mr · 2002. 2. 20. · adeler a tv erhver vsf or skerpr ojekt ef 698 imm-phd-2001-86 issn 0909-3192...
TRANSCRIPT
Hem
od
ynam
icsi
mu
lati
on
of
the
hea
rtu
sin
ga
2Dm
od
el
and
MR
dat
a
Per
nill
eT
ho
rup
Ad
eler
AT
VE
rhve
rvsf
ors
kerp
roje
ktE
F69
8
IMM
-PH
D-2
001-
86
ISS
N09
09-3
192
ISB
N87
-883
06-1
3-5
c
2001
byP
erni
lleT
horu
pA
dele
rP
rinte
dby
IMM
,D
TU
Boo
kbin
derH
ansM
eyer
Hem
odyn
amics
imul
atio
noft
hehe
artu
sing
a2D
mod
elan
dM
Rda
ta.
Per
nille
Tho
rup
Ade
ler
Aug
ust2
001
Pre
face
Thi
sth
esis
issu
bmitt
edin
part
ialf
ulfil
men
toft
here
quire
men
tsfor
the
Ph.
D.d
e-gr
eeat
the
Tech
nica
lUni
vers
ityof
Den
mar
k(D
TU
).T
hepr
ojec
ttoo
kpl
acea
san
indu
stria
lres
earc
hPh.
D.p
roje
ctad
min
iste
redb
yth
eD
anis
hA
cade
myo
fTe
ch-
nica
lSci
ence
s(AT
V)
and
finan
cial
lysu
ppor
tedb
yth
eD
anis
hA
genc
yfo
rT
rade
andI
ndus
tryi
nth
epe
riodA
ugus
t1st
1997
–A
ugus
t31s
t200
1dur
ing
whi
chth
ere
was
ato
talo
f10
.5m
onth
slea
ve.
Itw
asca
rrie
dou
twith
the
follo
win
gpa
rtne
rs:
Mat
h-Te
chA
ps,I
nfor
mat
ics
and
Mat
hem
atic
alM
odel
ling
(IM
M)
atD
TU
,an
dS
kejb
yS
ygeh
us,A
rhus
Uni
vers
ityH
ospi
tal.
Ack
now
ledg
emen
ts
Firs
tofa
llIw
ould
like
toth
ankm
ysu
perv
isor
s,who
alli
nth
eirv
ario
usw
aysh
ave
mad
ethi
spr
ojec
tpos
sibl
e.Ja
cobM
.Ja
cobs
en,R
oski
lde
Uni
vers
ity,h
asbe
ena
won
derf
ulan
din
spiri
ngsu
perv
isor
and
cow
orke
r.I
wou
ldno
thav
eob
tain
edas
good
akn
owle
dgeo
fthe
2Dm
odel
and
itspr
ogra
mm
ingc
odew
ithou
thim
.Je
s-pe
rLar
sen,
Mat
h-Te
ch,w
asth
ein
itiat
orof
the
proj
ecta
ndha
salw
ays
give
nm
eve
ryhe
lpfu
lgui
danc
eand
idea
sfor
furt
herw
ork.
Vin
cent
Alla
nB
arke
r,fo
rmer
lyof
IMM
has
give
nm
eve
ryth
orou
ghan
dus
eful
feed
back
onm
yw
ritte
nm
ate-
rial.
Writ
ing
the
thes
isw
ould
have
been
chao
ticw
ithou
this
advi
ce.E
rikM
orre
Ped
erse
n,Won
Yon
gK
iman
dK
imH
oulin
d,S
kejb
yS
ygeh
us,h
ave
help
edm
eun
ders
tand
the
MR
tech
niqu
eand
give
nm
eus
eful
feed
back
onth
em
odel
from
am
edic
alpo
into
fvi
ew.
Kim
Hou
lind
mus
tals
obe
than
ked
for
his
sugg
estio
nsco
ncer
ning
the
inve
stig
atio
noft
hedi
asto
licfu
nctio
nof
the
vent
ricle
.I
wou
ldlik
eto
expr
essm
ysp
ecia
lgra
titud
eto
Cha
rlesS
.Pes
kina
ndD
avid
M.
McQ
ueen
both
ofth
eC
oura
ntIn
stitu
teof
Mat
hem
atic
alS
cien
ces,N
ewY
ork
Uni
vers
ity.
Iam
inde
bted
toth
emfo
rgr
antin
gm
eac
cess
toth
eso
urce
code
ofth
eirh
eart
mod
elsb
utev
enm
ore
for
the
inva
luab
lehe
lp,g
uida
ncea
ndin
spira
tion
ii they
have
give
nm
eth
roug
hout
the
entir
epr
ojec
tand
part
icul
arly
durin
gm
yst
ayat
the
Cou
rant
Inst
itute
inth
efa
llof
1998
.I
also
wan
tto
than
kal
lmy
colle
ague
satM
ath-
Tech
and
IMM
for
givi
ngm
ew
onde
rful
wor
king
envi
ronm
ents
thro
ugho
utth
epa
stfo
urye
ars.
Fin
ally
,I
wan
tto
give
my
war
mes
tthan
ksto
my
fam
ily,
espe
cial
lym
yhu
s-ba
ndO
lean
dou
rson
Mag
nus,
for
allt
heir
love
and
supp
ort.
Bag
svæ
rd,A
ugus
t200
1
Per
nille
Tho
rup
Ade
ler
Abs
trac
t
Com
puta
tiona
lmod
elso
fthe
bloo
dflo
win
the
hear
tare
aus
eful
tool
for
stud
ying
the
func
tioni
ngof
the
hear
t.T
hepu
rpos
eof
this
thes
isis
toac
hiev
ea
bette
run
ders
tand
ingo
fhe
mod
ynam
icso
fth
eno
rmal
and
dise
ased
hear
tsth
roug
hth
eus
eofa
com
puta
tiona
lmod
elan
dm
agne
ticre
sona
nce(M
R)
data
.W
epr
esen
ta2D
com
puta
tiona
lmod
elof
the
bloo
dflo
win
the
left
side
ofth
ehe
art.
The
wor
kis
base
don
Pes
kina
ndM
cQue
en’s
2Dm
odel
dim
ensi
oned
toda
taon
the
dog
hear
t,w
hich
we
impr
ove
and
adju
stus
ing
phys
iolo
gica
lkno
wl-
edge
and
MR
velo
city
data
toac
hiev
ea
mod
elof
the
hum
anhe
art.
The
impr
ove-
men
tsre
quire
chan
ging
the
geom
etry,
the
timin
g,th
em
echa
nica
lactiv
atio
nof
the
hear
tmus
cula
ture
,and
the
afte
rload
.Fur
ther
mor
e,we
intr
oduc
eate
ther
ingo
fthe
othe
rwis
efre
ely
float
ing
hear
t.W
eev
alua
teth
em
odel
from
aco
mpu
tatio
nalan
dm
odel
ling
poin
tofv
iew
and
find
ase
tofr
easo
nabl
epar
amet
erva
lues
.Thi
sis
our
refe
renc
emod
el,w
hich
give
srep
rese
ntat
ivesi
mul
atio
nres
ults
.W
eco
mpa
rea
sim
ulat
ionu
sing
ourr
efer
ence
mod
elw
ithan
MR
velo
city
data
seto
btai
nedf
rom
ahe
alth
yhu
man
.T
heco
mpa
rison
isca
rrie
dou
tfor
the
in-
trav
entr
icul
arve
loci
tyfie
ldan
dth
eve
loci
tytim
ecu
rves
over
the
mitr
alrin
gan
dac
ross
the
aort
icou
tflow
trac
t.T
heco
mpa
rison
betw
eenv
eloc
ityfie
lds
show
sa
reas
onab
lyfai
rag
reem
entin
the
gene
ralfl
owpa
ttern
:aw
ide
inflo
wje
t,th
efo
r-m
atio
nof
anan
terio
rvor
tex
durin
gfil
ling,
and
anou
tflow
jett
hrou
ghth
eou
tflow
trac
t.T
here
ares
omed
isag
reem
entsin
the
deta
iledfl
owpa
ttern
,inpa
rtic
ular
with
rega
rdto
the
vort
expa
ttern
s.T
heve
loci
tytim
ecu
rves
from
the
sim
ulat
ion
show
good
agre
emen
twith
MR
data
.T
hetim
ing
inth
esi
mul
atio
nis
prac
tical
lyth
esa
mea
sin
the
MR
data
,whi
leth
erea
reso
med
iffer
ence
sbet
wee
nthe
shap
esan
dm
axim
umva
lues
ofth
eve
loci
tycu
rves
.W
eus
eou
r2D
mod
elto
perf
orm
inve
stig
atio
nsof
cert
ain
mec
hani
smsin
-vo
lved
inhe
artd
isea
sesa
ffect
ing
the
dias
tolic
func
tioni
ngof
the
hear
t.To
beab
leto
sim
ulat
epat
holo
gica
lcon
ditio
nsw
eim
prov
eth
em
odel
for
the
mec
hani
-ca
lact
ivat
ion
ofth
ehe
artm
uscl
e.W
efin
dth
atit
isno
tpos
sibl
eto
succ
essf
ully
iv sim
ulat
ean
isch
emic
apic
alre
gion
byle
tting
the
rela
xatio
nbe
slow
erin
the
apic
alre
gion
.How
ever
,we
are
able
tosu
cces
sful
lymod
ela
glob
alis
chem
icle
ftve
ntri-
cle
thro
ugha
slow
erre
laxa
tiono
fthe
entir
eve
ntric
lean
dto
mod
ela
myo
card
ial
infa
rctio
naf
fect
ing
the
apex
byle
tting
the
apic
alre
gion
bein
activ
e.In
both
ofth
esec
ases
the
sim
ulat
ion
resu
ltsco
mpa
rew
ellw
ithcl
inic
ally
obse
rved
data
ondo
gsan
dhu
man
s.W
epr
esen
tPes
kina
ndM
cQue
en’s3D
mod
elof
the
entir
ehum
anhe
arta
ndth
ene
arby
grea
tves
sels
.We
perf
orm
asi
mul
atio
nw
ithth
em
odel
,whe
rew
ead
just
the
timin
gto
beth
esa
mea
sin
ourr
efer
ence
2Dm
odel
.Unf
ortu
nate
ly,th
ere
sults
dono
tcom
pare
very
wel
lwith
MR
data
.In
part
icul
ar,t
heflo
wan
dve
loci
tyov
erth
em
itral
ring
are
noti
ngo
odag
reem
ent,a
ndth
epr
essu
rein
the
vent
ricle
sis
far
too
high
.F
urth
erm
ore,
the
3Dm
odel
isco
mpu
tatio
nally
very
dem
andi
ng.T
his,
toge
ther
with
the
disa
gree
men
twith
MR
data
,mak
esit
unfe
asib
leto
use
the
3Dm
odel
asa
tool
fori
nves
tigat
ingt
hehe
mod
ynam
icso
fthe
hear
t.How
ever
,the
3Dm
odel
give
sins
ight
into
the
vort
expa
ttern
inth
ele
ftve
ntric
le.A
clea
rvor
tex
ring
isfo
rmed
belo
wth
em
itral
valv
edu
ring
fillin
g,an
din
acu
t-aw
ayvi
ewth
isrin
gis
seen
astw
odi
stin
ctvo
rtic
es,s
imila
rto
the
vort
ices
form
edin
our2
Dm
odel
.
Res
ume
pada
nsk
Com
pute
rmod
elle
rafbl
odst
røm
ning
enih
jert
eter
etny
ttigt
værk
tøjti
lat
stud
ere
hjer
tets
funk
tion.
Form
alet
med
denn
eafh
andl
inge
rato
pna
enbe
dref
orst
aels
efo
rdet
rask
eog
dets
ygeh
jert
eshæ
mod
ynam
ikved
atan
vend
een
com
pute
rmod
elog
mag
netis
kres
onan
s(MR
)da
ta.
Vi
præ
sent
erere
n2D
com
pute
rmod
elafb
lods
trø
mni
ngen
ide
nve
nstr
esid
eaf
hjer
tet.
Arb
ejde
terb
aser
etpa
Pes
kino
gM
cQue
en’s2D
mod
eldi
men
sion
eret
tilda
tafo
rhun
dehj
erte
t.Den
nem
odel
forb
edre
rog
ænd
rerv
ived
brug
affy
siol
o-gi
skvi
den
ogM
Rha
stig
heds
datafo
rder
ved
atop
naen
mod
elaf
detm
enne
skelig
ehj
erte
.For
bedr
inge
rnek
ræve
ratg
eom
etrie
n,tid
sfor
løbe
t,den
mek
anis
keak
tive-
ring
afhj
erte
mus
kula
ture
nog
udlø
bsbe
tinge
lser
neænd
res.
Yde
rmer
eint
rodu
-ce
rerv
ien
fast
gøre
lsea
fdet
elle
rsfr
itfly
dend
ehje
rte.
Vie
valu
erer
mod
elle
nud
fra
etbe
regn
ings
-og
mod
elle
rings
mæ
ssig
tsyns
punk
tog
finde
rher
ved
etsæ
taf
para
met
ervæ
rdie
r.
Det
teer
vore
sref
eren
cem
odel
,derg
iver
repr
æse
ntat
ivesi
mu-
lerin
gsre
sulta
ter.
Vis
amm
enlig
neren
sim
uler
ingm
edre
fere
ncem
odel
lenm
edet
MR
hast
ighe
ds-
data
sætfra
etra
skm
enne
ske.S
amm
enlig
ning
enud
føre
sfor
deti
ntra
vent
rikul
ære
hast
ighe
dsfe
ltog
forh
astig
heds
kurve
rove
rmitr
alrin
geno
gpa
tvæ
rsaf
udlø
bstr
ag-
ten
ved
aort
a.S
amm
enlig
ning
enm
elle
mha
stig
heds
felte
rnevi
ser
enfo
rnuf
tigov
eren
sste
mm
elsei
detg
ener
elle
flow
mø
nste
r:en
bred
indl
øbs
jet,d
anne
lsen
afen
ante
riorh
virv
elve
dfy
ldni
ngog
enud
løbs
jetg
enne
mud
løbs
trag
tenv
edao
rta.
Der
erdo
gvi
sseu
over
enss
tem
mel
serid
etde
talje
rede
flow
mø
nste
r,spe
ciel
tmed
hens
yntil
hvirv
elm
øns
tret
.Has
tighe
dsku
rvern
efra
sim
uler
inge
nvis
ergo
dov
er-
enss
tem
mel
semed
MR
data
.Tid
sfor
løbe
tisi
mul
erin
gene
rsto
rtse
tdet
sam
me
som
iMR
data
,men
sder
eraf
vige
lser
mel
lem
form
enpa
ogm
axim
umvæ
rdie
rne
afha
stig
heds
kurve
rne.
Vib
enyt
terv
ores
2Dm
odel
tilat
udfø
reen
unde
rsø
gels
eafbe
stem
temek
anis
-m
er,d
erop
træ
derv
edhj
erte
sygd
omm
e,der
pavi
rker
den
dias
tolis
kefu
nktio
naf
hjer
tet.
Fora
tkun
nesi
mul
ered
isse
pato
logi
ske
tilst
ande
forb
edre
rvim
odel
lena
f
vi den
mek
anis
keak
tiver
ing
afhj
erte
mus
kula
ture
n.Vi
finde
r,at
deti
kke
erm
ulig
tsu
cces
fuld
tat
sim
uler
een
iskæ
mis
kapi
kalr
egio
nve
dat
lade
den
apik
ale
re-
gion
rela
kser
elan
gsom
mer
e.T
ilge
ngæ
ldka
nvi
med
succ
essi
mul
eree
ngl
obal
iskæ
mis
kven
stre
vent
rikel
ved
atla
dehe
leve
ntrik
len
rela
kser
elang
som
mer
e,og
vika
nsi
mul
ere
etm
yoka
rdie
infa
rkt,
derr
amm
erap
ex,
ved
atla
dede
nap
ikal
ere
gion
være
inak
tiv.
Ibe
gge
diss
etilf
æld
eer
derg
odov
eren
sste
mm
elsem
elle
msi
mul
erin
gsre
sulta
terog
klin
isk
obse
rvere
deda
tafr
aba
dehu
ndeo
gm
enne
sker.
Vi
præ
sent
ererP
eski
nog
McQ
ueen
’s3D
mod
elaf
hele
detm
enne
skelig
ehj
erte
ogde
nærm
estes
tore
kar.
Vi
udfø
rere
nsi
mul
erin
gm
edm
odel
len,
hvor
tidsf
orlø
bete
rdet
sam
mes
omi
vore
s2D
refe
renc
emod
el.D
esvæ
rree
rder
ikke
engo
dov
eren
sste
mm
elsem
elle
msi
mul
erin
gsre
sulta
terog
MR
data
.S
peci
elt
flow
etog
hast
ighe
deno
ver
mitr
alrin
gen
pass
erik
kego
dt,o
gtr
ykke
tive
ntrik
-le
rnee
ralt
for
højt.
Yde
rmer
eer3
Dm
odel
lenm
eget
bere
gnin
gskr
æve
nde.
Det
te,
sam
men
hold
tmed
den
darli
gesa
mm
enlig
ning
med
MR
data
,gø
rdet
umul
igta
tan
vend
e3D
mod
elle
nsom
etvæ
rktø
jtil
unde
rsø
gels
eafhj
erte
tshæ
mod
ynam
ik.
Til
geng
æld
kan
3Dm
odel
len
give
inds
igti
hvirv
elm
øns
tret
ide
nve
nstr
even
-tr
ikel
.D
erda
nnes
enty
delig
hvirv
elrin
gun
derm
itral
klap
penv
edfy
ldni
ng,o
gi
etsn
itge
nnem
hjer
tets
esde
nner
ing
som
todi
stin
kte
hvirv
ler,
svar
ende
tilde
tohv
irvle
r,de
rdan
nesi
2Dm
odel
len.
Con
tent
s
Pre
face
i
Abs
trac
tiii
Res
ume
pada
nsk
v
List
offig
ures
xi
1In
trod
uctio
n1
1.1
Hea
rtsi
mul
atio
ns.
..
..
..
..
..
..
..
..
..
..
..
..
.1
1.2
MR
scan
ning
..
..
..
..
..
..
..
..
..
..
..
..
..
..
41.
3U
sing
sim
ulat
ions
and
MR
data
toge
ther
..
..
..
..
..
..
..
41.
4O
utlin
eof
thes
is.
..
..
..
..
..
..
..
..
..
..
..
..
..
