nian
mo
delin
mo
men
tum
equ
ation
dim
ensio
nless
Navier-S
tokes
equ
ation
s:
!
"
"
#
$#
dim
ensio
nless
nu
mb
ers:
#
!"
"!
$#
"
%!
Stro
uh
alR
eyno
lds
Fro
ud
e
New
ton
ianfl
ow
inb
loo
dvessels
37T
U/e
Inco
mp
ressible
in-viscid
flow
:g
overnin
geq
uatio
ns
Bern
ou
lliequ
ation
salo
ng
streamlin
es:
Cau
chystress:
steadym
om
entu
man
dco
ntin
uity
equ
ation
s:
Eu
lereq
uatio
ns
streamlin
es(tan
gen
tp
arallelto
):
integ
ration
alon
gstream
lines
with
$
:
$&
Bern
ou
lliequ
ation
New
ton
ianfl
ow
inb
loo
dvessels
38T
U/e
Inco
mp
ressible
in-viscid
flow
:iro
tation
alflow
Bern
ou
llifor
the
com
plete
do
main
:
irotatio
nalflow
:
streamfu
nctio
n:
'
and
'
gives:
'
$&
Bern
ou
lli
wellkn
own
form
(Pito
ttu
be):
New
ton
ianfl
ow
inb
loo
dvessels
39T
U/e
Inco
mp
ressible
bo
un
dary
layerflo
w:
govern
ing
equ
ation
so
utsid
eb
ou
nd
arylayer:
Bern
ou
lligives
(
insid
eb
ou
nd
arylayer:
thickn
ess:
Æ!
con
tinu
ityeq
uatio
n:
"Æ!
(
u
uττ
mo
men
tum
equ
ation
s:
New
ton
ianfl
ow
inb
loo
dvessels
40T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
equ
ation
s
Navier-S
tokes
equ
ation
s:
incylin
dricalco
ord
inates
and
axi-symm
etric:
and
#
#)
# #
##
#
) #
# #
#
#
##
r
zv
r
r=a
v
z
New
ton
ianfl
ow
inb
loo
dvessels
41T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
equ
ation
s
con
tinu
ityeq
uatio
n:
#
##
fully
develo
ped
:
no
chan
ges
in
-directio
n:
con
tinu
ityeq
uatio
n:
##
with
on
lyad
missib
leso
lutio
n:
r
zv
r
r=a
v
z
New
ton
ianfl
ow
inb
loo
dvessels
42T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
equ
ation
s
mo
men
tum
equ
ation
s:
#
#)
# #
##
#
) #
# #
#
fully
develo
ped
:
i.e.
and
yields:
#
)#
##
#
r
zv
r
r=a
v
z
New
ton
ianfl
ow
inb
loo
dvessels
43T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
dim
ensio
nless
mo
men
tum
equ
ation
:
)#
##
#
scaledfo
rm:
Æ
characteristic
leng
th
#
#
Æ
characteristic
time
Æ
characteristic
velocity
"
"
yields:
"
)"
#
# #
#
or:
)
)"
#
# #
#
Æ
characteristic
pressu
re
)"
)"
New
ton
ianfl
ow
inb
loo
dvessels
44T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
dim
ensio
nless
scaledm
om
entu
meq
uatio
n:
)
#
# #
#
or:
#
# #
#
with
Wo
mersley
param
eter:
)
limitin
gvalu
es:
:in
ertiad
om
inated
Æ
large
vessels,hig
hfreq
uen
cy(ao
rta:)
:
friction
do
min
ated
Æ
smallvessels,low
frequ
ency
(capillaries:
)
New
ton
ianfl
ow
inb
loo
dvessels
45T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
velocity
pro
files
mo
men
tum
equ
ation
:
#
# #
#
linear
in
sosu
perp
ositio
no
fh
armo
nics
po
ssible
harm
on
icso
lutio
ns:
pressu
reg
radien
t(d
riving
force):
Re
velocity:
Re
#
mo
men
tum
equ
ation
:
#
#
# # #
#
New
ton
ianfl
ow
inb
loo
dvessels
46T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
velocity
pro
files
mo
men
tum
equ
ation
(arbitrary
):
#
# #
#
Sm
allWo
mersley
param
eter(frictio
nd
om
inated
):
#
# #
#
with
solu