5
2P
hysi
olog
yan
dan
atom
yof
the
hear
t7
2.1
Inte
rnal
anat
omy
..
..
..
..
..
..
..
..
..
..
..
..
..
82.
2F
unct
ioni
ngof
the
hear
t..
..
..
..
..
..
..
..
..
..
..
.10
2.3
Con
duct
ions
yste
mof
the
hear
t..
..
..
..
..
..
..
..
..
.11
2.4
Mus
cle
phys
iolo
gy.
..
..
..
..
..
..
..
..
..
..
..
..
13
32D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d17
3.1
Gen
eral
desc
riptio
noft
hem
odel
..
..
..
..
..
..
..
..
..
173.
2C
ontin
uous
form
ulat
ion
..
..
..
..
..
..
..
..
..
..
..
.18
3.2.
1S
ourc
esan
dsi
nks
..
..
..
..
..
..
..
..
..
..
..
203.
3D
iscr
etef
orm
ulat
ion
..
..
..
..
..
..
..
..
..
..
..
..
243.
3.1
Dis
cret
equa
ntiti
es.
..
..
..
..
..
..
..
..
..
..
243.
3.2
Dif
fere
nceo
pera
tors
and
disc
rete
funct
ion
..
..
..
.25
3.3.
3D
iscr
etiz
atio
nofe
quat
ions
ofm
otio
n.
..
..
..
..
..
27
viii
Con
tent
s
3.3.
4D
iscr
eteW
indk
esse
lmod
elfo
rth
eao
rtic
sink
..
..
..
.28
3.3.
5B
ound
aryf
orce
s.
..
..
..
..
..
..
..
..
..
..
.31
3.3.
6T
heac
tivat
ion
func
tion
..
..
..
..
..
..
..
..
..
.35
3.3.
7To
polo
gyan
din
itial
geom
etry
ofth
ehe
art.
..
..
..
..
383.
4O
utlin
eof
algo
rithm
..
..
..
..
..
..
..
..
..
..
..
..
40
4E
valu
atio
nof
the
2Dm
odel
434.
1S
imul
atio
nout
put.
..
..
..
..
..
..
..
..
..
..
..
..
.43
4.1.
1H
eart
boun
dary
confi
gura
tiona
ndflu
idm
arke
rs.
..
..
.44
4.1.
2V
eloc
ityfie
ld.
..
..
..
..
..
..
..
..
..
..
..
.44
4.1.
3P
ress
urefi
eld
..
..
..
..
..
..
..
..
..
..
..
..
494.
2T
hesc
alin
gfac
tor
...
..
..
..
..
..
..
..
..
..
..
..
504.
3P
erio
dicb
ound
aryc
ondi
tions
..
..
..
..
..
..
..
..
..
..
564.
4Te
ther
ingo
fthe
hear
t.
..
..
..
..
..
..
..
..
..
..
..
.60
4.5
The
Win
dkes
selm
odel
..
..
..
..
..
..
..
..
..
..
..
.66
4.6
Dis
trib
uted
sour
cesa
ndsi
nks
..
..
..
..
..
..
..
..
..
..
694.
7S
imul
atio
nove
rsev
eral
hear
tcyc
les
..
..
..
..
..
..
..
..
714.
8C
oncl
usio
ns.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
75
5C
ompa
ring
the
2Dm
odel
with
anM
Rda
tase
t77
5.1
The
MR
data
..
..
..
..
..
..
..
..
..
..
..
..
..
..
775.
2V
eloc
ityfie
lds
..
..
..
..
..
..
..
..
..
..
..
..
..
..
795.
3V
eloc
itytim
epr
ofile
s.
..
..
..
..
..
..
..
..
..
..
..
.85
5.4
Dis
cuss
iona
ndco
nclu
sion
s..
..
..
..
..
..
..
..
..
..
.89
6S
imul
atin
gpa
thol
ogic
alhe
artc
ondi
tions
936.
1M
edic
alba
ckgr
ound
..
..
..
..
..
..
..
..
..
..
..
..
946.
1.1
Mec
hani
smsd
urin
gea
rlyfil
ling
..
..
..
..
..
..
..
946.
1.2
Path
olog
ical
hear
tcon
ditio
ns.
..
..
..
..
..
..
..
.94
6.1.
3V
eloc
ityin
dexe
s.
..
..
..
..
..
..
..
..
..
..
.96
6.1.
4T
ime
cons
tant
ofre
laxa
tion.
..
..
..
..
..
..
..
..
996.
1.5
Reg
ions
ofth
eve
ntric
le.
..
..
..
..
..
..
..
..
.99
6.2
The
new
activ
atio
nfu
nctio
n.
..
..
..
..
..
..
..
..
..
..
100
6.2.
1T
heor
igin
alm
odel
..
..
..
..
..
..
..
..
..
..
.10
16.
2.2
The
new
impl
emen
tatio
n.
..
..
..
..
..
..
..
..
.10
26.
2.3
Tim
eco
nsta
ntof
rela
xatio
n..
..
..
..
..
..
..
..
.10
66.
3C
alib
ratin
gthe
mod
el.
..
..
..
..
..
..
..
..
..
..
..
.10
66.
3.1
Vel
ocity
profi
lesa
long
tran
sver
selin
es.
..
..
..
..
..
106
6.3.
2G
radu
alac
tivat
ion
..
..
..
..
..
..
..
..
..
..
.10
86.
3.3
Reg
iona
lact
ivat
ion
..
..
..
..
..
..
..
..
..
..
.11
4
Con
tent
six
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
..
..
..
..
..
..
120
6.4.
1S
low
rela
xatio
noft
heve
ntric
le.
..
..
..
..
..
..
.12
16.
4.2
Slo
wre
laxa
tiono
fapi
calr
egio
n.
..
..
..
..
..
..
.12
66.
4.3
Api
calr
egio
nin
activ
e.
..
..
..
..
..
..
..
..
..
129
6.5
Con
clus
ions
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.13
4
7T
he3D
hear
tmod
el13
77.
1D
escr
iptio
noft
hem
odel
..
..
..
..
..
..
..
..
..
..
..
137
7.1.
1E
quat
ions
ofm
otio
n.
..
..
..
..
..
..
..
..
..
.13
87.
1.2
Com
puta
tiona
lmet
hod
..
..
..
..
..
..
..
..
..
.14
07.
1.3
Initi
alge
omet
ry.
..
..
..
..
..
..
..
..
..
..
..
142
7.2
Res
ults
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
143
7.3
Dis
cuss
ion.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.14
8
8C
oncl
usio
nsan
dsu
gges
tions
149
8.1
Sum
mar
yofc
oncl
usio
ns.
..
..
..
..
..
..
..
..
..
..
.14
98.
2S
ugge
stio
nsfo
rfu
ture
wor
k.
..
..
..
..
..
..
..
..
..
..
151
App
endi
x15
3
AM
agne
ticre
sona
nceim
agin
g15
3A
.1B
asic
sofM
RI
..
..
..
..
..
..
..
..
..
..
..
..
..
..
153
A.1
.1E
xcita
tion
..
..
..
..
..
..
..
..
..
..
..
..
..
155
A.1
.2R
elax
atio
n.
..
..
..
..
..
..
..
..
..
..
..
..
156
A.1
.3P
ulse
sequ
ence
s..
..
..
..
..
..
..
..
..
..
..
.15
8A
.2Im
agin
gm
etho
ds.
..
..
..
..
..
..
..
..
..
..
..
..
.16
1A
.2.1
Slic
ese
lect
iong
radi
ent.
..
..
..
..
..
..
..
..
..
162
A.2
.2P
hase
enco
ding
grad
ient
..
..
..
..
..
..
..
..
..
163
A.2
.3R
ead-
outg
radi
ent.
..
..
..
..
..
..
..
..
..
..
.16
3A
.2.4
Imag
ing
sequ
ence
s..
..
..
..
..
..
..
..
..
..
.16
4A
.2.5
Mot
ion
artif
acts
..
..
..
..
..
..
..
..
..
..
..
.16
7A
.3V
eloc
itym
easu
rem
ents.
..
..
..
..
..
..
..
..
..
..
..
168
Bib
liogr
aph
y17
1
List
ofF
igur
es
2.1
The
-fou
rcha
mbe
redh
eart
isdi
vide
din
totw
opa
rts,
the
left
and
right
side
s.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.8
2.2
Dra
win
gof
ahe
arts
plit
perp
endi
cula
rtoth
ein
terv
entr
icul
arse
ptum
92.
3T
hero
otao
rtic
pres
sure
,vent
ricul
arpr
essu
re,a
tria
lpre
ssur
e,ar
-te
riali
nflow
,to
tali
nflow
toth
ehe
art,
atria
lvol
ume,
vent
ricul
arvo
lum
ean
dve
ntric
ular
outfl
ow.
..
..
..
..
..
..
..
..
..
112.
4S
chem
aticr
epre
sent
atio
noft
heco
nduc
tions
yste
mof
the
hear
t..
122.
5S
chem
aticr
epre
sent
atio
nofa
sect
iono
fare
laxe
dan
dco
ntra
cted
fibril
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.13
2.6
Act
ion
pote
ntia
land
cont
ract
ilem
echa
nica
lresp
onse
ofa
card
iac
mus
clefi
ber
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.14
3.1
Com
puta
tiona
ldom
ain
andi
nitia
lhea
rtbo
unda
ryco
nfigu
ratio
n
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
193.
2A
nel
ectr
ican
alog
ueto
the
linea
rres
ista
ncem
odel
..
..
..
..
233.
3D
iagr
amof
thre
e-el
emen
tWin
dkes
selm
odel
coup
ledt
odi
ode
valv
e24
3.4
Plo
toft
hefu
nctio
n
..
..
..
..
..
..
..
..
..
..
..
263.
5S
ketc
hof
the
mus
cula
rmod
el.
..
..
..
..
..
..
..
..
..
.34
3.6
Act
ivat
ion
asfu
nctio
nof
time
..
..
..
..
..
..
..
..
..
..
373.
7M
odifi
edin
itial
hear
tgeo
met
ry.
..
..
..
..
..
..
..
..
..
39
4.1
Hea
rtbo
unda
ryan
dflu
idm
arke
rsfr
omth
ere
fere
nces
imul
atio
n.
454.
2V
eloc
ityfie
ldan
dhea
rtbou
ndar
ycon
figur
atio
nfro
mth
ere
fere
nce
sim
ulat
ion
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
464.
3V
eloc
itytim
epr
ofile
atth
em
itral
valv
efo
rth
ere
fere
nces
imul
atio
n47
4.4
Vel
ocity
time
profi
leac
ross
the
outfl
owtr
actf
orth
ere
fere
nces
im-
ulat
ion
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
484.
5S
ourc
eflow
time
profi
lefo
rth
ere
fere
nces
imul
atio
n.
..
..
..
48
xii
List
offig
ures
4.6
Pre
ssur
etim
epr
ofile
for
the
refe
renc
esim
ulat
ion
..
..
..
..
.49
4.7
Tim
eva
riatio
nofv
entr
icul
arpr
essu
reand
activ
atio
nfo
rth
ere
fer-
ence
sim
ulat
ion
..
..
..
..
..
..
..
..
..
..
..
..
..
.50
4.8
Vel
ocity
field
and
hear
tbou
ndar
yat
s,w
ith
0.0
4,0.
1,0.
3an
d0.
5.
..
..
..
..
..
..
..
..
..
..
..
..
..
514.
9V
eloc
ityfie
ldan
dhe
artb
ound
arya
t
s,w
ith
0.0
4,0.
1,0.
3an
d0.
5.
..
..
..
..
..
..
..
..
..
..
..
..
..
524.
10V
eloc
ityfie
ldan
dhe
artb
ound
arya
t
s,w
ith
0.0
4,0.
1,0.
3an
d0.
5.
..
..
..
..
..
..
..
..
..
..
..
..
..
534.
11P
ress
ureti
me
profi
lefo
r
0.0
4,0.
1an
d0.
5.
..
..
..
..
..
544.
12V
eloc
itytim
epr
ofile
atth
em
itral
valv
efo
r
0.0
4,0.
1an
d0.
5.
554.
13V
eloc
itytim
epr
ofile
acro
ssth
eao
rtic
outfl
owtr
actf
or
0.0
4,0.
1an
d0.
5.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.55
4.14
Hea
rtbo
unda
ryfr
oma
refe
renc
esim
ulat
ion
and
from
asi
mul
a-tio
nw
itha
larg
eco
mpu
tatio
naldo
mai
n.
..
..
..
..
..
..
.57
4.15
Pre
ssur
etim
epr
ofile
from
asi
mul
atio
nwith
ala
rge
com
puta
tiona
ldo
mai
nand
from
the
refe
renc
esim
ulat
ion
..
..
..
..
..
..
.58
4.16
Vel
ocity
time
profi
leat
the
mitr
alva
lve
from
asi
mul
atio
nwith
ala
rge
com
puta
tiona
ldom
aina
ndfr
omth
ere
fere
nces
imul
atio
n.
.59
4.17
Vel
ocity
time
profi
leac
ross
the
aort
icsi
nkfr
oma
sim
ulat
ionw
itha
larg
eco
mpu
tatio
naldo
mai
nand
from
the
refe
renc
esim
ulat
ion
.59
4.18
Sou
rcefl
owtim
epr
ofile
sfr
oma
sim
ulat
ion
with
ala
rge
com
pu-
tatio
nald
omai
nand
from
are
fere
nces
imul
atio
n.
..
..
..
..
.60
4.19
Vel
ocity
field
and
hear
tbou
ndar
ycon
figur
atio
nfro
ma
sim
ulat
ion
with
note
ther
ingo
fthe
hear
t.
..
..
..
..
..
..
..
..
..
.61
4.20
Vel
ocity
field
andh
eartb
ound
aryc
onfig
urat
ionf
rom
the
refe
renc
esi
mul
atio
n.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.62
4.21
Pre
ssur
etim
epr
ofile
from
asi
mul
atio
nw
ithno
teth
erin
gof
the
hear
tand
from
the
refe
renc
esim
ulat
ion
..
..
..
..
..
..
..
634.
22V
eloc
itytim
epr
ofile
atth
em
itral
valv
efr
oma
sim
ulat
ionw
ithno
teth
erin
gand
from
the
refe
renc
esim
ulat
ion
..
..
..
..
..
..
644.
23S
ourc
eflow
time
profi
lefr
oma
sim
ulat
ion
with
note
ther
inga
ndfr
omth
ere
fere
nces
imul
atio
n..
..
..
..
..
..
..
..
..
..
654.
24In
itial
geom
etry
ofth
edo
ghe
art
..
..
..
..
..
..
..
..
..
674.
25P
ress
uret
ime
profi
lefr
oma
sim
ulat
ion
with
outt
heW
indk
esse
lm
odel
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
684.
26P
ress
ureti
me
profi
lefr
oma
sim
ulat
ionw
ithth
eW
indk
esse
lmod
el68
4.27
Pre
ssur
etim
epr
ofile
durin
g4
cycl
es.
..
..
..
..
..
..
..
.73
4.28
Vel
ocity
time
profi
ledu
ring
4cy
cles
atth
em
itral
ring
..
..
..
.73
List
offig
ures
xiii
4.29
Vel
ocity
time
profi
ledu
ring
4he
artc
ycle
sacr
osst
heao
rtic
sink
.74
4.30
Sou
rcefl
owtim
epr
ofile
durin
g4
hear
tcyc
les
..
..
..
..
..
.74
5.1
Vel
ocity
field
and
hear
tbou
ndar
yfro
mM
Rm
easu
rem
entsa
ndre
fere
nces
imul
atio
n..
..
..
..
..
..
..
..
..
..
..
..
.80
5.2
Vel
ocity
field
and
hear
tbou
ndar
yfro
mM
Rm
easu
rem
entsa
ndre
fere
nces
imul
atio
n..
..
..
..
..
..
..
..
..
..
..
..
.81
5.3
Vel
ocity
field
and
hear
tbou
ndar
yfro
mM
Rm
easu
rem
entsa
ndre
fere
nces
imul
atio
n..
..
..
..
..
..
..
..
..
..
..
..
.82
5.4
Vel
ocity
time
profi
leat
the
mitr
alrin
gfr
omM
Rda
taan
dsi
mul
atio
n87
5.5
Vel
ocity
time
profi
leac
ross
the
aort
icou
tflow
trac
tfro
mM
Rda
taan
dsi
mul
atio
n.
..
..
..
..
..
..
..
..
..
..
..
..
..
.88
6.1
Vel
ocity
profi
leal
ong
six
lines
from
ahe
alth
yper
son
..
..
..
.97
6.2
Pea
kve
loci
tyfr
omM
Rm
easu
rem
entso
fhe
alth
ype
rson
sand
AM
Ipa
tient
s.
..
..
..
..
..
..
..
..
..
..
..
..
..
.98
6.3
Vel
ocity
inde
xfr
omM
Rm
easu
rem
entso
fhe
alth
ype
rson
sand
AM
Ipa
tient
s.
..
..
..
..
..
..
..
..
..
..
..
..
..
.98
6.4
Tim
ede
layf
rom
MR
mea
sure
men
tsofh
ealth
yper
sons
and
AM
Ipa
tient
s.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
986.
5M
Rim
agew
ithth
efiv
ere
gion
soft
heve
ntric
ular
wal
lind
icat
ed.
996.
6Ill
ustr
atio
nof
the
num
berin
goft
helin
ksin
the
two
chai
nsin
the
atriu
man
dth
eve
ntric
le.
..
..
..
..
..
..
..
..
..
..
..
103
6.7
The
five
regi
onso
fthe
vent
ricul
arhe
artw
alla
tini
tialt
ime
..
..
106
6.8
The
six
lines
whe
resp
atia
lvel
ocity
profi
lesa
reco
mpu
tedd
urin
gdi
asto
le.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
108
6.9
Hea
rtbo
unda
ryfr
oma
sim
ulat
ion
with
grad
uala
ctiv
atio
nan
dfr
omth
ere
fere
nces
imul
atio
n.
..
..
..
..
..
..
..
..
..
.11
06.
10H
eart
boun
dary
from
asi
mul
atio
nw
ithgr
adua
lact
ivat
ion
and
from
the
refe
renc
esim
ulat
ion
..
..
..
..
..
..
..
..
..
..
111
6.11
Vel
ocity
time
profi
leat
the
mitr
alva
lve
from
asi
mul
atio
nw
ithgr
adua
lact
ivat
ion
and
from
the
refe
renc
esim
ulat
ion.
..
..
..
.11
26.