tion
:
#
00.25
0.50.75
1−
1 0 1
t/T
r/a0
0.250.5
0.751
−1 0 1
t/Tdp/dz
or:
#
Re
#
New
ton
ianfl
ow
inb
loo
dvessels
47T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
velocity
pro
files
mo
men
tum
equ
ation
(arbitrary
):
#
# #
#
Larg
eW
om
ersleyp
arameter
(inertia
do
min
ated):
#
with
solu
tion
:
or:
Re
00.25
0.50.75
1−
1 0 1
t/T
r/a0
0.250.5
0.751
−1 0 1
t/T
dp/dz
New
ton
ianfl
ow
inb
loo
dvessels
48T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
velocity
pro
files
firstg
uess:
mo
men
tum
equ
ation
(arbitrary
):
)#
##
#
"
)"Æ
instatio
nary
bo
un
dary
layerth
ickness:
" )"Æ
Æ )
or:
Æ
*
00.5
1−
1 0 1
dp/dz
00.5
1−
1 0 1
r/a0
0.51
−1 0 1
r/a
00.5
1−
1 0 1
t/T
r/a
New
ton
ianfl
ow
inb
loo
dvessels
49T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
velocity
pro
files
firstg
uess:
mo
men
tum
equ
ation
(arbitrary
):
#
# #
#
sub
stitute:
+ #
equ
ation
of
Besselfo
r
:
++
+
+
solu
tion
:
#
, #
,
00.5
1−
1 0 1
dp/dz
00.5
1−
1 0 1
r/a0
0.51
−1 0 1
r/a
00.5
1−
1 0 1
t/T
r/a
New
ton
ianfl
ow
inb
loo
dvessels
50T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
velocity
pro
files
Wo
mersley
pro
files:
#
Re
, #
,
−1
01
0 1 2 3 4 5dimesnionless velocity u [−]
radius r/a [−]
α=2
−1
01
0 1 2 3 4 5
radius r/a [−]
α=4
−1
01
0 1 2 3 4 5
radius r/a [−]
α=8
−1
01
0 1 2 3 4 5
radius r/a [−]
α=16
New
ton
ianfl
ow
inb
loo
dvessels
51T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
flow
defin
ition
:
-## -
$
with
:
$
,
,
.
..
.
inertia
do
min
ated(
):
-
$
friction
do
min
ated(
):
-
$
New
ton
ianfl
ow
inb
loo
dvessels
52T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
wallsh
earstress
defin
ition
:
/
#
$
$ /
inertia
do
min
ated(
*):
$
.
friction
do
min
ated(
0):
$
.
’exact’ approxim
ation
08
1624
32−π/4
−π/2
0
α [−]
argument of F10 [−]
’exact’ approxim
ation
08
1624
320
0.5 1
α [−]
modulus of F10 [−]
New
ton
ianfl
ow
inb
loo
dvessels
53T
U/e
Fu
llyd
evelop
edflo
win
straigh
ttu
bes:
Win
dkessel
win
dkesselm
od
el:
simp
leco
mp
liance:
con
stant
inertan
ce:
!
con
stant
periferalresistan
ce:
then
:
!
Fo
urier
transfo
rmatio
n:
!
aortic flow
00.2
0.40.6
0.81
−100 0
100
200
300
400
500
flow [ml/s]
aortic pressure
windkessel m
odel 2
windkessel m
odel
00.2
0.40.6
0.81
11 12 13 14 15 16 17
time [s]
pressure [kPa]
C
Rp
p0 ,q0
qa ,pa
New
ton
ianfl
ow
inb
loo
dvessels
54T
U/e
En
trance
flow
instraig
ht
tub
es:stead
yflo
w
bo
un
dary
layerd
evelop
men
t:V
L
δ
x2
x1
ord
ero
fm
agn
itud
es:
"!
Æ
""!
"!
)"!)"Æ
convective
forces
of
same
ord
ero
fvisco
us
forces:
)"Æ
"
!! Æ
")
steadyen
trance
leng
th(Æ
):
!
")
!
New
ton
ianfl
ow
inb
loo
dvessels
55T
U/e
En
trance
flow
instraig
ht
tub
es:u
nstead
yflo
w
bo
un
dary
layerd
evelop
men
t:V
L
δ
x2
x1
ord
ero
fm
agn
itud
es:
"!
Æ
""!
"!
)"!)"Æ
convective
forces
of
same
ord
ero
fvisco
us
forces:
)"Æ
"
!! Æ
")
un
steadyen
trance
leng
th(Æ
):
!
"
)
*
New
ton
ianfl
ow
inb
loo
dvessels
56T
U/e
Stead
yflo
win
curved
tub
es:en
trance
flow
coo
rdin
atesystem
s:
cylind
rical:1
toro
idal:
#1
defin
ition
s:
curvatu
re
radiu
s
curvatu
reratio
Æ
tub
e
#
-
1 -
innerw
alllow
erw
all
outerw
all
upperw
all
R0θ
Rz
O φ
New
ton
ianfl
ow
inb
loo
dvessels
57T
U/e
Stead
yflo
win
curved
tub
es:en
trance
flow
centralco
re,entran
ce(A
-B):
inviscid,
irotatio
nal
Bern
ou
lli:
c
mo
men
tum
equ
ation
inR
-directio
n:
(pressu
relarg
estat
ou
terw
all)
con
sequ
ently:
(velocity
largest
atin
-n
erw
all)inner
wall
lower
wall
outerw
all
upperw
all
BC
A
V
O
R0θ
Rz
O φ
New
ton
ianfl
ow
inb
loo
dvessels
58T
U/e
Stead
yflo
win
curved
tub
es:en
trance
flow
centralco
re,entran
ce(A
-B):
solu
tion
:
2
with2
"
Æ
Æ
(do
esn
ot
dep
end
on
)
innerw
alllow
erw
all
outerw
all
upperw
all
BC
A
V
O
R0θ
Rz
O φ
New
ton
ianfl
ow
inb
loo
dvessels
59T
U/e
Stead
yflo
win
curved
tub
es:en
trance
flow
develo
pm
ent
of
bo
un
dary
layer:
inb
ou
nd
arylayer:
02
"
0
incen
tralcore:
*2
"
*
inp
lane
of
symm
etry:acceleratio
n
flowfro
min
ner
wall
too
uter
wall
alon
gw
alls:flow
from
ou
terw
allto-
ward
sin
ner
wall
innerw
alllow
erw
all
outerw
all
upperw
all
BC
A
V
profilesaxialvelocity
secondaryvelocity
streamlines
O
R0θ
Rz
O φ
New
ton
ianfl
ow
inb
loo
dvessels
60T
U/e
Stead
yflo
win
curved
tub
es:d
imen
sion
lessp
arameters
mo
men
tum
equ
ation
sin
toro
idalco
ord
inate
system:
scaling
:
#
#
"
"
"
"
equ
ation
sin
r-directio
n:
##
#
Æ
Æ#
#
#
Æ
Æ#
#
##
two
param
eters:
Æ
")
New
ton
ianfl
ow
inb
loo
dvessels
61T
U/e
Stead
yflo
win
curved
tub
es:D
eann
um
ber
mo
men
tum
equ
ation
sin
toro
idalco
ord
inate
system:
equ
ation
sin
plan
eo
fsym
metry
(-:
#Æ
Æ#
#
#
#
#
better
scaling
for
Æ00:
#
#
Æ"
Æ"
Æ"
"
gives:
##
#
#
Æ
#
#
##
Dean
nu
mb
er:
Æ
New
ton
ianfl
ow
inb
loo
dvessels
62T
U/e
Stead
yflo
win
curved
tub
es:seco
nd
aryb
ou
nd
arylayer
mo
men
tum
equ
ation
sin
toro
idalco
ord
inate
system:
equ
ation
sin
-directio
n:
##
#
Æ
Æ#
#
Æ
#
Æ
Æ#
###
#
secon
dary
bo
un
dary
layer:at
#Æ
O(in
ertiafo
rces)=
O(visco
us
forces)
gives:
Æ
New
ton
ianfl
ow
inb
loo
dvessels
63T
U/e
Stead
yflo
win
curved
tub
es:fu
llyd
evelop
edflo
w
insu
mm
ary:
two
param
eters:
Æ
curvatu
reratio
:
Æ
Æ
Reyn
old
sn
um
ber:
")
smallvalu
eso
f
Æ:
Æ
Dean
nu
mb
er:
Æ
secon
dary
flow:
Æ
bo
un
dary
layer:
Æ
Dn=5000
Dn=5000
axialDn=600
Dn=600
secondary
Dn=2000
Dn=2000
New
ton
ianfl
ow
inb
loo
dvessels
64T
U/e
Un
steady
flow
incu
rvedtu
bes:
fully
develo
ped
instatio
nary
bo
un
dary
layer:
two
secon
dary
vortices:
Æ
viscou
sÆ
Æ
inviscid(see
entran
ce)
innerw
allouter
wall
innerw
allouter
wall
oscillatorysteady
New
ton
ianfl
ow
inb
loo
dvessels
65T
U/e
Un
steady
flow
incu
rvedtu
bes:
com
pu
tation
s/experim
ents
end
diasto
le:00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.2
0.4
0.6
0.8 1
1.2
1.4
exp
A
A
A
num
num
exp
oooexperimental
numerical
exp
num
exp
num
B
A
A
AA
B
A
B
B
New
ton
ianfl
ow
inb
loo
dvessels
66T
U/e
Un
steady
flow
incu
rvedtu
bes:
com
pu
tation
s/experim
ents
peak
systole:0
0.10.2
0.30.4
0.50.6
0.70.8
0.91
0
0.2
0.4
0.6
0.8 1
1.2
1.4
A
A
B
B
AA
B
numerical
B
AA
A
A
exp
num
exp
num
exp
num
num
exp
oooexperimental
New
ton
ianfl
ow
inb
loo
dvessels