12V
eloc
itytim
epr
ofile
atth
eao
rtic
outfl
owtr
actf
rom
asi
mul
atio
nw
ithgr
adua
lact
ivat
ion
and
from
the
refe
renc
esim
ulat
ion
..
..
.11
26.
13F
low
time
profi
lefo
rth
em
itral
sour
cean
daor
ticsi
nkfr
oma
sim
-ul
atio
nw
ithgr
adua
lact
ivat
ion
and
from
the
refe
renc
esim
ulat
ion
.11
36.
14V
eloc
itytim
epr
ofile
atth
em
itral
valv
efr
oma
sim
ulat
ion
whe
re
"!#$ % &'() an
dfr
omth
ere
fere
nces
imul
atio
n..
..
..
..
115
6.15
Vel
ocity
profi
leal
ongs
ixlin
esfr
oma
sim
ulat
iono
fahe
alth
yhum
an11
6
xiv
List
offig
ures
6.16
Pea
kvel
ocity
from
asi
mul
atio
nofa
heal
thyp
erso
n.
..
..
..
.11
86.
17V
eloc
ityin
dex
from
asi
mul
atio
nofa
heal
thyp
erso
n.
..
..
..
118
6.18
Tim
ede
layf
rom
asi
mul
atio
nofa
heal
thyp
erso
n.
..
..
..
..
118
6.19
Vel
ocity
field
atth
etim
eof
PV
atth
eM
LT
line
..
..
..
..
..
119
6.20
Tim
eva
riatio
nof
pres
sure
diff
eren
cebe
twee
natr
ium
and
vent
ri-cl
efr
oma
sim
ulat
iono
fahe
alth
yhum
an.
..
..
..
..
..
..
120
6.21
Vel
ocity
profi
lesf
rom
asi
mul
atio
nwhe
reve
ntric
ular
rela
xatio
nis
slow
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
122
6.22
Pea
kvel
ocity
from
asi
mul
atio
nw
here
the
vent
ricul
arac
tivat
ion
issl
ow.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.12
36.
23V
eloc
ityin
dex
from
asi
mul
atio
nwhe
reth
eve
ntric
ular
rela
xatio
nis
slow
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
123
6.24
Tim
ede
layf
rom
asi
mul
atio
nw
here
the
vent
ricul
arre
laxa
tioni
ssl
ow.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
123
6.25
Vel
ocity
field
atth
etim
eof
PV
atth
eM
LT
line
from
asi
mul
atio
nw
here
the
vent
ricul
arre
laxa
tioni
ssl
ow.
..
..
..
..
..
..
.12
46.
26T
ime
varia
tion
ofpr
essu
redi
ffer
ence
betw
eena
triu
man
dve
ntri-
cle
from
asi
mul
atio
nwhe
reth
eve
ntric
ular
rela
xatio
nis
slow
..
.12
56.
27V
eloc
itypr
ofile
sfro
ma
sim
ulat
ionw
here
the
apic
alre
laxa
tioni
ssl
ow.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
127
6.28
Pea
kvel
ocity
from
asi
mul
atio
nwhe
reth
eap
ical
rela
xatio
nis
slow
128
6.29
Vel
ocity
inde
xfr
oma
sim
ulat
ionw
here
the
apic
alre
laxa
tioni
ssl
ow12
86.
30T
ime
dela
yfro
ma
sim
ulat
ionw
here
the
apic
alre
laxa
tioni
ssl
ow.
128
6.31
Vel
ocity
field
atth
etim
eof
PV
atth
eM
LT
line
from
asi
mul
atio
nw
here
the
apic
alre
laxa
tioni
ssl
ow.
..
..
..
..
..
..
..
..
129
6.32
Tim
eva
riatio
nof
pres
sure
diff
eren
cebe
twee
natr
ium
and
vent
ri-cl
efr
oma
sim
ulat
ionw
here
the
apic
alre
laxa
tioni
ssl
ow.
..
..
129
6.33
Vel
ocity
field
and
hear
tbou
ndar
yfro
ma
sim
ulat
ion
whe
reth
eap
ical
regi
onis
inac
tive.
..
..
..
..
..
..
..
..
..
..
..
130
6.34
Vel
ocity
profi
lesf
rom
asi
mul
atio
nw
here
the
apic
alre
gion
isin
-ac
tive
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
131
6.35
Pea
kvel
ocity
from
asi
mul
atio
nwhe
reth
eap
ical
regi
onis
inac
tive
132
6.36
Vel
ocity
inde
xfr
oma
sim
ulat
ionw
here
the
apic
alre
gion
isin
activ
e132
6.37
Tim
ede
layf
rom
asi
mul
atio
nwhe
reth
eap
ical
regi
onis
inac
tive
.13
26.
38T
ime
varia
tion
ofpr
essu
redi
ffer
ence
betw
eena
triu
man
dve
ntri-
cle
from
asi
mul
atio
nwhe
reth
eap
ical
regi
onis
inac
tive
..
..
..
134
7.1
Initi
alge
omet
ryof
the
3Dhe
art.
..
..
..
..
..
..
..
..
..
142
7.2
Cut
-aw
ayvi
ewof
the
initi
alfib
erco
nfigu
ratio
n.
..
..
..
..
.14
3
List
offig
ures
xv
7.3
Flo
wtim
epr
ofile
atth
em
itral
and
aort
icva
lve
..
..
..
..
..
.14
57.
4V
eloc
itytim
epr
ofile
atth
eao
rtic
valv
e.
..
..
..
..
..
..
..
146
7.5
Vel
ocity
time
profi
leat
the
mitr
alva
lve
..
..
..
..
..
..
..
.14
67.
6C
ut-a
way
view
ofth
ehe
artd
urin
gea
rlyve
ntric
ular
dias
tole
..
..
147
7.7
Sec
tiont
hrou
ghth
ele
ftve
ntric
lesh
owin
gst
ream
lines
ofve
ntric
-ul
arfil
ling
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
147
A.1
Asp
inni
ngnu
cleu
swith
itsdi
pole
vect
or* whe
nno
exte
rnal
mag
-ne
ticfie
ldis
appl
ied
..
..
..
..
..
..
..
..
..
..
..
..
.15
4A
.2P
rece
ssio
nof
the
dipo
leve
ctor
ofa
spin
ning
nucl
eusa
bout
anex
tern
alm
agne
ticfie
ld
+ ..
..
..
..
..
..
..
..
..
..
.15
4A
.3S
pira
lling
mot
ion
ofth
em
agne
tizat
ionv
ecto
rdur
ing
anR
Fpu
lse
.15
5A
.4T
hetim
eva
riatio
nof
the
leng
thof
the
mag
netiz
atio
nvec
tord
ur-
ing
rela
xatio
n.
..
..
..
..
..
..
..
..
..
..
..
..
..
.15
7A
.5D
epha
sing
ofsp
insa
ftera
90
, puls
e.
..
..
..
..
..
..
..
.15
7A
.6T
ime
varia
tion
ofth
eF
ID.
..
..
..
..
..
..
..
..
..
..
.15
8A
.7T
hesp
inec
hose
quen
ce.
..
..
..
..
..
..
..
..
..
..
..
159
A.8
Tim
eva
riatio
nof
the
MR
sign
alam
plitu
dedu
ring
asp
inec
hose
quen
ces.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.16
0A
.9T
rans
vers
erel
axat
ionc
urve
for
two
tissu
esw
ithdi
ffer
ent
-/. values
161
A.1
0A
sinc
wav
epu
lsea
ndits
Four
iert
rans
form
..
..
..
..
..
..
162
A.1
1M
Rsp
ectr
uman
dco
rres
pond
ingg
rey
scal
eim
age
..
..
..
..
.16
4A
.12
The
grad
ient
echo
sequ
ence
..
..
..
..
..
..
..
..
..
..
.16
6A
.13
Imag
esfr
oma
card
iacM
RI
with
velo
city
enco
ding
..
..
..
..
169
Cha
pter
1
Intr
oduc
tion
Hem
odyn
amics
imul
atio
nsof
the
hear
tbas
edon
com
puta
tiona
lmod
elsc
anbe
aus
eful
tool
for
stud
ying
the
norm
alan
dab
norm
alfu
nctio
ning
ofth
ehe
art.
Inth
isap
proa
chm
agne
ticre
sona
nceim
agin
g(M
RI)
can
beus
edbo
thas
am
eans
for
obta
inin
gin
putp
aram
eter
sand
hear
tbou
ndar
ycon
ditio
nsto
the
com
puta
tiona
lm
odel
and
for
eval
uatin
gthe
sim
ulat
ionr
esul
ts.
The
purp
oseo
fth
ew
ork
pres
ente
dinth
isth
esis
isto
achi
eve
abe
tteru
nder
-st
andi
ngof
the
hem
odyn
amic
sof
the
norm
alan
ddi
seas
edhe
arts
usin
ga
com
-pu
tatio
nalm
odel
and
MR
mea
sure
men
ts.The
star
ting
poin
tis
the
exis
ting
two
and
thre
edi
men
sion
alhe
artm
odel
sdev
elop
edby
Cha
rlesS
.Pes
kina
ndD
avid
M.M
cQue
en.
1.1
Hea
rtsi
mul
atio
ns
Pes
kini
nitia
ted
his
wor
kon
hear
tmod
ellin
gin
his
Ph.
D.t
hesi
s(P
eski
n,19
72a)
with
atw
odi
men
sion
al(2
D)
mod
elof
the
bloo
dflo
win
the
left
side
ofa
dog’
she
art.
He
intr
oduc
eda
new
met
hod
for
fluid
-str
uctu
reco
uplin
gca
lled
the
im-
mer
sedb
ound
arym
etho
d.T
heid
eaof
the
imm
erse
dbou
ndar
ymet
hodi
sth
atth
ehe
artis
tota
llyim
mer
sedi
nflu
id,
i.e.t
hehe
artb
ound
aryh
asflu
idon
both
side
san
dth
ebo
unda
ryis
repr
esen
tedin
term
soft
hefo
rces
itex
erts
onth
eflu
id.
Thu
sth
ebo
unda
ryis
sim
ply
asp
ecia
lized
regi
onof
the
fluid
whe
reex
tra
forc
esar
eap
-pl
ied.
The
fluid
dyna
mic
aleq
uatio
nsar
esol
ved
usin
gC
horin
’spr
ojec
tionm
etho
d(C
horin
,196
8).A
draw
back
ofth
ism
etho
dis
that
alll
engt
hsan
dtim
esha
veto
bere
duce
dto
achi
eve
num
eric
alst
abili
ty,w
ithth
ere
sult
that
the
Rey
nold
snu
mbe
ris
redu
cedt
oan
unph
ysio
logi
calva
lue.
2In
trod
uctio
n
Pes
kinh
asde
velo
peda
ndst
udie
dthe
2Dm
odel
inco
llabo
ratio
nwith
othe
rs,
inpa
rtic
ular
Dav
idM
.McQ
ueen
.Thi
sw
ork
isdo
cum
ente
dby
ase
rieso
fpap
ers:
Pes
kin(
1972
b),P
eski
n(19
75),
Pes
kin(
1977
),P
eski
nand
McQ
ueen
(198
0),P
e-sk
in(1
981)
,McQ
ueen
etal
.(19
82)a
ndP
eski
nand
Prin
tz(1
993)
.The
sepa
pers
,w
hich
desc
ribet
hem
odel
ofa
dog’
she
art,f
orm
the
back
grou
ndfo
rour
wor
kw
ithth
e2D
hear
tmod
el.
Pes
kina
ndM
cQue
en’s
orig
inal
purp
osei
ncr
eatin
gahe
artm
odel
was
toes
-ta
blis
ha
com
pute
rtoo
lfor
stud
ying
the
func
tioni
ngof
the
mitr
alva
lve
(Yel
linet
al.,
1981
;Mei
sner
etal
.,19
85)a
ndfo
rde
sign
inga
ndin
vest
igat
ingp
rost
hetic
hear
tval
ves
for
repl
acem
ento
fth
ena
tura
lmitr
alva
lve
(McQ
ueen
and
Pes
kin,
1983
;McQ
ueen
and
Pes
kin,
1985
).To
geth
erw
ithot
hers
,Pes
kin
has
also
stud
ied
vario
usal
tern
ative
num
eric
alsc
hem
esfo
rth
eso
lutio
nof
the
flow
equa
tions
orfo
rth
etr
eatm
ento
fthe
coup
ling
ofth
ebo
unda
ryfo
rces
toth
eflu
id(M
cCra
cken
and
Pes
kin,
1980
;Bor
gers
and
Pes
kin,
1987
;Tu
and
Pes
kin,
1992
;May
oan
dP
eski
n,19
93).
The
seal
tern
ative
met
hods
are
noti
mpl
emen
tedin
ourv
ersi
onof
the
com
puta
tiona
lmod
el.
The
com
puta
tiona
lmet
hodo
fth
e2D
hear
tmod
elha
ssi
nce
been
expa
nded
to3D
(Gre
enbe
rget
al.,
1987
;Pes
kin
and
McQ
ueen
,198
9;M
cQue
enan
dP
e-sk
in,1
989)
,and
the
wor
kof
Pes
kina
ndM
cQue
enha
scul
min
ated
ina
3Dm
odel
ofth
een
tire
hum
anhe
arta
ndne
arby
grea
tves
sels(
Pes
kin,
1989
;Pes
kina
ndM
c-Q
ueen
,199
2;P
eski
nand
McQ
ueen
,199
3b;P
eski
nand
McQ
ueen
,199
3a;P
eski
nan
dM
cQue
en,1
994;
Pes
kina
ndM
cQue
en,1
995;
Pes
kina
ndM
cQue
en,1
996;
Dav
is,1
995;
McQ
ueen
and
Pes
kin,
2000
).In
the
late
stw
ork
with
the
3Dm
odel
(McQ
ueen
and
Pes
kin,
1997
;McQ
ueen
and
Pes
kin,
2000
)the
num
eric
alm
etho
dfo
rso
lvin
gth
eflu
iddy
nam
ical
equa
tions
hasb
eenc
hang
ed,e
limin
atin
gth
ere
-du
ctio
noft
heR
eyno
ldsn
umbe
r.Sin
ceth
ein
trod
uctio
noft
hefu
ll3D
hear
tmod
el,
Pes
kina
ndM
cQue
enha
vest
oppe
dwor
kon
the
2Dm
odel
and
are
com
plet
elyf
o-cu
sedo
nth
e3D
mod
el.T
he3D
mod
el,h
owev
er,i
sve
ryde
man
ding
with
resp
ect
toco
mpu
tatio
nalpo
wer
(McQ
ueen
and
Pes
kin,
1997
).In
fact
,itw
ould
have
been
too
dem
andi
ngfo
rth
epr
ojec
trep
orte
din
this
thes
is,a
ndit
was
deci
dedt
ous
em
ainl
yth
e2D
mod
el.
The
imm
erse
dbou
ndar
ymet
hodh
asbe
enst
udie
dand
used
byot
hers
.Bey
eran
dLe
Veq
ue(1
992)
stud
ied
the
accu
racy
ofva
rious
num
eric
alap
prox
imat
ions
toth
eflu
idbo
unda
ryco
uplin
gin
aon
e-di
men
sion
almod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d,an
din
Bey
er(1
992)
the
met
hodw
asus
edfo
ra
mod
elof
the
coch
lea(
inne
rear
).S
tock
iean
dW
etto
n(1
995)
pres
ente
dast
abili
tyan
alys
isof
the
equa
tions
for
the
imm
erse
dbou
ndar
ymet
hod.
Lem
mon
,Yog
anat
hana
ndco
wor
kers
have
used
the
prin
cipl
esof
the
imm
erse
dbo
unda
rym
etho
dto
cons
truc
tthei
row
n3D
mod
elof
the
left
side
ofth
ehe
art.
In
1.1
Hea
rtsi
mul
atio
ns3
Ves
iera
ndY
ogan
atha
n(199
2)th
em
etho
dis
valid
ated
byco
mpa
rison
betw
eent
heco
mpu
tatio
nalre
sults
for
am
odel
ofth
eflo
wth
roug
ha
flexi
ble
tube
and
anap
-pr
oxim
atea
naly
ticso
lutio
n.In
Yog
anat
hane
tal.
(199
4)th
e3D
mod
elof
the
flow
inth
ele
ftve
ntric
ledu
ring
part
ofsy
stol
eis
pres
ente
d,and
som
ecom
paris
onsw
ithex
perim
enta
lresu
ltsar
ere
port
ed.T
his
mod
elis
used
inY
ogan
atha
neta
l.(1
995)
tost
udyt
hesy
stol
ican
terio
rmitr
alva
lve
mot
ion
synd
rom
e.In
allo
fthe
sepa
pers
the
grou
pus
esth
esa
men
umer
ical
met
hoda
sorig
inal
lyus
edby
Pes
kina
ndM
c-Q
ueen
.As
men
tione
dabo
ve,t
his
met
hodi
sno
tabl
eto
com
pute
atph
ysio
logi
cal
Rey
nold
snu
mbe
rs.I
nLe
mm
onan
dY
ogan
atha
n(19
98)t
hegr
oup
chan
gedt
henu
mer
ical
met
hodt
ore
duce
the
com
puta
tiona
ldem
ands
ofth
em
odel
and
tobe
able
toco
mpu
teat
phys
iolo
gica
lRey
nold
snum
bers
.Thi
sw
asfu
rthe
rexp
loite
din
Lem
mon
and
Yog
anat
han(2
000b
)whe
rea
3Dm
odel
ofth
eflo
wth
roug
hthe
left
hear
tdur
ing
dias
tole
was
pres
ente
dand
eval
uate
dby
com
paris
onsto
clin
ical
data
.T
his
dias
tolic
3Dm
odel
isus
edin
Lem
mon
and
Yog
anat
han(2
000a
)tosi
mul
ate
path
olog
ical
cond
ition
sand
ther
ebyi
nves
tigat
ethe
dias
tolic
func
tion
ofth
ehe
art.
The
imm
erse
dbou
ndar
ymet
hodi
ssp
ecia
linth
ew
ayit
mod
elst
hein
tera
ctio
nbe
twee
nthe
fluid
flow
and
the
tissu
esof
the
hear
t.A
noth
erap
proa
chis
tole
tth
em
ovem
ento
fth
ebo
unda
rybe
give
nin
adva
nce
inde
pend
ently
ofth
eflo
w.