67T
U/e
Un
steady
flow
incu
rvedtu
bes:
com
pu
tation
s/experim
ents
end
systole:
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.2
0.4
0.6
0.8 1
1.2
1.4
A
A
B
B
AA
B
exp
B
AA
A
A
oooexperimental
exp
numerical
num
exp
num
exp
num
num
New
ton
ianfl
ow
inb
loo
dvessels
68T
U/e
Flo
win
bran
ched
tub
es:in
tro
same
ph
eno
men
aas
incu
rvedtu
bes:
New
ton
ianfl
ow
inb
loo
dvessels
69T
U/e
Flo
win
bran
ched
tub
es:co
mp
utatio
nalan
dexp
erimen
tal
steadyfl
ow:
num
exp
I
I
I
I
I
AA
exp
BB
VA
A
A
B
B
A
A
A
B
B
B
B
I
I
I
num
exp
num
exp
num
N
ewto
nian
flo
win
blo
od
vessels70
TU
/e
Flo
win
bran
ched
tub
es:co
mp
utatio
nalresu
lts
New
ton
ianfl
ow
inb
loo
dvessels
71T
U/e
Flo
win
bran
ched
tub
es:co
mp
utatio
nalresu
lts
Mech
anics
of
the
vesselwall
72T
U/e
Th
evesselw
all:m
orp
ho
log
y
intim
a:
end
oth
elialcells
sub
end
oth
eliallayer
med
ia:
elastine
smo
oth
mu
sclecells
adventitia:
elastine
+co
llagen
fib
res
vasovaso
rem
Mech
anics
of
the
vesselwall
73T
U/e
Vesselw
allmech
anics:
intro
du
ction
inh
om
og
eneo
us:
effectievem
od
ulu
s
aniso
trop
ic:
circum
ferential
and
lon
gitu
din
almo
du
lus
no
n-lin
ear:
linearisatio
n
incre-
men
talmo
du
lus
visco-elastic:
com
plex
mo
du
lus
lon
gitu
din
altheterin
g:
3
inco
mp
ressible:
strain
stresskPa
elastin
strain
stressMPa
collagen
EOMPa
EOGPa
longitudinal
circumferantial
stretchratio
CauchystressMPa
Mech
anics
of
the
vesselwall
74T
U/e
Inco
mp
ressible
elasticd
eform
ation
:g
overnin
geq
uatio
ns
con
stitutive
equ
ation
:
Cau
chystress:
4
and
mo
men
tum
and
con
tinu
ityeq
uatio
ns:
dim
ensio
nfu
llequ
ation
s:
4
no
nd
imen
sion
alvariables:
!
,
,
(5!
,
"(5!
dim
ensio
nless
equ
ation
s:
5!
5!
%
! 54!
Mech
anics
of
the
vesselwall
75T
U/e
Inco
mp
ressible
elasticd
eform
ation
:g
overnin
geq
uatio
ns
mo
men
tum
and
con
tinu
ityeq
uatio
ns:
dim
ensio
nless
equ
ation
s:
5!
5!
%
! 54!
ord
erso
fm
agn
itud
e:
!
6
+
5
gives:
4
Mech
anics
of
the
vesselwall
76T
U/e
Inco
mp
ressible
elasticd
eform
ation
:g
overnin
geq
uatio
ns
mo
men
tum
and
con
tinu
ityeq
uatio
ns:
dim
ensio
nless
equ
ation
s:
4
sligh
tlyco
mp
ressible:
,"
"
volu
metric
and
isoch
oric
defo
rmatio
n:
,
7, 4
with
,
and
7=co
mp
ression
mo
du
lus
Mech
anics
of
the
vesselwall
77T
U/e
Inco
mp
ressible
elasticd
eform
ation
:sm
allstrains
smallstrain
s:00)
defo
rmatio
nten
sor:
volu
me
ratio:
, #
isoch
oric
part:
,
#
#
isoch
oric
Fin
ger
tenso
r:
,
#
#
Mech
anics
of
the
vesselwall
78T
U/e
Inco
mp
ressible
elasticd
eform
ation
:sm
allstrains
smallstrain
s:00)
Cau
chystress
tenso
r:
7, 4
7# 4 #
7 4# 4
infi
nitesim
alstrain:
gives:
7
4# 4
Mech
anics
of
the
vesselwall
79T
U/e
Inco
mp
ressible
elasticd
eform
ation
:sm
allstrains
smallstrain
s:00)
infi
nitesim
alstrain:
#
with
:com
pressio
nm
od
ulu
s
7
shear
mo
du
lus
4
or
Po
isson
ratio
74
74
You
ng
’sm
od
ulu
s
74
74
Mech
anics
of
the
vesselwall
80T
U/e
Inco
mp
ressible
elasticd
eform
ation
:sm
allstrains
tensile
tests:
89
9
:
88
:
and
con
sequ
ently:
:
8
8
8
h
h
h
l
l
l
zz
Mech
anics
of
the
vesselwall
81T
U/e
Sm
allstrains:
wallm
otio
n
thin
wall:
:
zd
irection
:
3 ;
: :
axialrestraint:
3
::
1
directio
n:
3
; 1 1
#1
;
: : :
gives:
:
;
z
r
u
r=a
h
r
vz
v
r
o
Mech
anics
of
the
vesselwall
82T
U/e
Sm
allstrains:
wallm
otio
n
mo
men
tum
inr
directio
n(n
oin
ertia):
1: 1
11
and
:
gives:
;
with
:-; -- ;-; -- ;
com
plian
ceo
rd
istensib
ility:
-
zdz
z
pzt
a
d
dz
Wave
ph
eno
men
ain
blo
od
vessels83
TU
/e
Wave-p
hen
om
ena:
pressu
rean
dfl
ow
win
dkesselm
od
el:
differen
tialequ
ation
:
harm
on
icso
lutio
ns:
imp
edan
ce:
no
taccu
rate,no
waves
00.5
1−500 0
500
00.5
110 15 20
05
100 50
100
05
10−4 −2 0
05
100 10 20
05
10−4 −2 0
05
100 0.1 0.2
05
10−0.5 0 0.5
argqa
argpa
aorticow
absqa
aorticpressure
abspa
absZ
argZ
Wave
ph
eno
men
ain
blo
od
vessels84
TU
/e
Wave
ph
eno
men
a:
& :
travelling
waves
travelling
waves:
com
plex
wave
nu
mb
er:
222
6
com
plex
amp
litud
e:
actualp
ressure:
&<+2 1
attenu
ation
2
wavelen
gth
2
00.2
0.40.6
0.81
10 12 14 16 18
time [s]
pressure [kPa]
abdominal
ascending
Wave
ph
eno
men
ain
blo
od
vessels85
TU
/e
Wave
ph
eno
men
a:fl
uid
flo
w
cylind
ricalcoo
rdin
ates:
#
#)
# #
##
#
) #
# #
#
#
##
scaling
:
#
#
2
"
2 "
2 &
"&
Wave
ph
eno
men
ain
blo
od
vessels86
TU
/e
Wave
ph
eno
men
a:fl
uid
flo
w
dim
ensio
nless
form
:
"&
#
2
#
#
#
# # 2
"&
#
#
# #
# 2
#
# #
dim
ensio
nless
gro
up
s:
)
"&
2 -
Wave
ph
eno
men
ain
blo
od
vessels87
TU
/e
Wave
ph
eno
men
a:fl
uid
flo
w
equ
ation
so
fm
otio
n:
#
)#
# #
#
#
##
waves:
2
#
gives
(seerig
idtu
be):
#
2 , #
,
Wave
ph
eno
men
ain
blo
od
vessels88
TU
/e
Wave
ph
eno
men
a:w
avep
rop
agatio
n
z
r
qzt
v
azt
t
Azt
integ
ratedco
ntin
uity
equ
ation
:
integ
ratedm
om
entu
meq
uatio
n:
/
linearized
disp
ersion
relation
:
=
Wave
ph
eno
men
ain
blo
od
vessels89
TU
/e
Wave
ph
eno
men
a:w
avesp
eedan
datten
uatio
n
harm
on
icso
lutio
ns:
wave
nu
mb
er:
2&
5
22
wave
speed
&2
wave
leng
th
-2
attenu
ation
5-2 2
Wave
ph
eno
men
ain
blo
od
vessels90
TU
/e
Wave
ph
eno
men
a:larg
eW
om
ersleyn
um
ber
disp
ersion
relation
(
no
friction
):
2
2
solu
tion
:
2
&
wavesp
eed:
&
Mo
ens-K
ortew
eg
attenu
ation
:
5
adm
ittance:>
2
&
Wave
ph
eno
men
ain
blo
od
vessels91
TU
/e
Wave
ph
eno
men
a:sm
allWo
mersley
nu
mb
er
disp
ersion
relation
(
no
inertia):
2
2
solu
tion
:2
&
2
wavesp
eed:
&
&
attenu
ation
:
5-
adm
ittance:>
&
>
Wave
ph
eno
men
ain
blo
od
vessels92
TU
/e
Wave
ph
eno
men
a:arb
itraryW
om
ersleyn
um
ber
disp
ersion
relation
:
2
2 =
solu
tion
:
2&
$ 2
$
wavesp
eed:
&2
attenu
ation
:
5-2 22
adm
ittance:
>22
> $
&
Wave
ph
eno
men
ain
blo
od
vessels93
TU
/e
Wave
ph
eno
men
a:arb
itraryW
om
ersleyn
um
ber
wavesp
eed:
&&2
2
adm
ittance:
>>
2
22
large
,viscoelastic:
=
2&
= 2
=
010
2030
400
0.5 1
wavespeed c/c0
010
2030
400
0.5 1
alpha
attenuation gamma/2pi
10−
210
010
20
0.5 1
wavespeed c/c0
10−
210
010
20
0.5 1
alpha
attenuation gamma/2pi
010
2030
400
0.5 1
abs(Y/Y0)
010
2030
400 10 20 30 40 50
alpha
arg(Y/Y0)
10−
210
010
20
0.5 1
abs(Y/Y0)10
−2
100
102
0 20 40 60
alphaarg(Y/Y0)
Wave
ph
eno
men
ain
blo
od
vessels94
TU
/e
Wave
ph
eno
men
a:p
rop
agatio
n
01
2−
0.2 0
0.2
0.4
0.6
0.8 1
1.2
time t [s]
distance z [m]
Pressure w
ave, elastic tube
01
2−
0.2 0
0.2
0.4
0.6
0.8 1
1.2
time t [s]
distance z [m]
Pressure w
ave, visco−elastic tube
viscoelastictube
p
p
elastictube
Wave
ph
eno
men
ain
blo
od
vessels95
TU
/e
Wave
ph
eno
men
a:w
averefl
ection
con
tinu
ity:
! ! !