Inth
isap
proa
chon
lyth
ere
sulti
ngflo
wof
the
fluid
has
tobe
com
pute
d.T
hem
etho
dof
pred
eter
min
edbo
unda
rym
otio
nw
asca
rrie
dou
tin
Geo
rgia
dis
etal
.(1
992)
fora
nid
ealiz
edve
ntric
ular
geom
etry
inth
efo
rmof
aco
ntra
ctin
gelli
psoi
d.In
Tayl
oret
al.(
1993
),Ta
ylor
and
Yam
aguc
hi(1
995b
),Ta
ylor
and
Yam
aguc
hi(1
995a
)and
Tayl
oret
al.(
1996
)it
was
done
for
a3D
geom
etry
cons
truc
tedu
sing
aca
stof
ado
g’s
hear
ttake
nin
the
dias
tolic
phas
ewith
anas
sum
edm
ovem
ento
fth
eve
ntric
ular
wal
l.F
urth
erm
ore,
Sch
oeph
oers
teret
al.
(199
3;19
94)u
sed2
Dci
ne-a
ngio
grap
hicim
ages
tode
term
inet
hehe
artw
allb
ound
aryi
nbo
tha
heal
thy
and
adi
seas
ed(c
oron
arya
rter
ydis
ease
)left
vent
ricle
and
subs
eque
ntlyc
ompu
ted
the
flow
durin
gsy
stol
e.T
hey
expa
nded
thei
rm
etho
dto
asi
mpl
esp
heric
al3D
vent
ricle
inG
onza
leza
ndS
choe
phoe
rste
r(199
6),w
here
they
also
inve
stig
ated
the
effe
ctof
abno
rmal
wal
lmot
ion
onth
eflo
wpa
ttern
.In
Din
gan
dS
choe
phoe
rste
r(1
997)
they
used
cine
-ang
iogr
amsto
cons
truc
ta3D
syst
olic
mod
elof
the
left
vent
ricle
assu
min
gcirc
ular
cros
ssec
tions
.T
hem
etho
dof
pred
eter
min
edbo
unda
rym
otio
nha
sals
obe
enus
edin
com
-bi
natio
nw
ithM
Rsc
anni
ngda
ta.J
ones
and
Met
axas
(199
8)us
edM
RI
SPA
MM
togi
veth
ehe
artb
ound
aryc
onfig
urat
ionf
ora
flow
mod
elof
the
left
vent
ricle
.M
RI
SPA
MM
allo
ws
trac
king
ofm
ater
ialp
oint
son
the
hear
tand
give
san
accu
-ra
tere
pres
enta
tiono
fthe
left
vent
ricul
arm
otio
nfr
omen
d-di
asto
leto
end-
dias
tole
(Par
ket
al.,
1996
).S
aber
etal
.(20
01)u
seda
nato
mic
aldat
aobt
aine
dby
MR
scan
-ni
ngto
cons
truc
tam
odel
ofth
ele
ftve
ntric
leus
ing
mov
ing
mes
hes.T
heir
mod
el
4In
trod
uctio
n
sim
ulat
esth
ebl
ood
flow
durin
ga
full
card
iacc
ycle
exce
ptfo
rth
ela
tedi
asto
le.
1.2
MR
scan
ning
The
curr
ents
tate
ofth
em
agne
ticre
sona
nce(
MR
)te
chni
quea
llow
sth
ede
ter-
min
atio
nof
3Dve
loci
tyfie
lds
for
the
flow
inth
ehu
man
hear
t(e.
g.H
oulin
det
al.,
1994
;Kim
etal
.,19
95;W
alke
reta
l.,19
96).
InH
oulin
det
al.(
1994
)and
Kim
etal
.(1
995)
the
thre
eco
mpo
nent
sof
the
bloo
dve
loci
tyfie
ldin
asi
ngle
slic
e(t
helo
ng-a
xis
plan
e)w
asob
tain
edin
patie
ntsw
ithis
chem
iche
artd
isea
sean
din
norm
alsu
bjec
ts,re
spec
tivel
y.F
rom
thes
evel
ocity
data
they
extr
actv
eloc
-ity
time
curv
esat
cert
ain
posi
tions
,e.g
.ove
rthe
mitr
alrin
g.F
urth
erm
ore,
Kim
etal
.(19
95)q
uant
ified
the
left
vent
ricle
vort
exse
endu
ring
dias
tole
.Wal
kere
tal.
(199
6)m
easu
redth
eth
ree
dim
ensi
onal
bloo
dve
loci
tyfie
ldin
mul
tiple
slic
esin
the
hum
anhe
art.
Inal
loft
hese
thre
epap
ersq
ualit
ative
desc
riptio
nsof
the
bloo
dflo
war
eal
sogi
ven.
Rec
ently
,MR
scan
ning
hasb
eenu
sedi
nth
eas
sess
men
tofth
edi
asto
licfu
nc-
tioni
ngof
the
left
vent
ricle
.Hou
lind
etal
.(19
99)i
nves
tigat
edth
eag
e-de
pend
ent
chan
gesin
the
bloo
dve
loci
tydi
strib
utio
ndu
ring
early
fillin
g.In
trav
entr
icul
arflo
wha
sbe
enco
rrel
ated
with
dise
asea
ndis
ofim
port
ance
toca
rdio
logi
stas
apo
ssib
lem
eans
ofdi
agno
sis(
Wal
kere
tal.,
1996
).In
Hou
lind
(200
1b)t
hech
ange
sinea
rlydi
asto
licbl
ood
flow
patte
rnsi
ndi
s-ea
sedh
eart
s(e.
g.fr
omac
utem
yoca
rdia
linfa
rctio
n)ar
einv
estig
ated
ands
oar
ethe
chan
gesin
velo
city
profi
leso
btai
neda
long
tran
sver
selin
esin
the
left
vent
ricle
attim
esju
stpr
iort
oan
dafte
rthe
occu
rren
ceof
the
peak
velo
city
atth
em
itral
leafl
ettip
s.E
xam
inin
gth
epe
akve
loci
ty(t
hem
agni
tude
and
the
time
ofoc
curr
ence
)at
such
lines
give
sins
ight
into
the
dias
tolic
func
tioni
ngof
the
hear
t.E
spec
ially
the
rela
xatio
nof
the
left
vent
ricle
(whi
chis
redu
cedd
urin
gce
rtai
nhe
artc
ondi
tions
)in
fluen
cest
hedi
asto
licfu
nctio
ning
ofth
ehe
art,
and
this
isre
flect
edin
the
flow
patte
rndu
ring
fillin
g.H
owev
er,a
tpre
sent
note
very
mec
hani
smre
spon
sibl
efor
the
obse
rved
chan
gesin
the
bloo
dflo
wpa
ttern
iskn
own.
1.3
Usi
ngsi
mul
atio
nsan
dM
Rda
tato
geth
er
Com
puta
tiona
lmod
elso
fth
ehe
arta
reve
ryus
eful
assi
mul
atio
nto
ols,
whe
reth
eus
ere.
g.ca
nte
stth
ena
tura
land
path
olog
ical
func
tioni
ngof
the
hear
t.In
aco
mpu
tatio
nalm
odel
the
effe
ctso
fdiff
eren
tpar
amet
ersc
anbe
inve
stig
ated
inde
-pe
nden
tlyan
dco
mbi
ned.
But
for
thes
emod
elto
betr
ustw
orth
yth
ere
sults
have
tobe
valid
ated
.The
type
ofM
Rm
easu
rem
entsp
rese
nted
ine.
g.K
imet
al.(
1995
)
1.4
Out
line
ofth
esis
5
are
idea
lfor
valid
atio
nof
mod
elso
fth
ebl
ood
flow
inth
ehe
art,
sinc
eth
eyar
em
easu
rem
entso
fthe
bloo
dve
loci
tyfie
ldw
hich
isth
ety
pica
lout
putf
rom
aco
m-
puta
tiona
lmod
el.N
atur
ally,
deriv
edqu
antit
ies,
such
asve
loci
tytim
ecu
rves
,can
also
beco
mpa
redb
etw
eenm
odel
and
MR
mea
sure
men
ts.C
ompu
tatio
nalm
od-
els
ofth
ebl
ood
flow
inth
ehe
arth
ave
not,
toou
rkn
owle
dge,
prev
ious
lybe
enva
lidat
edin
deta
ilby
com
paris
onsto
MR
velo
city
field
data
.T
hepo
ssib
ility
ofco
mbi
ning
MR
mea
sure
men
tsand
com
puta
tiona
lhear
tmod
-el
sto
obta
ina
com
pute
rtool
fors
tudy
ingt
hefu
nctio
ning
ofth
ehu
man
hear
tisth
em
otiv
atio
nfo
rou
rwor
k.W
est
artw
ithP
eski
nand
McQ
ueen
’s2D
hear
tmod
elof
ado
g’s
hear
tand
thro
ugh
the
use
ofM
Rda
taan
dge
nera
lphy
siol
ogic
alan
dan
atom
ical
know
ledg
eadj
ustit
tore
pres
enta
hum
anhe
art.
The
n,th
em
odel
and
itssi
mul
atio
nres
ults
are
eval
uate
dand
com
pare
dwith
MR
data
.Thi
s2D
hum
anhe
artm
odel
can
then
beus
edto
stud
ycer
tain
mec
hani
smsa
ffect
ing
the
dias
tolic
func
tioni
ngof
the
hear
tund
erpa
thol
ogic
alco
nditi
ons.
1.4
Out
line
ofth
esis
The
thes
isis
orga
nize
dasf
ollo
ws:
Cha
pter
1T
his
intr
oduc
tion.
Cha
pter
2D
escr
iptio
noft
heph
ysio
logy
and
anat
omyo
fthe
hear
t.T
hech
apte
rin
trod
uces
the
read
erto
the
nece
ssar
yphy
siol
ogic
alba
ckgr
ound
know
ledg
ean
dte
rmsu
sedi
nth
esu
bseq
uentc
hapt
ers.
Cha
pter
3O
ur2D
mod
elof
the
left
side
ofth
ehu
man
hear
tisde
scrib
edin
both
the
cont
inuo
usan
dth
edi
scre
teca
se,a
ndth
eco
mpu
tatio
nalp
roce
dure
isgi
ven.
The
impr
ovem
entsa
ndch
ange
snee
dedto
tran
sfor
mP
eski
nand
Mc-
Que
en’s
orig
inal
2Dm
odel
ofa
dog’
she
artt
oth
atof
ahu
man
hear
tare
argu
edfo
ran
dde
scrib
ed.
Cha
pter
4E
valu
atio
nof
the
2Dm
odel
from
Cha
pter
3.C
erta
inco
mpu
tatio
nal
andm
odel
linga
spec
tsoft
hem
odel
aree
valu
ated
andt
hepr
oper
adju
stm
ent
ofm
odel
para
met
ersis
foun
d.
Cha
pter
5C
ompa
rison
ofth
e2D
mod
elfr
omC
hapt
er4
with
anM
Rda
tase
tof
the
velo
city
field
inth
ehu
man
hear
t.T
hein
trav
entr
icul
arve
loci
tyfie
ldan
dve
loci
tytim
ecu
rves
atth
em
itral
ring
and
atth
eao
rtic
outfl
owtr
acta
reco
mpa
redw
ithco
rres
pond
ingM
Rda
ta.
6In
trod
uctio
n
Cha
pter
6T
he2D
mod
elfr
omC
hapt
er4
isus
edto
sim
ulat
ecer
tain
mec
hani
sms
ofea
rlyfil
ling
ofth
ele
ftve
ntric
leun
derp
atho
logi
calc
ondi
tions
.Thi
sre
-qu
iresi
mpr
ovin
gth
em
odel
for
the
rela
xatio
nand
cont
ract
iono
fth
ehe
art
wal
l.T
heim
prov
edm
odel
isth
ense
tup
tosi
mul
atea
heal
thyh
uman
,and
path
olog
ical
cond
ition
sare
sim
ulat
edby
alte
ring
the
rela
xatio
nan
dco
n-tr
actio
nof
the
vent
ricle
.
Cha
pter
7D
escr
iptio
nof
Pes
kina
ndM
cQue
en’s
3Dm
odel
ofth
een
tire
hear
t.T
here
sults
from
asi
mul
atio
nwith
the
mod
elar
ebrie
flyco
mpa
redw
ithM
Rda
ta.
Cha
pter
8T
heco
nclu
sion
sof
the
thes
isar
esu
mm
ariz
edan
dre
com
men
datio
nsfo
rfu
ture
wor
kar
epr
esen
ted.
App
endi
xA
Intr
oduc
tion
toth
em
agne
ticre
sona
nceim
agin
gte
chni
que.
Thi
sap
pend
ixpr
ovid
esth
ere
ader
with
back
grou
ndin
form
atio
non
how
the
MR
data
used
inth
eth
esis
isob
tain
ed.
Cha
pter
2
Phy
siol
ogy
and
anat
omy
ofth
ehe
art
Thi
sch
apte
rpro
vide
san
intr
oduc
tiont
oth
eph
ysio
logy
and
anat
omyo
fthe
hear
tes
sent
ialfo
rund
erst
andi
ngth
esu
bseq
uentc
hapt
erso
fthi
sth
esis
.Itis
part
lyba
sed
onO
lufs
enet
al.(
2001
),an
dw
eus
eGan
ong(
1993
),To
rtor
a(1
999)
and
Guy
ton
(199
1)as
refe
renc
es.
The
hear
tisth
epu
mp
that
driv
esth
ebl
ood
flow
inth
eca
rdio
vasc
ular
syst
emby
perio
dic
cont
ract
iona
ndre
laxa
tiono
fth
ehe
artm
uscl
es.T
heca
rdio
vasc
ular
syst
emco
nsis
tsof
two
part
s,th
esy
stem
ican
dpu
lmon
aryc
ircui
ts,c
onne
cted
via
the
left
and
right
hear
tcha
mbe
rs.F
rom
the
left
side
ofth
ehe
artb
lood
ispu
mpe
din
toth
esy
stem
icci
rcui
tth
roug
hth
eao
rta.
The
syst
emic
circ
ulat
ion
tran
spor
tsan
ddi
strib
utes
esse
ntia
lsubs
tanc
esto
the
tissu
esan
dit
rem
oves
bypr
oduc
tsof
met
abol
ism
.Inth
em
uscl
esan
dtis
sues
oxyg
enis
part
ially
exch
ange
dwith
carb
ondi
oxid
e.A
sa
resu
ltsth
ebl
ood
ispa
rtly
deox
ygen
ated
whe
nit
retu
rnst
oth
erig
htsi
deof
the
hear
tthro
ught
wo
big
vein
s(th
esu
perio
rand
infe
riorv
enac
ava)
.The
right
side
ofth
ehe
artp
umps
the
bloo
din
toth
epu
lmon
aryc
ircul
atio
n.H
ere
the
part
lyde
oxyg
enat
edbl
ood
isca
rrie
dto
the
lung
tissu
esw
here
carb
ondi
oxid
eis
exch
ange
dwith
oxyg
enan
dth
ebl
ood
then
retu
rnst
oth
ele
ftsi
deof
the
hear
tth
roug
hfou
rpul
mon
aryv
eins
(tw
ofr
omea
chlu
ng).
8P
hysi
olog
yand
anat
omy
ofth
ehe
art
2.1
Inte
rnal
anat
omy
The
hear
tcon
sist
sof
four
cham
bers
assh
own
inF
igur
e2.
1.T
hele
ftve
ntric
lean
dle
ftat
rium
cons
titut
ethe
left
side
ofth
ehe
arta
ndth
erig
htve
ntric
lean
drig
htat
rium
cons
titut
ethe
right
side
ofth
ehe
art.
The
left
and
right
side
sof
the
hear
tare
sepa
rate
dby
ase
ptum
.T
heat
riaar
edi
vide
dby
the
inte
ratr
ials
eptu
man
dth
eve
ntric
lesa
rese
para
tedb
yth
ein
-te
rven
tric
ular
sept
um.T
heat
riove
ntric
ular
open
ing
conn
ects
each
atriu
mto
itsco
rres
pond
ingv
entr
icle
.Thi
sop
enin
gis
surr
ound
edby
arin
gof
fibro
sust
issu
e,th
ean
nulu
sfibr
osus
.The
low
erpa
rtof
the
hear
tisca
lledt
heap
exw
hile
the
uppe
rbr
oade
ndof
the
hear
tisca
lled
the
base
,see
Fig
ure
2.2.
The
hear
twal
lsco
nsis
tofm
uscl
etis
sued
enot
edth
em
yoca
rdiu
m.T
heth
ick-
ness
ofth
em
yoca
rdiu
mva
rieso
vert
hehe
arta
ndin
crea
sesw
ithw
orkl
oad.
The
atria
dole
ssw
ork
than
the
vent
ricle
s.C
onse
quen
tlyth
eat
rialw
alls
are
thin
ner,
appr
oxim
atel
y2m
m.
The
right
vent
ricle
isex
pose
dto
the
pres
sure
ofth
epu
l-
Fig
ure
2.1:
The
four
-cha
mbe
redh
eart
isdi
vide
din
totw
opa
rts,
the
left
and
right
side
s.B
oth
part
scon
sist
ofa
vent
ricle
and
anat
rium
.The
left
side
ofth
ehe
artis
anat
omic
ally
larg
erth
anth
erig
htsi
de.F
rom
Rid
eout
(199
1).
2.1
Inte
rnal
anat
omy
9
Fig
ure
2.2:
Dra
win
gof
ahe
arts
plit
perp
endi
cula
rtoth
ein
terv
entr
icul
arse
ptum
.A
dapt
edfr
omN
ette
r(19
91).
mon
ary
circ
ulat
ion
and
has
aw
all
thic
knes
sof
abou
t5m
m.
The
grea
test
wal
lth
ickn
ess,
abou
t15
mm
,bel
ongs
toth
ele
ftve
ntric
lew
hich
ejec
tsbl
ood
into
the
entir
esy
stem
icci
rcul
atio
n.T
hebl
ood
flow
inth
ehe
arti
sun
idire
ctio
nald
ueto
two
sets
ofva
lves
,the
atrio
vent
ricul
arva
lves
and
the
sem
iluna
rval
ves.
The
atrio
vent
ricul
arva
lves
are
loca
tedb
etw
eene
acha
triu
man
dve
ntric
le.T
hese
milu
narv
alve
sare
foun
din
the
open
ingb
etw
eene
achv
entr
icle
and
itsco
rres
pond
ingo
utflo
war
tery
.V
alve
sare
notf
ound
atth
eve
nous
inle
tsin
toth
eat
ria,s
eeF
igur
e2.
2.T
heat
riove
ntric
ular
valv
esco
nsis
tofl
eafle
ts(c
usps
)with
atr
iang
ular
shap
e.T
herig
hthe
artc
ham
berc
onta
inst
hree
leafl
etsa
ndco
nseq
uent
lythe
valv
eis
de-
note
dthe
tric
uspi
dva
lve.