! ! !
adm
ittance:
>
2
reflectio
nan
dtran
smissio
n:
!
!> >
> >
? !
!
>
> >
pr
t
pip
elastictube
Wave
ph
eno
men
ain
blo
od
vessels96
TU
/e
Wave
ph
eno
men
a:w
averefl
ection
transitio
n:
0!
*!
?
Wave
ph
eno
men
ain
blo
od
vessels97
TU
/e
Wave
ph
eno
men
a:w
averefl
ection
01
2−
0.2 0
0.2
0.4
0.6
0.8 1
1.2
time t [s]
distance z [m]
h(z>0.5) =
h(z<0.5)/2
020
4060
0
0.1
0.2
0.3
angular frequency [1/s]
reflection coef. [−]
020
4060
0
0.5 1
1.5 2
2.5
angular frequency [1/s]
transmission coef. [−]
00.5
1−0.2 0
0.2
0.4
0.6
0.8 1
1.2
time t [s]
distance z [m]
h(z>0.447) = h(z<0.447)/2
pr
t
pr
t
pip
elastictube
pip
elastictube
Wave
ph
eno
men
ain
blo
od
vessels98
TU
/e
Wave
ph
eno
men
a:w
averefl
ection
bifu
rcation
:
! ! ! !
! ! ! !
!
!
> > >
> > >
? !
!
>
> > >
? !
!?
Wave
ph
eno
men
ain
blo
od
vessels99
TU
/e
Wave
ph
eno
men
a:w
averefl
ection
00.5
11.5
2-0.2 0
0.2
0.4
0.6
0.8 1
1.2
time t [s]
distance z [m]
a0:a1:a2=1:1:1
00.5
11.5
2-0.2 0
0.2
0.4
0.6
0.8 1
1.2
time t [s]
distance z [m]
a0:a1:a2=3:2.1:1.8
pt
pi
pr
discretebifurcation
pt
Wave
ph
eno
men
ain
blo
od
vessels100
TU
/e
Wave
ph
eno
men
a:effective
adm
ittance
jun
ction
n:
>
>
>
>
?
>
>
>
jun
ction
m:
>
>
>
>
?
>
>
>
effective
adm
ittance:
>
!
! >
mn
1
Nm
1
j
Nn
Wave
ph
eno
men
ain
blo
od
vessels101
TU
/e
Wave
ph
eno
men
a:card
iacw
ork
cardiac
wo
rk:
imp
edan
ce:
gives:
leftven
tricle1400
200
righ
tven
tricle155
73
No
n-N
ewto
nian
flo
win
blo
od
vessels102
TU
/e
Pro
perties
of
blo
od
:m
orp
ho
log
y
plasm
a:
den
sity2%6
viscosity
@+
m
m
m
blo
od
cells:
cellsn
um
ber
un
stressedsh
ape
volu
me
%an
dd
imen
sion
inb
loo
dp
er
eryth
rocytes
bico
ncave
disc
458x1-3
leuco
cytes
rou
gh
lysp
herical
7-221
platelets
rou
nd
edo
roval
2-4
No
n-N
ewto
nian
flo
win
blo
od
vessels103
TU
/e
Visco
metric
pro
peties
of
blo
od
:stead
y
steadysh
ear:
No
n-N
ewto
nian
flo
win
blo
od
vessels104
TU
/e
Visco
metric
pro
peties
of
blo
od
:tran
sient
stepin
shear:
shear stress [Pa]
blood
time
[s]
New
tonianfluid
No
n-N
ewto
nian
flo
win
blo
od
vessels105
TU
/e
New
ton
ianm
od
els:co
nstitu
tiveeq
uatio
ns
gen
eral:
pow
erseries:
= = = =
Cayley
Ham
ilton
:
AAA
AAA
with
:A
#
AA
A #
AAA
No
n-N
ewto
nian
flo
win
blo
od
vessels106
TU
/e
New
ton
ianm
od
els:co
nstitu
tiveeq
uatio
ns
Cayley
Ham
ilton
:
= % % %
with
:%% AAAAAA
results:
% % %
New
ton
ian:
%
%
%
yields:
No
n-N
ewto
nian
flo
win
blo
od
vessels107
TU
/e
Gen
eralizedN
ewto
nian
mo
dels:
con
stitutive
equ
ation
s
Gen
eralizedN
ewto
nian
:
%
% AA
%
with
:AA
#
and
5
we
get:
5
No
n-N
ewto
nian
flo
win
blo
od
vessels108
TU
/e
Gen
eralizedN
ewto
nian
mo
dels:
con
stitutive
equ
ation
s
pow