The
valv
ein
the
left
cham
berc
onta
inst
wo
leafl
etsa
ndis
thus
refe
rred
toas
the
bicu
spid
valv
e(o
rm
ore
com
mon
lyth
em
itral
valv
e).
Leafl
ette
ndon
s,al
soca
lled
the
chor
daet
endi
neae
,are
foun
dun
dere
achv
alve
.T
hey
orig
inat
efro
mth
epa
pilla
rym
uscl
esth
atar
elo
cate
don
the
inne
rsur
face
ofea
chve
ntric
le.W
hen
the
atriu
mco
ntra
ctst
hele
aflet
shan
gsla
ckin
the
vent
ricle
,bu
twhe
nth
eve
ntric
leco
ntra
ctst
hele
aflet
sare
push
edto
geth
ercl
osin
gth
eat
ri-ov
entr
icul
arop
enin
g.T
hech
orda
eten
dine
aese
cure
the
leafl
ets,
prev
entin
gth
emfr
omm
ovin
gin
toth
eat
rium
durin
gth
eco
ntra
ctio
noft
heve
ntric
le.
The
sem
iluna
rval
ves
prev
entb
lood
from
flow
ing
back
into
the
hear
tafte
rej
ectio
nint
oth
ear
terie
s.E
ach
ofth
ese
milu
narv
alve
scon
sist
soft
hree
cres
cent
shap
edle
aflet
s.T
hese
leafl
etsh
ave
noch
orda
eten
dine
ae.T
hey
are
dila
ted
with
10P
hysi
olog
yand
anat
omy
ofth
ehe
art
bloo
dan
dcl
oset
heop
enin
gwhe
nth
epr
essu
rein
the
arte
ryex
ceed
sthe
pres
sure
inth
eve
ntric
le.T
herig
htse
milu
narv
alve
isca
lled
the
pulm
onar
yval
ve,w
hile
the
left
sem
iluna
rval
veis
calle
dth
eao
rtic
valv
e.
2.2
Fun
ctio
ning
ofth
ehe
art
The
time
cour
seof
ahe
artp
erio
dcan
roug
hlyb
edi
vide
din
toan
activ
eph
asea
nda
rela
xed
phas
e,te
rmed
syst
olea
nddi
asto
le,re
spec
tivel
y.T
herig
htan
dlef
tsid
esof
the
hear
twor
kal
mos
tsyn
chro
nous
ly,m
eani
ngth
atth
eyco
ntra
ctan
drel
axal
mos
tat
the
sam
etim
e.W
hen
dias
tole
begi
nsth
epu
lmon
arya
ndao
rtic
valv
esar
ecl
osed
,bec
ause
the
pres
sure
inth
ear
terie
sexc
eeds
the
pres
sure
inth
eve
ntric
les.
The
tric
uspi
dan
dm
itral
valv
ear
eop
en,a
ndth
eve
nous
bloo
dflo
ws
thro
ugh
the
atria
and
fills
the
rela
xed
vent
ricle
s.T
hent
heat
riaco
ntra
ct(a
tria
lsys
tole
)thus
incr
easi
ngth
eflo
wto
the
vent
ricle
san
dca
usin
gthe
pres
sure
inth
eve
ntric
lest
oris
e.S
hort
lyaf
ter,
vent
ricul
arsy
s-to
lese
tsin
resu
lting
ina
pres
sure
rise
that
caus
esth
etr
icus
pid
and
mitr
alva
lves
tocl
ose.
Thu
sth
eve
ntric
lesc
ontr
actis
ovol
umic
ally
.T
his
ends
whe
nth
eve
n-tr
icul
arpr
essu
reeq
uals
the
arte
rialp
ress
ure,
atw
hich
poin
tthe
pulm
onar
yand
aort
icva
lves
open
and
bloo
dflo
ws
into
the
arte
rials
yste
m.T
hepu
lmon
arya
ndao
rtic
valv
esdo
noto
pen
atth
esa
met
ime
inth
etw
osi
deso
fth
ehe
art.
The
pulm
onar
yval
vein
the
right
side
open
swhe
nth
epr
essu
rein
the
right
vent
ricle
exce
eds1
0m
mH
gan
dth
eao
rtic
valv
ein
the
left
side
open
swhe
nth
epr
essu
rein
the
left
vent
ricle
exce
eds8
0m
mH
g.T
heop
enin
goft
hepu
lmon
arya
ndao
rtic
valv
esm
arks
the
onse
tofe
ject
ion.
The
pulm
onar
yand
aort
icva
lves
clos
ewhe
nve
ntric
ular
pres
sured
rops
belo
war
teria
lpre
ssur
eand
initi
ates
the
next
isov
olum
ical
rela
xatio
npha
sean
da
furt
her
fall
inve
ntric
ular
pres
sure
.Whe
nth
eve
ntric
ular
pres
sure
falls
belo
wth
eat
rial
pres
sure
,the
tric
uspi
dan
dm
itral
valv
esop
enan
dth
ebl
ood
that
hase
nter
edth
eat
rium
durin
gve
ntric
ular
syst
olefl
ows
into
the
vent
ricle
s(th
ispe
riod
isca
lledt
hera
pidl
yfil
ling
phas
e),a
nda
new
hear
tcyc
lest
arts
.T
heam
ount
ofbl
ood
ejec
tedb
yea
chve
ntric
lepe
rstr
oke
isab
out7
0m
l.T
heej
ectio
nfra
ctio
n(t
hepe
rcen
tage
ofth
eve
ntric
ular
volu
me
ejec
teda
teac
hstr
oke)
isab
out6
5%(G
anon
g,19
93).
Car
diac
outp
utis
defin
edas
the
amou
ntof
bloo
dej
ecte
dfro
mth
ele
ftve
ntric
lein
toth
eas
cend
inga
orta
.In
are
stin
gad
ultt
his
isap
prox
imat
ely5
l/min
(70
ml
0 72be
ats/
min
)(G
anon
g,199
3).
The
pum
ping
hear
trep
eatst
his
cour
sedu
ring
each
beat
with
atim
epe
riod
ofap
prox
imat
ely0
.8si
nre
stin
ghum
ans.
Fig
ure
2.3
show
sve
ntric
ular
pres
sure
and
2.3
Con
duct
ion
syst
emof
the
hear
t11
Fig
ure
2.3:
The
root
aort
icpr
essu
re,ve
ntric
ular
pres
sure
,atria
lpre
ssur
e,art
eria
lin
flow
,tot
alin
flow
toth
ehe
art,a
tria
lvol
ume,
vent
ricul
arvo
lum
ean
dve
ntric
ular
outfl
ow.
The
vert
ical
fully
draw
nan
dda
shed
lines
indi
cate
valv
ecl
osur
ean
dop
enin
gs,re
spec
tivel
y.F
rom
Noo
rder
graa
f(19
78).
flow
curv
esth
atar
ere
pres
enta
tivefo
ra
norm
alhu
man
left
vent
ricle
.
2.3
Con
duct
ion
syst
emof
the
hear
t
The
cont
ract
iono
fthe
hear
tisin
itiat
edby
elec
tric
alst
imul
atio
n.T
hese
elec
tric
alim
puls
esar
espr
eadt
oal
lpar
tsof
the
myo
card
ium
byth
eco
nduc
tions
yste
m.T
heco
nduc
tions
yste
mco
nsis
tof
the
sino
atria
l(S
A)
node
,the
atrio
vent
ricul
ar(A
V)
node
,the
atrio
vent
ricul
arbu
ndle
(orb
undl
eof
His
),th
ebu
ndle
bran
ches
,and
the
cond
uctin
gfibe
rsca
lledP
urki
njefi
bers
,see
Fig
ure2
.4.A
llpa
rtso
fthe
cond
uctio
nsy
stem
are
able
tose
ndou
tper
iodi
cim
puls
esw
ithou
tneu
rals
timul
atio
n.T
his
isca
lled
auto
mat
ism
.T
heS
Ano
deis
asm
allm
asso
fmus
cula
rfibe
rslo
cate
din
the
myo
card
ium
ofth
erig
htat
rium
near
the
inflo
wof
the
supe
riorv
enac
ava,
seeF
igur
e2.
4.T
heS
A
12P
hysi
olog
yand
anat
omy
ofth
ehe
art
Fig
ure
2.4:
Sch
emat
icrep
rese
ntat
iono
fthe
cond
uctio
nsys
tem
ofth
ehe
art.F
rom
Tort
ora
(199
9).
node
gene
rate
simpu
lses
fast
erth
anth
eot
herp
arts
ofth
eco
nduc
tions
yste
man
dth
usin
itiat
esea
chhe
artb
eatan
dco
ntro
lsth
efr
eque
ncy
ofth
ehe
artr
ate.
The
impu
lses
from
the
SA
node
reac
hthe
atria
first
,cau
sing
them
toco
ntra
ct.
The
velo
city
ofco
nduc
tioni
nth
eat
rialm
uscl
eis
betw
een0
.3m
/san
d1.
0m
/s(G
uyto
n,19
91).
Late
rthe
impu
lses
reac
hthe
AV
node
.The
AV
node
islo
cate
din
the
inte
ratr
ials
eptu
m,s
eeF
igur
e2.
4.T
heA
Vno
deco
nduc
tssl
owly
,i.e
.the
impu
lses
are
dela
yed(
appr
oxim
atel
y0.1
3s,
Guy
ton,
1991
)in
the
AV
node
.Thi
sal
low
sfo
rth
eat
riato
empt
ythe
irbl
ood
(by
cont
ract
ion)
into
the
vent
ricle
sbef
ore
the
vent
ricle
ssta
rtto
cont
ract
.Fro
mth
eA
Vno
deth
eim
puls
esar
ese
ntth
roug
hth
ebu
ndle
ofH
is.
The
bund
leof
His
isth
eon
lyel
ectr
ical
conn
ectio
nbet
wee
nth
eat
riaan
dve
ntric
les.
The
bund
leof
His
bran
ches
into
two,
one
bran
chgo
ing
toth
erig
htve
ntric
lean
dth
eot
herg
oing
toth
ele
ftve
ntric
le.B
oth
bran
ches
goin
the
dire
ctio
nof
the
apex
ofth
ehe
arta
ndsu
bdivi
dein
toa
com
plex
netw
ork
ofP
urki
nje
fiber
s,se
eFig
ure
2.4.
The
latte
rstim
ulat
esve
ntric
ular
cont
ract
ion.
The
cont
ract
ions
tart
satt
heap
exan
dth
ensp
read
ssup
erio
rlyto
war
dth
eba
seof
the
hear
t.T
heco
nduc
tionv
eloc
ityin
the
Pur
kinj
efib
ers
isve
ryfa
st(a
ppro
xim
atel
y1.
5–4.
0m/s
)and
the
elec
tric
alim
puls
esre
acht
heen
tire
inne
rsur
face
ofth
eve
n-tr
icle
sal
mos
tsim
ulta
neou
sly.
The
nth
eim
puls
esm
ove
thro
ugh
the
vent
ricul
arm
uscl
esto
war
dth
eou
ters
urfa
ceof
the
hear
twith
ave
loci
tyof
appr
oxim
atel
y0.
3-0.
5m/s
.
2.4
Mus
cle
phys
iolo
gy13
2.4
Mus
cle
phys
iolo
gy
The
hear
tpum
psby
the
cont
ract
iono
fits
mus
clefi
bers
.The
hear
thas
aco
mpl
exge
omet
ricst
ruct
urew
ithm
uscl
esly
ing
ina
com
plic
ated
patte
rn.T
heca
rdia
cmus
-cl
eits
elfc
onsi
stso
fin
divi
dual
mus
clefi
bers
,or
myo
cyte
s,w
hich
are
the
smal
l-es
tfun
ctio
nalu
nits
inth
em
uscl
estr
uctu
re.E
ach
mus
clefi
beri
sap
prox
imat
ely
# * m
long
with
adi
amet
erof
# * m
.T
hem
uscl
efibe
rsbr
anch
and
inte
rdig
itate
,but
each
fiber
isa
com
plet
euni
tsur
roun
dedb
ya
cell
mem
bran
e.M
uscl
efib
ers
cont
ain
anu
mbe
rof
fibril
spl
aced
inpa
ralle
l,eac
hof
whi
chis
subd
ivide
dby
the
Z-li
nes
into
bloc
ksca
lleds
arco
mer
es.The
sarc
omer
essho
wal
-te
rnat
ingl
ight
and
dark
band
s,du
eto
para
llelb
undl
esof
thic
kan
dth
infil
amen
ts.
The
top
pane
lof
Fig
ure
2.5
show
sa
piec
eof
afib
rilw
ithits
char
acte
ristic
stria
ted
patte
rn,a
nda
sarc
omer
egoi
ngfr
omon
eZ
line
toan
othe
r.A
tth
eZ
Thi
n fil
amen
t
Thi
ck fi
lam
ent
Z li
neZ
line
Sar
com
ere
Z li
neZ
line
Z li
neZ
line
Rel
axed
Con
trac
ted
Fib
ril −
rel
axed
Fig
ure
2.5:
Top
pane
lsho
ws
ase
ctio
nof
are
laxe
dfib
rilw
ithits
char
acte
ristic
strip
edpa
ttern
.T
hest
riped
patte
rnfo
llow
sfr
omth
ebu
ndle
sof
inte
rdig
itatin
gth
ick
and
thin
filam
ents
how
nin
the
mid
dle
pane
l.T
helo
wer
pane
lsho
ws
the
cont
ract
edst
atew
here
the
Zlin
esm
oves
clos
erto
geth
er.
14P
hysi
olog
yand
anat
omy
ofth
ehe
art
lines
the
sarc
otub
ular
syst
emis
loca
ted.
Itco
nsis
tsof
the
sarc
opla
smicr
etic
ulum
(whi
chco
ntai
nsa
Ca. 1 re
servo
ir)an
dth
eT
syst
em.B
oth
ofth
esea
rees
sent
ial
for
cont
ract
iono
fthe
mus
clefi
bers
,see
belo
w.
Inth
em
iddl
epa
nelin
Fig
ure
2.5
the
inte
rdig
itatin
gth
ick
and
thin
filam
ents
are
show
nw
hen
the
fiber
isin
are
laxe
dst
ate.
Whe
na
mus
clefi
berc
ontr
acts
the
thin
filam
ents
slid
eov
erth
eth
ick
filam
ents
.The
leng
thof
the
thic
kfil
amen
tsis
cons
tant
,whe
reas
the
Zlin
esm
ove
clos
erto
geth
erw
hen
the
mus
clec
ontr
acts
and
fart
hera
part
whe
nit
isst
retc
hed.
As
the
mus
cles
hort
ens,t
heth
infil
amen
tsfr
omth
eop
posi
teen
dsof
the
sarc
omer
eapp
roac
heac
hoth
eran
dev
entu
ally
over
lap.
The
botto
mpa
nelo
fF
igur
e2.
5sh
ows
the
slid
edth
ick
and
thin
filam
ents
ina
cont
ract
edst
ate,
whe
reth
eth
infil
amen
tsar
eno
tyet
over
lapp
ing.
Con
trac
tiono
fth
em
uscl
efib
ers
isin
itiat
edby
ach
ange
inth
epo
tent
ialo
fth
ece
llm
embr
anes
urro
undi
ngth
efib
er.
Car
diac
cells
resp
ondt
oan
elec
tric
alst
imul
usby
prod
ucin
gcha
nges
inth
eio
nco
nduc
tanc
esand
inth
epo
tent
ialo
fthe
irm
embr
ane.
Ifth
est
imul
usis
suffi
cien
tlyst
rong
anac
tion
pote
ntia
lisge
nera
ted,
whi
chin
aca
rdia
cmus
clei
sre
spon
sibl
efor
initi
atin
gco
ntra
ctio
n(D
espo
poul
osan
dS
ilber
nagl
,199
1).
Fig
ure
2.6
show
sa
typi
cala
ctio
npo
tent
ialo
fa
card
iac
mus
cle.
Car
diac
cells
have
are
stin
gpo
tent
ialo
fab
out-
90m
V.A
suffi
cien
tlyst
rong
010
020
030
040
0
0100
−900
time
[ms]
over
shoo
t
depolarization
repolarization
plat
eau
mec
hani
cal
resp
onse
contraction [%]
membrane potential [mV]
Fig
ure
2.6:
Act
ion
pote
ntia
l(so
lidcu
rve)
and
cont
ract
ilem
echa
nica
lresp
onse
(dot
tedc
urve
)ofa
card
iacm
uscl
efibe
rplo
tted
onsa
met
ime
axis
.
2.4
Mus
cle
phys
iolo
gy15
stim
ulus
prod
uces
anac
tion
pote
ntia
lby
first
redu
cing
the
rest
ingm
embr
anep
o-te
ntia
lto
ale
ssne
gativ
eva
lue
(dep
olar
izat
ion)
.Dep
olar
izat
ionp
roce
edsr
apid
lyan
dex
hibi
tsan
over
shoo
t,see
Fig
ure
2.6.
Thi
sis
follo
wed
bya
plat
eaub
efor
ethe
mem
bran
epot
entia
lret
urns
toth
eba
selin
e.T
hede
pola
rizat
ionl
asta
bout
2m
s,bu
tthe
plat
eaup
hase
and
repo
lariz
atio
nlas
t200
ms
orm
ore,
seeF
igur
e2.
6.T
hepr
oces
sby
whi
chde
pola
rizat
iono
fthe
mus
clefi
beri
nitia
tesc
ontr
actio
nis
calle
dex
cita
tion-
cont
ract
ionc
oupl
ing.
The
actio
npo
tent
ialis
tran
smitt
edto
all
fibril
sin
the
fiber
via
the
Tsy
stem
and
ittr
igge
rsth
ere
leas
eof
Ca. 1 fr
omth
esa
rcop
lasm
icret
icul
um.T
heC
a. 1 initi
ates
cont
ract
ion.
The
cont
ract
ilere
spon
seof
card
iacm
uscl
ebeg
ins
just
afte
rthe
star
tof
depo
lariz
atio
nand
last
sabo
ut1.
5tim
esas
long
asth
eac
tion
pote
ntia
l(Gan
ong,
1993
).F
igur
e2.
6sh
ows
the
actio
npo
tent
iala
ndth
eco
rres
pond
ingm
echa
nica
lresp
onse
plot
ted
onth
esa
met
ime
axis
.
Cha
pter
3
2Dhe
artm
odel
usin
gth
eim
mer
sed
boun
dary
met
hod
Thi
sch
apte
rpre
sent
sour
2di
men
sion
al(2
D)
mat
hem
atic
almod
elof
the
bloo
dflo
win
the
hear
t.T
heor
igin
al2D
mod
el(P
eski
n,19
77)w
asdi
men
sion
edto
data
onth
edo
ghe
art.
InJa
cobs
enet
al.(
2001
)and
Ade
lera
ndJa
cobs
en(2
001)
the
mod
elw
asad
apte
dto
the
hum
anhe
artt
hrou
ghth
eus
eof
MR
data
,apr
oced
ure
that
requ
ired
fittin
gpa
ram
eter
sfor
geom
etry,
timin
g,m
uscu
larc
ontr
actio
nand
afte
rload
.Her
ew
efo
llow
the
pres
enta
tionin
Ade
lera
ndJa
cobs
en(2
001)
and,
inad
ditio
n,in
trod
ucem
odifi
catio
nsre
gard
ingt
ethe
ringa
ndth
em
odel
ofth
epa
pil-
lary
mus
clea
ndch
ords
.
3.1
Gen
eral
desc
riptio
nof
the
mod
el
The
mod
elre
pres
entsa
2Dse
ctio
noft
hele
ftsi
deof
the
hear
t.Thi
sse
ctio
nlie
sin
apl
ane,
the
long
axis
plan
e,w
hich
pass
esth
roug
hthe
apex
ofth
ehe
art,b
isec
tsth
ele
aflet
soft
hem
itral
valv
ean
dcu
tsth
eao
rtic
outfl
owtr
act.
The
mod
elde
scrib
esth
ebl
ood
flow
inan
dth
em
ovem
ento
fthe
left
atriu
man
dth
ele
ftve
ntric
ledu
ring
aca
rdia
ccyc
le,i
nclu
ding
the
mov
emen
tof
the
mitr
alva
lve
conn
ectin
gthe
two
cham
bers
.T
heve
nous
inflo
wto
the
left
hear
tism
odel
ledb
ya
sour
cepl
aced
atth
ece
n-tr
eof
the
atriu
m,a
ndth
eao
rtic
outfl
owis
allo
wed
thro
ugh
asi
nkpl
aced
atth
eou
tflow
trac
tof
the
vent
ricle
.The
mod
elre
pres
entst
hetis
sues
ofth
ehe
art,b
oth
activ
em
uscl
etis
sues
and
pass
iveva
lve
tissu
es,a
sinfi
nite
lyth
in,m
assl
esse
last
ic
182D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
boun
darie
stota
llyim
mer
sedin
bloo
d,i.e
.with
bloo
don
both
side
s.T
hese
boun
d-ar
iest
hen
man
ifest
them
selve
sto
the
fluid
byex
ertin
gfo
rces
onit
and
are
them
-se
lvesm
oved
byth
eflu
idw
ithth
elo
calfl
uid
velo
city
.T
hebe
atin
gof
the
mod
elhe
artis
indu
cedb
ysu
cces
sive
rela
xatio
nand
cont
ract
iono
fthe
elas
ticbo
unda
ries
repr
esen
tingth
ehe
artm
uscl
es.
The
mod
elus
esa
time-
step
ping
proc
edur
eof
the
follo
win
gty
pe:
Fro
ma
know
nhe
artb
ound
aryc
onfig
urat
ion,
the
boun
dary
forc
esar
eup
date
d.O
nth
eba
siso
fthe
sefo
rces
andt
hefo
rces
inth
eflu
id,t
heve
loci
tyan
dpre
ssur
efiel
dsar
eup
date
d.F
inal
ly,t
heve
loci
tyfie
ldis
used
toup
date
the
hear
tbou
ndar
ycon
figu-
ratio
n.
3.2
Con
tinuo
usfo
rmul
atio
n
Inth
ism
odel
the
hear
tisim
mer
sedi
na
box
fille
dw
itha
fluid
ofth
esa
med
ensi
tyan
dvi
scos
ityas
bloo
d.T
hehe
artb
ound
ary(
mus
cula
rwal
land
valv
ele
aflet
s)ar
etr
eate
dasm
assl
esso
ne-d
imen
sion
albou
ndar
iesto
tally
imm
erse
din
fluid
(blo
od)
and
capa
bleo
fex
ertin
gfo
rces
onth
eflu
id.
The
fluid
inth
eca
vitie
sco
nfine
dby
the
hear
tbou
ndar
iesr
epre
sent
sblo
odin
the
hear
tcha
mbe
rs,w
hile
that
outs
ide
repr
esen
tssur
roun
ding
tissu
eand
mas
sofh
eartm
uscl
e.T
hehe
artm
odel
incl
udes
aso
urce
inth
eat
rium
toac
coun
tfor
the
pulm
onar
yven
ousi
nflow
and
asi
nkat
the
aort
icou
tflow
trac
tto
allo
wfo
rej
ectio
nof
bloo
ddu
ring
syst
ole.
Mor
eove
r,si
nce
the
hear
texp
ands
durin
gdi
asto
lean
dco
ntra
ctsd
urin
gsy
stol
e,th
em
odel
also
incl
udes
anex
terio
rsin
k/so
urce
,pla
ceda
long
the
vert
ical
boun
darie
sof
the
dom
ain,
tom
ake
room
for
the
varia
tion
into
talh
eart
volu
me
durin
ga
hear
tbea
t.T
hepo
into
fde
part
uref
orth
eco
ntin
uous
form
ulat
ion
isth
eN
avie
r-S
toke
seq
uatio
nfor
inco
mpr
essi
blefl
uid
flow
and
the
cont
inui
tyeq
uatio
n,in
whi
chth
eun
know
nsar
eth
eflu
idve
loci
tyfie
ld
243 an
dth
esc
alar
pres
sure
field
5 3 ,
with
3 6
deno
tingt
hepo
sitio
nve
ctor
and
deno
tingt
hetim
eva
riabl
e:
78 9 2 9 : 2 ;< 2= < 5:*<.2:>
(3.1
)
< ;2 ? 3
(3.2
)
The
cons
tant
s
7 and
* denot
ethe
fluid
dens
ityan
dvi
scos
ity,r
espe
ctive
ly.
>43
deno
test
hefo
rce
dens
ity(f
orce
peru
nita
reai
n2D
)ex
erte
don
the
fluid
byth
ehe
artb
ound
ary,
and
? 3 re
pres
entst
hear
eaflo
wra
tede
nsity
dist
ribut
ion
ofth
eso
urce
sand
sink
s.T
hefo
rce
field
> isth
eon
lyre
pres
enta
tiono
fth
ehe
art
boun
dary
’sin
fluen
ceon
the
fluid
.T
heco
mpu
tatio
nald
omai
n is
take
nto
be
3.2
Con
tinuo
usfo
rmul
atio
n19
asq
uare
inth
em
odel
,and
the
boun
dary
cond
ition
isth
ere
quire
men
tthat
the
velo
city
and
pres
sure
field
sbe
perio
dicw
ithth
isdo
mai
nasa
perio
d.E
quat
ions
(3.1
)and
(3.2
)are
base
don
assu
mpt
ions
conc
erni
ngth
em
omen
tum
ofth
eflu
idth
aten
ters
orle
aves
the
mod
elat
the
sour
cesa
ndsi
nks.
Flu
idle
avin
gth
em
odel
atsi
nks
take
sits
mom
entu
mw
ithit
asit
leav
esth
em
odel
,whi
leflu
iden
terin
gthe
mod
elat
sour
cesh
asth
esa
mev
eloc
ity(i.
e.sa
mem
omen
tum
)ast
heflu
idat
the
poin
twhe
reit
ente
rs(J
akob
sena
ndN
iss,
2000
).T
hese
assu
mpt
ions
conc
erni
ngso
urce
sand
sink
will
bedi
scus
sedf
urth
erin
the
eval
uatio
nof
the
mod
el(s
eeS
ectio
n4.6
).T
heex
tern
alflu
idfo
rce
dens
ity
> 3 is
ade
lta-f
unct
ionl
ayer
give
nby
>43
@A B 43 C
(3.3
)
whe
reth
efu
nctio
n
, D DE ,
para
met
eriz
esth
ehe
artb
ound
ary(
acl
osed
curv
e)an
d
B de
note
sthe
boun
dary
forc
ede
nsity
with
resp
ectto
,i.e
.
B C is
the
forc
eex
erte
don
the
fluid
bya
piec
eoft
hebo
unda
ryof
leng
th
C abo
ut
.From
this
equa
tioni
tis
seen
that
the
forc
efie
ld
> 3 is
zero
ever
y-w
here
exce
ptat
the
hear
tbou
ndar
ywhe
reit
iseq
ualto
the
forc
ede
nsity
exer
ted
onth
eflu
idby
the
hear
tbou
ndar
y.F
igur
e3.
1sh
ows
the
com
puta
tiona
ldom
ain
with
the
initi
alhe
artb
ound
ary
F dr
awn.
The
choi
ceof
initi
alge
omet
ryw
illbe
disc
usse
dinS
ectio
n3.3
.7.
The
hear
tbou
ndar
yissi
mul
ated
eith
eras
pass
ivetis
suef
ollo
win
gth
eflu
idm
otio
n
0
x
Ω
y
X(s
,t )
aort
a
mitr
alva
lve
vent
ricle
atriu
m
Fig
ure
3.1:
Com
puta
tiona
ldom
ain
and
initi
alhe
artb
ound
aryc
onfig
urat
ion
.
202D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
oras
activ
em
uscl
etis
sue.
Fort
hepa
ssive
elas
ticpa
rtso
fthe
boun
dary,
the
boun
d-ar
yfo
rce
dens
ity
B depe
ndso
nly
onth
ecu
rren
tcon
figur
atio
n
,w
hile
for
the
activ
epa
rts(
sim
ulat
ingm
uscl
es)it
also
depe
ndso
nth
ehi
stor
yof
thro
ught
hetim
ede
velo
pmen
tofi
nter
nalv
aria
bles
(var
iabl
eres
tingl
engt
hs)w
hich
dete
rmin
eth
eco
ntra
ctile
prop
ertie
sof
the
boun
dary.
The
time
deve
lopm
ento
fth
ein
tern
alva
riabl
esis
dete
rmin
edfr
oma
give
nfu
nctio
nof
time,
the
activ
atio
nfu
nctio
n
G
(see
Sec
tions
3.3.
5an
d3.
3.6)
.The
spec
ificf
orm
ofth
ebo
unda
ryfo
rce
dens
ity
B w
illbe
spec
ified
inth
edi
scre
tefo
rmul
atio
nof
the
mod
elin
Sec
tion3
.3.5
.T
heflu
idis
subj
ectto
the
no-s
lipco
nditi
onat
the
hear
tbou
ndar
yand
ther
efor
eth
ebo
unda
rym
oves
with
the
loca
lflui
dve
loci
ty
9 9 2
@ H2 3 3 C 3
(3.4
)
The
mot
ion
ofth
ehe
artb
ound
aryi
sno
tkno
wn
inad
vanc
e,an
dth
usth
eab
ove
equa
tionc
onst
itute
san
equa
tiono
fmot
ion
for
the
hear
tbou
ndar
y.T
heve
loci
tyis
assu
med
tobe
cont
inuo
usac
ross
the
boun
dary
asa
part
ofth
eno
-slip
cond
ition
.T
heab
ove
equa
tions
desc
ribet
heflu
idve
loci
tyfie
ld(a
ndre
late
dqua
ntiti
es)
inan
Eul
eria
nw
ay,
whi
lequ
antit
iesr
elat
edto
the
hear
tbou
ndar
yare
desc
ribed
ina
Lagr
angi
anw
ay.
Itis
seen
that
the
funct
ion
isus
edto
coup
leE
uler
ian
quan
titie
swith
Lagr
angi
anqu
antit
ies.
Ineq
uatio
n(3.
3)th
e
funct
ion
spre
ads
the
Lagr
angi
anhe
artb
ound
aryf
orce
B toth
eE
uler
ian
forc
efie
ld
> ,w
hile
ineq
uatio
n(3
.4)
the
funct
ion
isus
edto
com
pute
the
Lagr
angi
anflu
idve
loc-
ityat
the
boun
dary
2
from
the
Eul
eria
nflu
idve
loci
tyfie
ld
2 3 .
Thi
sco
uplin
gbe
twee
nflui
dan
dbo
unda
ryis
the
hear
toft
heim
mer
sedb
ound
ary
met
hod.
Ital
low
sfo
rth
ein
tera
ctio
nbet
wee
naflu
idan
dan
imm
erse
dbou
ndar
y,w
ithou
tkno
win
gth
em
otio
nof
eith
erin
adva
nce.
Fur
ther
mor
e,by
intr
oduc
ing
this
coup
ling
itis
poss
ible
tose
tup
the
equa
tions
ofm
otio
nfo
rth
eflu
id(t
heN
avie
r-S
toke
sequ
atio
n3.1
and
the
cont
inui
tyeq
uatio
n3.2
)with
outd
istin
guis
h-in
gbe
twee
nEul
eria
nqua
ntiti
esan
dLa
gran
gian
hear
tbou
ndar
yqua
ntiti
es.
3.2.
1S
ourc
esan
dsi
nks
Ven
ousi
nflow
toth
ehe
artis
mod
elle
dby
aso
urce
inth
eat
rium
,whi
leou
tflow
durin
gej
ectio
nis
mod
elle
dby
asi
nkat
the
aort
icou
tflow
trac
t.T
hehe
artc
hang
esar
ea(d
efine
das
the
area
insi
deth
ecl
osed
hear
tbou
ndar
y)du
ring
ahe
artc
ycle
,an
dth
usit
isne
cess
aryto
incl
ude
aso
urce
/sin
kout
side
the
hear
tbou
ndar
yto
acco
mm
odat
ethe
chan
gesin
area
durin
ga
hear
tcyc
le.T
heso
urce
inth
eat
rium
and
the
sink
atth
eao
rtic
outfl
owar
ere
ferr
edto
asth
ein
terio
rsou
rces
and
sink
s,w
hile
the
sour
ce/s
inko
utsi
deth
ehe
artb
ound
aryi
sre
ferr
edto
asth
eex
terio
rso
urce
/sin
k.
3.2
Con
tinuo
usfo
rmul
atio
n21
Den
sity
dist
ribut
ion
The
sour
cesa
ndsi
nks
are
defin
edas
area
swhe
reflu
idca
nen
tero
rle
ave
the
mod
el.T
hey
arei
ncor
pora
tedin
the
mod
elth
roug
hthe
dens
itydi
strib
utio
n
?43
inth
eco
ntin
uity
equa
tion(
3.2)
.T
hede
nsity
dist
ribut
ion
isze
roex
cept
inth
ear
easa
tthe
sour
cesa
ndsi
nks.
Itis
assu
med
tobe
ofth
efo
rm
?43 IKJL M JL 3 N JL :
.PO Q R)I Q MQ 3 N Q
whe
re
S # co
rres
pond
stoth
eso
urce
inth
eat
rium
and
S co
rres
pond
stoth
esi
nkin
the
aort
icou
tflow
trac
t.T
hein
dex
T refer
sto
the
exte
riors
ourc
e/si
nk.
Fur
ther
,
I Q
and
IKJL de
note
the
area
flow
rate
thro
ugh
the
sour
ceor
sink
,po
sitiv
e
I mea
ning
inflo
wan
dne
gativ
e
I mea
ning
outfl
ow.C
onve
rsio
nbet
wee
nar
eaan
dvo
lum
eis
defin
edby
spec
ifyin
ga’t
hick
ness
’oft
heot
herw
ise2
dim
en-
sion
aldo
mai
n
.
M Q and
M JLar
ew
eigh
tfun
ctio
ns(in
dim
ensi
onpe
runi
tare
a)re
pres
entin
gthe
flow
dist
ribut
ion
acro
ssth
eso
urce
sand
sink
s(as
sum
edto
betim
ean
dflo
win
depe
nden
t),an
d
N Q and
N JL de
note
the
curr
entp
ositi
onso
fthe
cent
reof
the
sour
cesa
ndsi
nks(
whi
chm
ove
with
the
hear
t).
The
mat
hem
atic
alreq
uire
men
ton
M Q and
M JL
isth
atth
eybe
cont
inuo
usno
n-ne
gativ
efu
nctio
nssa
tisfy
ing
@ HM Q 3 C 3 # U VW
@ HM JL 3 C 3#
(3.5
)
Inth
em
odel
the
inte
riors
ourc
esan
dsi
nks
are
defin
edon
aqu
adra
ticdo
mai
nof
wid
th
X andth
ew
eigh
tfun
ctio
nsar
ech
osen
tobe
prop
erly
scal
ed‘c
osin
eha
t’fu
nctio
nsof
the
form
M Q 6
Y[Z\ Z]) ^_` # : ab'c d ^e .f
g` #: ab'c d ^e .6fg fo
r
6 hi ^ .^ .j.
othe
rwis
e
The
exte
riorw
eigh
tfun
ctio
nM JL
isde
fined
ona
rect
angu
lard
omai
nof
wid
th
Xk
and
ahe
ight
l equa
ltoth
ehe
ight
ofth
eco
mpu
tatio
naldo
mai
n.
M JL
isa
‘cos
ine
ridge
’oft
hefo
rm
M JL 6
YmZ\ Z4]) ^no` # : ab'c d ^ne .fg
for
6 hi ^n . ^n .j 0i lj
othe
rwis
e
222D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
Sin
ceth
eve
loci
tyfie
ldis
requ
iredt
obe
perio
dict
here
isno
neti
nflow
toth
edo
mai
n
thro
ugh
the
boun
darie
s,so
the
inco
mpr
essi
bilit
yof
the
fluid
impl
ies
that
the
neti
nflow
thro
ught
heso
urce
sand
sink
smus
tals
obe
zero
,i.e.
@ H</;2C3
@ H?43 C 3 IKJL :
.PO Q R)I Q
By
the
abov
eeq
uatio
nwe
have
IKJL
. O Q R)I Q
Thu
swe
can
writ
e
?43
. O Q R)I Q ? Q 3
(3.6
)
whe
re
? Q43 MQ 3 N Q M JL
43 N JL
(3.7
)
The
flow
rate
I Q
isas
sum
edto
bea
func
tion
ofth
esp
acea
vera
geof
the
pres
sure
p Q atth
eso
urce
orsi
nk.