erlaw
mo
del:
5
Carreau
-Yasud
am
od
el:
5
Casso
nm
od
el:
/
5
6:
Bin
gh
am
6:
Casso
n0
12
0
0.005
0.01
0.015
0.02
0.025
shear rate
viscosity
New
tonian
shear thickening
shear thinning
yield stress
01
20
0.01
0.02
0.03
0.04
0.05
shear rate
tau
No
n-N
ewto
nian
flo
win
blo
od
vessels109
TU
/e
Po
wer
lawm
od
el:stead
yfl
ow
ina
tub
e
mo
men
tum
equ
ation
:
#
##/
integ
ration
:
/
#
−1
01
0
0.5 1
1.5 2
2.5
r/a [−]
v/vmax [−]
Pow
er−law
fluid in a tube
n=1
n=1/2
n=1/3
n=1/4
n=1/5
viscosity
mo
del:
/
#
#
#
#
yields:
#
#
integ
ration
:
#
#
No
n-N
ewto
nian
flo
win
blo
od
vessels110
TU
/e
Visco
elasticm
od
els:u
pp
erco
nvectedM
axwell
UC
Mm
od
el:
with
:
!
"
! "
dim
ensio
nless
form
:
with
:
Deb
orah
nu
mb
er
"!
Weissen
berg
nu
mb
er
No
n-N
ewto
nian
flo
win
blo
od
vessels111
TU
/e
Visco
elasticm
od
els:lin
earM
axwell
:
oscillatin
gsh
earrate:
55
corresp
on
din
gstress:
//
Æ
com
plex
viscosity:
/ 5
Æ
with
: / 5 &<+Æ
/ 5 +Æ
00.5
10
0.1
0.2
0.3
shear [−]
steps in shear
00.5
10 5 10
shear rate [1/sec]
00.5
10
0.1
0.2
t [sec]
shear stress * 1/G0 [−]
00.5
10
0.1
0.2
0.3steps in shear rate
00.5
10
0.5 1
1.500.5
10
0.1
0.2
0.3
t [sec]
00.5
1−
0.5 0
0.5O
scillating Shear
00.5
1−
5 0 5
elastic solid viscoelastic fluidviscous fluid
00.2
0.40.6
0.8−
0.2 0
0.2
t [sec]
viscosity [Pas]
p
lasma
frequency[H
z]
No
n-N
ewto
nian
flo
win
blo
od
vessels112
TU
/e
Visco
elasticm
od
els:o
scillating
flo
win
atu
be
mo
men
tum
equ
ation
:
#
##/
UC
Mm
od
el:
/
/
5
/
5//
harm
on
icfu
nctio
ns:
# #
/ #/ #
yields:
)
)
)
with
No
n-N
ewto
nian
flo
win
blo
od
vessels113
TU
/e
Visco
elasticm
od
els:o
scillating
flo
win
atu
be
solu
tion
:
#
, #
,
−1
01
0 1 2 3 4 5u
r/a−
10
10 1 2 3 4 5u
r/a−
10
10 1 2 3 4 5u
r/a−
10
10 1 2 3 4 5u
r/a
−1
01
0 1 2 3 4 5u
r/a−
10
10 1 2 3 4 5u
r/a−
10
10 1 2 3 4 5u
r/a−
10
10 1 2 3 4 5u
r/a
24
816
De=
0
De=
0.5
No
n-N
ewto
nian
flo
win
blo
od
vessels114
TU
/e
Rh
eolo
gy
of
susp
ensio
ns:
con
stitutive
equ
ation
s
rigid
sph
eres-
lowco
ncen
tration
:
1 1
Ein
stein:
10
1 11
Batch
elor:
10
defo
rmab
lesp
heres
-low
con
centratio
n:
1 1
Taylor:
10
rigid
asymm
etricp
articles-
lowco
ncen
tration
:
15 1
2
2 2
55
Qu
emad
a
defo
rmab
leasym
metric
particles
-low
con
centratio
n:
orien
tation
and
defo
rmatio
n
decrease
invisco
sitylarg
ed
eform
ation
viscoelasticity
No
n-N
ewto
nian
flo
win
blo
od
vessels115
TU
/e
Rh
eolo
gy
of
susp
ensio
ns:
con
stitutive
equ
ation
s
asymm
etricp
articles-
hig
hco
ncen
tration
:
lowshear
:ag
greg
ation
,structu
red
evelop
men
t
increase
invisco
sityhigh
shear:
orien
tation
and
defo
rmatio
n
decrease
invisco
sity
time
con
stants:
agg
regatio
n:
O(10s)
defo