The
pres
sure
can
only
bede
term
ined
upto
aco
nsta
nt,th
usit
only
mak
esse
nset
odi
scus
spre
ssur
ediff
eren
cesr
athe
rtha
nab
solu
tepr
essu
re.T
here
fere
ncep
ress
ureis
chos
ento
beth
esp
acea
vera
geof
the
pres
sure
atth
eex
terio
rsin
k.T
hesp
acea
vera
geof
the
pres
sure
with
resp
ectto
the
wei
ghtf
unct
ionM
Q isthen
defin
edas
follo
ws:
p Q
@ H5 3 M Q 3NQ C3
@ H543 M JL 3 N JL C 3
@ H5 3 ? Q 3 C 3
q 5 ? Q r
Fore
achs
ourc
eand
sink
asp
ecifi
crel
atio
nbet
wee
n
I Q
and
p Q isgi
ven,
mak
ing
the
flow
depe
ndup
onth
epr
essu
re
I Q 5 .
Hen
cein
trod
ucin
gth
isin
equa
tion(
3.6)
the
sour
ce/s
inkte
rm
? can
now
beex
pres
seda
sa
time
vary
ing
func
tion
s 5 of
the
time
vary
ing
pres
sure
field
5 asfo
llow
s:
? 3 s 5 3
. O Q R)I Q 5 ? Q43
(3.8
)
3.2
Con
tinuo
usfo
rmul
atio
n23
Flo
wan
dpr
essu
rere
latio
nfo
rat
rial
sour
ce
The
spec
ific
rela
tion
betw
een
I )
and
p ) for
the
sour
cein
the
atriu
mis
chos
ento
beth
atof
alin
earr
esis
tanc
emod
elpu
mpi
ngag
ains
taco
nsta
ntpr
essu
rere
servo
ir.F
igur
e3.
2sh
ows
anel
ectr
ican
alog
ueof
alin
earr
esis
tanc
emod
el.T
hego
vern
ing
equa
tiona
tthe
atria
lsou
rcei
s
I ) put!vp ) w t!v
(3.9
)
whe
re
p t!v deno
testh
elo
adpr
essu
reatt
heso
urce
and
w t!v den
otes
the
resi
stan
ceag
ains
tthe
flow
thro
ught
heso
urce
.
P 1R
src
P src
Q1
Fig
ure
3.2:
An
elec
tric
anal
ogue
toth
elin
earr
esis
tanc
emod
el.
Flo
wan
dpr
essu
rere
latio
nfo
rao
rtic
sink
Inth
eor
igin
alm
odel
deve
lope
dby
Pes
kint
heao
rtic
sink
was
mod
elle
dby
alin
ear
resi
stan
cem
odel
pum
ping
agai
nsta
cons
tant
pres
sure
rese
rvoir
com
bine
dwith
adi
ode-
like
valv
e,w
hich
open
edth
esi
nkw
hen
the
pres
sure
atth
esi
nkex
ceed
eda
give
nva
lue
for
the
pres
sure
inth
eao
rtaa
ndcl
osed
itag
ainw
hen
the
pres
sure
fell
belo
wth
atva
lue.
We
find
that
the
linea
rres
ista
ncem
odel
with
aco
nsta
ntpr
essu
rere
servo
irfo
rth
ere
latio
nbe
twee
nthe
outfl
owra
tean
dth
epr
essu
regi
vesa
rath
erun
real
is-
ticsh
apeo
fth
eas
soci
ated
pres
sure
vers
ustim
epr
ofile
,with
the
pres
sure
alm
ost
clam
peda
tthe
pres
crib
edco
nsta
ntpr
essu
rein
the
aort
a(se
eSec
tion4
.5).
Am
ore
real
istic
curv
efo
rmis
obta
ined
byco
mbi
ning
the
diod
e-lik
esi
nkw
itha
sim
-pl
eth
ree-
elem
entW
indk
esse
lmod
elof
the
arte
rialim
peda
nce.
The
Win
dkes
sel
mod
elis
also
alin
earr
esis
tanc
emod
elbu
twith
atim
eva
ryin
gpr
essu
rere
servo
ir.S
eeF
igur
e(3
.3)f
orth
eel
ectr
ican
alog
ueof
the
Win
dkes
selm
odel
.Le
t
p . deno
teth
epr
essu
reat
the
aort
icou
tflow
trac
tand
I . the
outfl
owra
teof
the
vent
ricle
.The
inflo
wto
the
Win
dkes
selm
odel
isth
en
I . ,a
ndw
eha
ve
I .
YxZZ\ ZZ4]p . putyQ
w tyQ )
if
p . zputyQ
if
p . putyQ
(3.1
0)
242D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
Rsn
k2
Rsn
k1P s
nk
P 2
C
−Q
2
Fig
ure
3.3:
Dia
gram
ofth
ree-
elem
entW
indk
esse
lmod
elco
uple
dto
diod
eva
lve.
whe
re
w tyQ )
repr
esen
tsthe
prox
imal
arte
rialr
esis
tanc
eto
the
flow
and
whe
reth
epr
essu
re
putyQ m
aybe
view
edas
the
pres
sure
ata
repr
esen
tative
prox
imal
loca
tion
inth
ear
teria
lsys
tem
.Thi
sre
latio
nbe
twee
nflow
and
pres
sure
issi
mila
rto
the
rela
tionf
orth
eat
rials
ourc
e(se
eequ
atio
n3.9
),ex
cept
that
the
load
pres
sure
(
p"tyQ )for
the
aort
icsi
nkis
time
depe
nden
t.
p tyQ is
gove
rned
byth
edi
ffer
entia
lequ
atio
n
|C p tyQ
C I .
p tyQ w tyQ .
(3.1
1)
whe
re
w tyQ .
repr
esen
tsthe
perip
hera
lresi
stan
cean
d
| repr
esen
tsthe
arte
rial
com
plia
nce.
The
com
puta
tiono
fthe
boun
dary
forc
esin
volv
esho
wac
tive
andp
assiv
etis
sue
ism
odel
led.
Thi
sis
notd
escr
ibed
onth
eco
ntin
uous
leve
lbut
onth
edi
scre
tele
vel
(see
Sec
tion3
.3.5
).
3.3
Dis
cret
efo
rmul
atio
n
The
solu
tion
ofth
eco
ntin
uous
mod
elde
scrib
edin
the
prev
ious
sect
ioni
sdo
neby
disc
retiz
inga
llqu
antit
iesa
ndeq
uatio
nsof
mot
ion
and
then
solv
ing
the
disc
rete
equa
tions
bya
time-
step
ping
proc
edur
e.
3.3.
1D
iscr
ete
quan
titie
s
We
disc
retiz
ethe
squa
redo
mai
n
usin
ga
squa
re
0 gr
idof
poin
tsw
ith
~ 6 de
notin
gth
edi
stan
cebe
twee
nnei
ghbo
urin
gpoi
nts.
Tim
eis
divi
ded
into
inte
rval
sof
leng
th
,an
dth
epa
ram
eter
inte
rval
for
the
boun
dary
isdi
vide
din
tosu
bint
erva
lsof
leng
th
.Thu
sth
ein
depe
nden
tvaria
bles
are
3.3
Dis
cret
efo
rmul
atio
n25
rest
ricte
dto
the
valu
es
3 3 ~ ~ ,
an
d S ,w
here
,
,
and
S are
inte
gers
.H
ence
atea
chtim
ele
vel
w
eha
veth
efie
ldqu
antit
ies
2y ,
5y and
>y
defin
edon
the
grid
poin
ts,i.
e., 2y
2 ~ ~
5y 5 ~ ~
>y > ~ ~
for
# .
The
boun
dary
confi
gura
tion
y attim
ele
vel
isde
fined
asa
finite
se-
quen
ceof
poin
ts,n
otne
cess
arily
coin
cidi
ngw
ithth
egr
idpo
ints
,with
asso
ciat
edbo
unda
ryfo
rce
dens
ities
.Thu
s y Q S
By Q$ B S
for
S # ,wh
ere
denot
esth
enu
mbe
rof
boun
dary
poin
tsin
the
dis-
cret
izat
iona
ndw
here
By Q deno
tesa
boun
dary
forc
era
ther
than
afo
rce
dens
ity.
3.3.
2D
iffer
ence
oper
ator
san
ddi
scre
te
K func
tion
The
diff
eren
tiale
quat
ions
inth
eco
ntin
uous
form
ulat
ion
ared
iscr
etiz
edby
repl
ac-
ing
the
diff
eren
tialo
pera
tors
bydi
ffer
ence
oper
ator
s.T
hece
nter
ed,fo
rwar
dan
dba
ckwa
rddi
ffer
ence
sinth
e
and
6 direct
ions
are
used
.The
sear
ede
fined
for
afu
nctio
n
onth
edo
mai
ngrid
asfo
llow
s:
L
1) () ~
1) ()~
1 L
1) ~
1
1) ~
( L
() ~
(
() ~
Inad
ditio
nto
thes
e,a
diff
eren
ceop
erat
or
L ap
prox
imat
ingt
heco
ntin
uous
oper
ator
< isne
eded
.Thi
sdi
ffer
ence
oper
ator
isob
tain
edth
roug
hthe
useo
fthe
follo
win
gap
prox
imat
ion
tothe
-fun
ctio
n(P
eski
nand
Prin
tz,1
993)
:
3 6
for3
6
(3.1
2)
262D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
-3h
-2h
-h
h2h
3h
0.1
0.2
0.3
0.4
0.5
Fig
ure
3.4:
Plo
toft
hefu
nctio
n
.
whe
re
# ~` # : ab' ~g if
D~
if
z~
(3.1
3)
Fig
ure
3.4
show
sa
plot
ofth
e
func
tion.
Whe
nth
e
func
tion
isus
edth
ear
gum
entis
ofth
efo
rm
3 ,w
here
isa
boun
dary
poin
ttha
tdoe
snot
nece
ssar
ilyco
inci
dew
itha
grid
poin
t.T
hevi
rtue
sof
this
choi
ceof
func
tion
are
give
nin
Pes
kin(
1977
).In
Pe-
skin
and
McQ
ueen
(199
6),B
eyer
and
LeV
eque
(199
2)an
dB
eyer
(199
2)va
rious
alte
rnat
ivest
oth
isch
oice
have
been
stud
ied.
The
vect
ordi
ffer
ence
oper
ator
L th
atap
prox
imat
esth
eco
n-tin
uous
oper
ator
< can
bede
fined
asfo
llow
s(s
eeP
eski
nand
Prin
tz(1
993)
and
Tho
rup
(199
7)): L
6
O Ln nk
6k k
66k
(3.1
4)
6
O Ln nk
6k k
66k
(3.1
5)
with
:N Re . R (e .
@e . (e . :N CN
6 6: Re . R (e .
6 @e . (e . 6: C
3.3
Dis
cret
efo
rmul
atio
n27
The
oper
ator
wor
ksin
the
follo
win
gw
ay:
5y L5y 5y ;2y Ly L: y
whe
re
2y y Ly .
3.3.
3D
iscr
etiz
atio
nof
equa
tions
ofm
otio
n
The
Nav
ier-
Sto
kese
quat
ion(
3.1)
and
the
cont
inui
tyeq
uatio
n(3.
2)ar
edis
cret
ized
follo
win
ga
sche
mea
dapt
edfr
omA
.Cho
rin(1
968;
1969
),se
ePes
kin(
1972
b),P
e-sk
in(1
972a
)and
Pes
kin(
1977
).T
his
intr
oduc
esth
efr
actio
nals
tepv
eloc
ityfie
lds
2y1)/ ,
2y1) ) an
d
2y1)/. fo
und
for
give
n
2y and
>y byso
lvin
gin
succ
essi
onth
eeq
uatio
ns
72y1) 2y >y
(3.1
6)
78 2y1)/) 2y1)/
: y L L2y1) )= *1 L( L2y1)/)
(3.1
7)
78 2y1)/. 2y1)/)
: y 2y1)/.= *1 ( 2y1)/.
(3.1
8)
The
n2y1)
and
5y1)ar
efo
und
byso
lvin
gth
esy
stem
78 2y1) 2y1) .
= : 5y1)
(3.1
9)
;2y1) sy 5y1)
(3.2
0)
Her
e
isth
eve
ctor
diff
eren
ceop
erat
orde
fined
abov
e,an
d
s y 5y1) is
obta
ined
from
adi
scre
tizat
iono
fequ
atio
n(3.
8),
sy 5y1)
. O Q R)Iy Q 5y1) ?y Q
(3.2
1)
whe
re
? y Q isfo
und
byth
edi
scre
tizat
iono
fequ
atio
n(3.
7)
?y QM Qy Q M JL y JL
(3.2
2)
Fort
heso
urce
inth
eat
rium
we
have
acco
rdin
gto
equa
tion(
3.9)
Iy ) 5y1)
put!vpy ) w t!v
put!vq 5y1) ? y)r w t!v
(3.2
3)
282D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
base
don
the
disc
rete
inne
rpro
duct
q 5 ?r
O ~. 5 ?
whe
reth
esu
mm
atio
nis
over
the
grid
poin
tsof
the
dom
ain.
We
note
that
the
sour
ce/s
inkfu
nctio
n
? isde
fined
andc
ontin
uous
ever
ywhe
rein
,whi
leth
epr
es-
sure
5 isonly
defin
edon
the
grid
poin
ts.W
eal
sono
teth
atth
ech
oice
ofw
eigh
tfu
nctio
ns
M Q is
such
that
the
disc
rete
anal
ogue
ofth
eno
rmal
izat
ionc
ondi
tion
(3.5
)hol
dsfo
ral
ltra
nsla
teso
fM Q .Fo
rthe
trea
tmen
toft
heao
rtic
sink
seeS
ectio
n3.
3.4. Equ
atio
n(3
.3)
for
the
coup
ling
ofth
ebo
unda
ryfo
rces
toth
eflu
idis
dis-
cret
ized
usin
gth
eap
prox
imat
iont
oth
ede
ltafu
nctio
ngi
ven
by(3
.12)
and
(3.1
3).
The
disc
retiz
edve
rsio
nofe
quat
ion(
3.3)
then
read
s
>y
O QBy S 43 y S
O QBy Q 43 y Q
(3.2
4)
The
calc
ulat
iono
fthe
boun
dary
forc
es
By Q isdi
scus
sedin
Sec
tion3
.3.5
.F
inal
ly,
equa
tion(
3.4)
coup
ling
the
fluid
mot
ion
toth
ebo
unda
rym
otio
nis
disc
retiz
edus
ing
the
sam
eapp
roxi
mat
iont
oth
ede
ltafu
nctio
n,i.e
.
y1) Qy Q: 2y1) Q
y Q: O 2y1) 3 y Q~.
(3.2
5)
Inso
lvin
gth
edi
scre
teeq
uatio
nsit
isne
cess
aryto
redu
ceal
llen
gths
and
times
bya
fact
or
( D# )
toen
sure
num
eric
alst
abili
ty.
The
scal
ingr
educ
esth
eR
eyno
lds
num
ber,
whi
chis
equi
vale
ntto
rais
ing
the
visc
osity
ofth
eflu
idby
afa
ctor
of
#$ .Ano
ther
effe
ctof
the
scal
ingi
sth
atth
ebo
unda
ryla
yero
fthe
fluid
isra
ised
bya
fact
orof
#$ .The
scal
ingi
sdi
scus
sedin
Sec
tion4
.2.
Fur
ther
deta
ilsof
the
solu
tion
ofth
edi
scre
tized
equa
tions
ofm
otio
nca
nbe
foun
din
Tho
rup
(199
7).
3.3.
4D
iscr
ete
Win
dkes
selm
odel
for
the
aort
icsi
nk
The
rela
tion
betw
eenfl
owan
dpr
essu
refo
rth
eW
indk
esse
lmod
elat
the
aort
icou
tflow
isfo
und
bydi
scre
tizin
gthe
equa
tions
ofth
ism
odel
inS
ectio
n3.2
.1.
3.3
Dis
cret
efo
rmul
atio
n29
Equ
atio
n(3.
11)f
orth
epr
essu
re
p t!v isdi
scre
tized
acco
rdin
gto
anim
plic
itE
uler
met
hoda
sfol
low
s:
py tyQpy () tyQ
I y . |p y tyQ | w ty
Q .W
eus
e
#
and
here,b
ecau
sew
ene
edan
expr
essi
onfo
r
I y . .Thi
seq
uatio
nm
aybe
solv
edfo
r
p y tyQ : py tyQ
py () tyQ# : $| w tyQ .
$ |# : $| w tyQ .Iy .
(3.2
6)
Inse
rtin
g(3.
10)i
nto
(3.2
6)an
dso
lvin
gfo
r
p y tyQ we
get
py tyQY[ZZZZZ\ ZZZZZ]py () tyQ: ¡ ¢£p y .
# : £
if
py .zpy tyQ
py () tyQ # : ¡ _£
if
py .py tyQ
(3.2
7)
whe
re
w () w () tyQ ):w () tyQ . a
nd
py .q 5y1) ? y .r .F
rom
rela
tions
(3.1
0)an
d(3
.26)
we
dedu
ceth
at
py .zpy tyQ
ifan
don
lyif
py .zpy () tyQ # :
/ ¡ _£
(3.2
8)
Inse
rtin
g(3
.27)
and
(3.2
8)in
to(3
.10)
we
obta
inth
efo
llow
ing
rela
tion
betw
een
I y . and
p y . ata
give
ntim
ele
vel
:
Iy .
YmZZ4ZZZZZZZZ\ ZZZZZZZ4ZZZ]p y .py () tyQ # :
/ ¡ _£
w tyQ )# : £
# : / ¡ _£
if
p y .zpy () tyQ # :
¡ _£
if
p y .py () tyQ # :
¡ _£
(3.2
9)
Hen
ceat
each
time
leve
l
forwhi
chth
eva
lve
isop
en,th
erei
sa
linea
rpre
ssur
e-flo
wre
latio
nbe
twee
n
py . and
I y . (com
pare
with
equa
tion(
3.23
)for
the
sour
ce
302D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
inth
eat
rium
).T
heap
pare
ntre
sist
ance
toth
eflo
w(t
hede
nom
inat
orin
the
abov
eex
pres
sion
)isco
nsta
ntin
time,
butt
hepr
essu
re
p y tyQ aga
inst
whi
chth
ehe
artis
pum
ping
varie
swith
time.