rmatio
n:
O(0.05s)
deformation
viscosity [Pas]
shearrate
[s]
erythrocytesin
Ringer
hardenederythrocytes
normalblood
aggregation
Flo
wp
atterns
inth
em
icro-circu
lation
116T
U/e
Micro
-circulatio
n:
dim
ensio
ns
dim
ensio
ns:
! "
smallarteries
70-50010
405
0.251
arterioles
10-702
50.1
0.0050.1
capillaries
4-101
10.005
0.00030.01
venu
les10-110
24
0.10.005
0.15sm
allveins
110-50010
203
0.151
dim
ensio
nless
gro
up
s:
")
)
mo
men
tum
equ
ation
:
#
##
#
Flo
wp
atterns
inth
em
icro-circu
lation
117T
U/e
Micro
-circulatio
n:
& :
smallarteries
and
veins
con
centratio
n:
& &
0#0
0#0
viscosity:
0#0
0#0
a
cc
ep
ece
t c
ac
z
r
Flo
wp
atterns
inth
em
icro-circu
lation
118T
U/e
Micro
-circulatio
n:
& :
smallarteries
and
veins
velocity:
# #
# #
0#0
# #
# #
0#0
integ
ration
:
##
0#0
##
0#0
dim
ensio
nless:
B#,
B
B
0B0
B
B
0B0
−1
−0.8
−0.6
−0.4
−0.2
00.2
0.40.6
0.81
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
r/a [−]
v/vp [−]
plasma
0.5
0.6
0.7
0.8
0.9
1.0
Flo
wp
atterns
inth
em
icro-circu
lation
119T
U/e
Micro
-circulatio
n:
:sm
allarteriesan
dvein
s
flow
:
-
effectivevisco
sity:
#
#
con
centratio
n:
&$ &
&
&
#
#
mean
con
centratio
n:
&
-
$
-#&##
& ##&
0.50.6
0.70.8
0.90 2 4viscosity [−]
100
101
102
0 2 4
0.50.6
0.70.8
0.90
0.5 1
concentration [−]
0.50.6
0.70.8
0.91
1.5 2
2.5
core diameter ac/a [−
]
cell velocity [−]
100
101
102
0
0.5 1100
101
102
1
1.5 2
diameter a [10−
3 mm
]
corecore
mean
mean
cellvelocity:
%%
%
&
&
&&
Flo
wp
atterns
inth
em
icro-circu
lation
120T
U/e
Micro
-circulatio
n:
&
& :
arterioles
and
venu
les
con
centratio
n:
& &
0#0
0#0
viscosity:
0#0
0#0
0
a
cc
ep
ece
t c
z
r
ac
Flo
wp
atterns
inth
em
icro-circu
lation
121T
U/e
Micro
-circulatio
n:
&
& :
arterioles
and
venu
les
velocity:
B
0B0
B
B
0B0
−1
−0.8
−0.6
−0.4
−0.2
00.2
0.40.6
0.81
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
r/a [−]
v/vp [−]
plasma
0.5
0.6
0.7
0.8
0.9
Flo
wp
atterns
inth
em
icro-circu
lation
122T
U/e
Micro
-circulatio
n:
&
& :
arterioles
and
venu
les
flow
:
-
effectivevisco
sity:
con
centratio
n:
&$ &
&
&
mean
con
centratio
n:
&&
0.50.6
0.70.8
0.90 2 4 6viscosity [−]
100
101
102
0 5
0.50.6
0.70.8
0.91
1.5 2
2.5
core diameter ac/a [−
]
cell velocity [−]
0.50.6
0.70.8
0.90
0.5 1
concentration [−]
100
101
102
0
0.5 1100
101
102
1
1.5 2
diameter a [10−
3 mm
]
corecore
mean
mean
cellvelocity:
%%
%
&
&
&&
Flo
wp
atterns
inth
em
icro-circu
lation
123T
U/e
Micro
-circulatio
n:
& :
capillaries
assum
ptio
ns:
parab
olic
cellshap
e
2.
locatio
no
fw
all
film
thickn
ess
. 2
thin
film
app
roximatio
n:
a
z
r
cell
wall
h
cell
wallr
z
ap
ac p
p
Flo
wp
atterns
inth
em
icro-circu
lation
124T
U/e