Whe
nth
eva
lues
of
p y . and
I y . are
com
pute
dfro
mth
ein
tera
ctio
nwith
the
hear
tmod
elth
enth
epr
essu
re
p"tyQ sho
uldb
eup
date
dacc
ordi
ngto
the
follo
win
gfo
rmul
a
py tyQY[ZZZZZ\ ZZZZZ]py .:w tyQ )Iy .
if
p y .zpy () tyQ # :
/ ¡ _£
py () tyQ # : / ¡ _£
if
p y .py () tyQ # :
/ ¡ _£
(3.3
0)
The
abov
em
odel
for
the
aort
icsi
nkgi
vesr
ise
toth
efo
llow
ing
algo
rithm
for
upda
tingt
hepr
essu
rean
dou
tflow
rate
from
time
leve
l
#
totim
ele
vel
andfo
rco
ntro
lling
the
open
inga
ndcl
osin
gof
the
valv
e
1.C
ompu
tea
prel
imin
aryu
pdat
eof
p tyQ acc
ordi
ngto
p y tyQpy () tyQ # :
/ ¡ _£.
2.C
ompu
teth
epr
essu
re
p y . from
the
hear
tmod
elus
ing
rela
tions
hip(
3.29
)be
twee
npre
ssur
eand
flow
ineq
uatio
n(3.
21)f
orth
eso
urce
/sin
kterm
s :If
the
valv
eis
open
use
Iy .
p y .p y tyQ
w tyQ ) # : $ w|$ #: $ w tyQ .|
othe
rwis
e
Iy .
3.If
p y .zp y tyQ a
ndth
eva
lve
isno
tope
n,th
enth
eva
lve
shou
ldbe
open
.T
huso
pent
heva
lve
byob
tain
ingt
heflo
w
I y . throu
gha
reco
mpu
tingo
fthe
pres
sure
p y . from
the
hear
tmod
elus
ing
the
first
ofth
etw
opr
essu
re-fl
owre
latio
nshi
psin
(3.2
9).
4.If
p y .p y tyQ a
ndth
eva
lve
isop
en,th
enth
eva
lve
shou
ldbe
clos
ed.C
lose
the
valv
eby
reco
mpu
tingt
hepr
essu
re
p y . usin
gth
ese
cond
pres
sure
-flow
rela
tions
hipi
n(3
.29)
.
3.3
Dis
cret
efo
rmul
atio
n31
5.C
ompu
teth
eco
rrec
tval
ueof
p y tyQ acc
ordi
ngto
(3.3
0):
Ifth
eva
lve
isop
enth
en
py tyQpy .:w tyQ )Iy .
Oth
erw
isen
ofu
rthe
rupd
atin
gis
need
ed.
The
actu
alch
oice
ofW
indk
esse
lpar
amet
ers(
| ,w tyQ )
and
w tyQ . )ar
edi
s-cu
ssed
inS
ectio
n4.5
.
3.3.
5B
ound
ary
forc
es
Now
we
turn
toth
ede
scrip
tiono
fthe
boun
dary
forc
esat
the
disc
rete
leve
l.
The
link
form
alis
m
The
posi
tion
ofth
ehe
artb
ound
aryi
sde
fined
byits
confi
gura
tionv
ecto
r
) ¤ ¥
cons
istin
gof
the
posi
tion
vect
ors
Q forth
ebo
unda
rypo
ints
.T
hebo
unda
ryfo
rces
are
mod
elle
dasa
risin
gfr
omel
astic
sprin
gs,c
alle
dlin
ks,c
onne
ctin
gthe
sebo
unda
rypo
ints
.T
hese
tofa
lllin
ksis
deno
ted
¦ .Let
deno
tea
link
conn
ectin
gbou
ndar
ypoi
nt
S whi
chha
spos
ition
vect
or
Q with
boun
dary
poin
t
§ whi
chha
spos
ition
vect
or¨ .Th
enit
isa
prop
erty
ofth
em
odel
that
the
forc
e
B Q on
the
poin
t
S due
toth
elin
k
,ata
nygi
ven
time
leve
l,is
ofth
efo
rm
B QB Q -c ¨ Qf
¨ Q
¨ Q
(3.3
1)
whe
reth
efu
nctio
n
- give
sthe
tens
ioni
nlin
k
vers
usits
leng
th
E ¨ Q .
Not
eth
atsu
pers
crip
t does
notd
enot
eatim
est
ep.T
heel
astic
prop
ertie
sof
the
boun
dary
are
thus
defin
edat
each
time
leve
lby
ase
tof
leng
th-t
ensi
onre
latio
ns
- - E ,
one
for
each
link.
The
mod
elfo
rth
esel
engt
h-te
nsio
nrel
atio
nsis
give
nbe
low
.T
heto
talb
ound
aryf
orce
By Q onbo
unda
rypo
int
S isno
wgi
ven
by
By QB Q y O © ª
« ¡¬B Qy
(3.3
2)
whe
re
¦ Q denot
esth
ese
tofa
lllin
ksw
hich
have
the
boun
dary
poin
t
S ason
eof
itsen
dpo
ints
.Mor
eth
anon
elin
kca
nco
nnec
tagi
ven
pair
ofpo
ints
.
322D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
How
ever
,the
boun
dary
forc
esac
tual
lyap
plie
dto
the
fluid
are
notc
ompu
ted
dire
ctly
byev
alua
ting
equa
tion(
3.32
).In
stea
dan
impl
icit
met
hod,
actu
ally
anim
plic
itfr
actio
nals
tep
met
hod,
isus
edto
ensu
renu
mer
ical
stab
ility
.T
hefo
rces
are
notc
ompu
tedf
rom
the
final
boun
dary
confi
gura
tion
y1)
butf
rom
aco
nfig-
urat
ion
defin
edim
plic
itly
asfo
llow
s:
Qy Q: 2 Q
2 2y : >
whe
re
2 QO 2 43 y Q~.
(3.3
3)
> O QB Q 3 y Q
(3.3
4)
B QB Q ) ¤¥
Not
eth
ateq
uatio
n(3.
33)i
sof
the
sam
efor
mas
the
last
term
ineq
uatio
n(3.
25)
and
equa
tion(
3.34
)is
ofth
esa
mef
orm
aseq
uatio
n(3.
24).
By
subs
titut
ioni
nth
eab
ove
five
equa
tions
we
find
the
follo
win
gno
n-lin
ear
syst
em(P
eski
n,19
77;T
horu
p,19
97)
Qk Q:®B Q ) ¤¥ S #
(3.3
5)
whe
re
k Qy Q: 2y Q
(whi
chis
know
nat
time
leve
l
:# )an
dw
here
® isch
osen
soth
at
Q isan
appr
oxim
atio
nto
y1) Q .InP
eski
n(19
81)
® isch
osen
tobe
; $ ~. . .F
inal
ly,e
quat
ion(
3.35
)can
beso
lved
byN
ewto
n’s
met
hod
for
Q (Pes
kin
and
McQ
ueen
,198
0;T
horu
p,19
97),
and
the
forc
esap
plie
dto
the
fluid
are
give
nby
B QB Q ) ¤¥ ,i
.e.i
neq
uatio
n(3.
24)i
nser
t
B Qin
stea
dof
By Q .P
assiv
etis
sue
mod
el
The
hear
tval
vesa
ndth
eou
tflow
trac
tare
esse
ntia
llyel
astic
tissu
eand
thes
epar
tsof
the
hear
tbou
ndar
yare
ther
efor
emod
elle
dby
linea
rspr
ings
.The
leng
th-t
ensi
onre
latio
nin
the
links
mod
ellin
gth
eva
lves
is(P
eski
n,19
77)
- E ¯ °± E E E zE
E E
3.3
Dis
cret
efo
rmul
atio
n33
whe
re°±
isth
est
iffne
ssan
d
E isthe
rest
ing
leng
th.I
tis
seen
that
the
tens
ion
isne
ver
nega
tive,
soth
eel
emen
tissl
ack
whe
nits
leng
this
less
than
the
rest
-in
gle
ngth
.T
his
mea
nsth
atth
ele
aflet
scan
resi
stte
nsio
nbu
tno
tcom
pres
sion
(McQ
ueen
etal
.,19
82).
InP
eski
nand
McQ
ueen
(198
0)pr
osth
etic
hear
tval
ves
wer
em
odel
ledb
ych
angi
ngth
ele
ngth
tens
ionr
elat
ion
toal
low
for
resi
stan
ceto
com
pres
sion
.T
hete
nsio
n
- ina
link
betw
eent
hebo
unda
rypo
ints
Q and
¨ inth
ebo
unda
ryco
nfigu
ratio
n
is
- ¯ °± E E E zE
E E
(3.3
6)
whe
reth
ein
stan
tane
ousle
ngth
ofth
elin
kis
E ¨ Q .T
here
stin
glen
gth
E iscom
pute
dast
hele
ngth
ofth
elin
kat
the
initi
altim
e.
Mus
cula
rm
odel
Alin
kre
pres
entin
gam
uscu
larfi
beri
sm
odel
leda
stw
olin
ears
prin
gsin
para
llel,
one
ofw
hich
,the
activ
epa
rt,h
asa
varia
ble
rest
ing
leng
th,s
eeF
igur
e3.
5.T
heva
riabl
ere
stin
gle
ngth
ism
odel
led
bya
cont
ract
ileel
emen
tinse
riesw
ithth
elin
ears
prin
g.T
his
rest
ing
leng
this
cont
rolle
din
asp
ecifi
cway
byth
ele
ngth
ofth
elin
kE ,i
tste
nsio
n
- ,and
agi
ven
func
tion
oftim
e
G mod
ellin
gthe
activ
atio
nle
velo
fth
em
uscl
e.T
hesp
ring
inth
eac
tive
part
isde
note
dthe
serie
sele
men
t.T
hesp
ring
inpa
ralle
lwith
the
activ
epa
rt,a
ndw
hich
cons
titut
esth
epa
ssive
part
,is
calle
dth
epa
ralle
lele
men
t.T
hete
nsio
n
- ina
mus
cula
rlink
inth
ebo
unda
ryco
nfigu
ratio
n
isth
esu
mof
the
pass
ivean
dth
eac
tive
part -
-³²:- ^
(3.3
7)
The
tens
ioni
nth
epa
ssive
part
isfo
und
sim
ilarly
toth
epa
ssive
tissu
emod
elab
ove.
Thu
s
-³²¯ ° ²J E E ²J
if
E zE ²J
if
E E ²J
(3.3
8)
whe
re° ²J
isth
est
iffne
ss,
E isth
ele
ngth
ofth
em
uscu
larli
nkan
d
E ²J isthe
rest
ingl
engt
hof
the
para
llele
lem
ent.° ²J
and
E ²J arefix
edin
time.
The
tens
ioni
nth
eac
tive
part
isgi
ven
by
- ^¯ ° tJ E Ey1) vJ
if
E zE y1) vJ
if
E E y1) vJ
(3.3
9)
342D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
L ce
Par
alle
lel
emen
t
Ser
ies
elem
ent
Con
trac
tile
elem
ent
L ce
L−
L
Fig
ure
3.5:
Ske
tch
ofth
em
uscu
larm
odel
with
two
linea
rspr
ings
inpa
ralle
l,one
ofw
hich
isco
nnec
tedin
serie
sto
aco
ntra
ctile
elem
ent.
whe
re° tJ
isth
est
iffne
ssof
the
serie
sele
men
tand
E vJ isth
ele
ngth
ofth
eco
n-tr
actil
eel
emen
t.W
hen
we
com
pare
this
expr
essi
onw
ithth
eex
pres
sion
for
the
tens
ioni
nth
epa
ssive
part
abov
e,w
ese
etha
tthe
leng
thof
the
cont
ract
ileel
emen
t
E vJ has
the
func
tion
ofa
rest
ingl
engt
h.S
ince
E vJ varie
sin
time,
we
can
thin
kof
the
activ
epa
rtas
alin
ears
prin
gw
itha
varia
bler
estin
glen
gth
give
nby
E vJ .In
the
cont
inuo
usfo
rmul
atio
nof
the
mod
elth
eve
loci
tyof
cont
ract
iono
fthe
cont
ract
ileel
emen
tisgo
vern
edby
the
follo
win
gco
mbi
natio
nof
adi
ffer
entia
leq
uatio
nand
anal
gebr
aice
quat
ion
YmZZ\ ZZ]C E vJ C
´"µ^L G ;G ° E - ^
° E :- ^
if
G ¶
E vJE
if
G
(3.4
0)
whe
re
´"µ^L and°
are
mus
clep
aram
eter
sand
G isa
give
nfu
nctio
nof
time
calle
dth
eac
tivat
ion
func
tion.
For
deta
ilson
the
activ
atio
nfu
nctio
nse
eS
ec-
tion
3.3.
6.A
tco
nsta
ntac
tivat
ion
G andle
ngth
E the
equi
libriu
mva
lue
ofth
eac
tive
ten-
sion
is
- ^G° E °tJ EE vJ ,so
the
max
imum
valu
eof
G compa
tible
with
non-
nega
tive
rest
ingl
engt
h
E vJ is
G °tJ$ °
.Cha
ract
eris
ticof
the
mod
elis
that
the
appr
oach
toth
iseq
uilib
rium
isfa
ster
atsm
alle
ract
ivat
ion
leve
lsdu
eto
the
G
inth
ede
nom
inat
orin
the
expr
essi
onin
(3.4
0).
Equ
atio
n(3.
40),
with
the
activ
atio
nde
pend
ence
,isa
gene
raliz
atio
nof
Hill
’seq
uatio
nfor
the
velo
city
ofco
ntra
ctio
nof
ate
tani
zedm
uscl
e.T
hem
odel
abov
e
3.3
Dis
cret
efo
rmul
atio
n35
iskn
own
asH
ill’s
thre
eele
men
tmod
el,s
eech
apte
rs9
and
10of
Fun
g(1
993)
.In
the
case
of
Gy1) ¶ th
ete
nsio
n
- ^ inth
elin
kca
nbe
foun
dfr
omth
esy
stem
- ^° tJ E Ey1) vJ
Ey1) vJEy vJ ´ µ^L Gy1)Gy1) °E - ^
° E :- ^
Com
bini
ngth
esee
quat
ions
and
solv
ing
for
- ^ gives
- ^# ° tJ` | · + ¸ ·:+:|.· |g
(3.4
1)
whe
re
· ´ µ^L8 # Gy1):#=
+ ° E
° tJ| E Ey vJ: ´ µ^L
Sin
cebo
th· an
d
+ are
posi
tive,
itis
seen
that
for
| D th
ete
nsio
n
- ^D .
Thi
sco
rres
pond
sto
E y1) vJzE in
whi
chca
seth
ete
nsio
n
- ^ shoul
dbe
zero
(see
equa
tion(
3.39
)).T
hus
we
can
set
- ^ if
| D w
ithou
thav
ing
toco
mpu
te
E y1) vJdi
rect
ly.
3.3.
6T
heac
tivat
ion
func
tion
The
activ
atio
nfu
nctio
n
G whi
chw
asin
trod
uced
for
the
mus
cula
rlink
abov
eis
the
driv
ing
forc
eof
the
hear
tmod
el.T
heac
tivat
ion
func
tion
mod
elst
here
leas
eof
Ca. 1 (P
eski
n,20
01),
whi
chin
itiat
esth
eco
ntra
ctio
nofh
eartm
uscl
easm
entio
ned
inS
ectio
n2.
4.T
hus
the
activ
atio
nfu
nctio
nde
term
ines
the
time
cour
sein
the
mod
el,i.
e.th
eco
ntra
ctio
nand
rela
xatio
noft
hehe
artm
uscl
e.W
ede
scrib
eher
eth
efo
rmof
the
activ
atio
nfu
nctio
nas
itw
asin
the
orig
inal
mod
elby
Pes
kina
ndM
cQue
en.W
eha
veal
soim
plem
ente
dane
wfo
rmof
the
activ
atio
nfu
nctio
n,bu
tth
isis
the
topi
cof
Cha
pter
6an
dw
illno
tbe
disc
usse
dhe
re. The
time
cour
seof
the
hear
tcyc
leis
dete
rmin
edby
the
follo
win
gpa
ram
eter
s:he
artr
ate,
onse
tand
dura
tion
times
ofat
riala
ndve
ntric
ular
syst
ole
rela
tive
tohe
artp
erio
d,an
dex
cita
tion
and
rela
xatio
nrat
eco
nsta
nts(s
eebe
low
).T
hem
odel
sets
the
initi
altim
eof
the
sim
ulat
ion
inth
eea
rlyve
ntric
ular
dias
tole
durin
gth
e
362D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
isov
olum
icre
laxa
tionp
hase
ofth
eve
ntric
le.T
heac
tivat
ion
func
tion
G isde
-fin
edin
sepa
ratew
ays
indi
asto
lean
dsy
stol
eand
isdi
ffer
entf
orth
elin
ksin
the
atriu
m,t
heve
ntric
lean
dth
epa
pilla
rym
uscl
e.D
urin
gdi
asto
leth
eac
tivat
ion
func
tion
G isde
fined
asa
solu
tion
toa
dif-
fere
ntia
lequ
atio
noft
hefo
rm
C G C "!G
whe
re
u! denot
esa
rela
xatio
nrat
eco
nsta
nt.D
urin
gsy
stol
e
G isde
fined
asa
solu
tion
toa
casc
aded
seto
fdiff
eren
tiale
quat
ions
ofth
efo
rm
C G C J G µ^LG
C G C J G G
whe
re
J denot
esan
exci
tatio
nra
teco
nsta
ntan
d
G µ^L
deno
testh
elim
iting
valu
eof
the
activ
atio
nfu
nctio
n.T
heac
tivat
ion
func
tion
G insyst
olei
sch
osen
asth
eso
lutio
nto
ase
cond
orde
req
uatio
nbec
ause
itgi
ves
asm
ooth
tran
sitio
nto
the
syst
olic
phas
e,w
here
asth
eau
xilia
ryfu
nctio
n
G hasa
mor
eab
rupt
upst
roke
.In
addi
tion,
the
activ
atio
nfu
nc-
tion
isre
quire
dto
beco
ntin
uous
for
the
who
lehe
artp
erio
d.T
here
isa
diff
eren
tse
tof
cons
tant
sfor
the
atriu
man
dth
eve
ntric
le,b
utth
esa
mes
etis
used
for
the
vent
ricle
and
the
papi
llary
mus
cle.
The
initi
alva
lues
ofth
eac
tivat
ion
func
tion
are
diff
eren
tfor
allt
hree
type
sofl
inks
.In
Pes
kina
ndM
cQue
en’sor
igin
alm
odel
the
cons
tant
sfort
heac
tivat
ion
func
-tio
nw
ere
sett
om
atch
ado
g’s
hear
t.S
ince
itis
oura
imto
mak
ea
sim
ulat
ion
ofth
em
odel
rese
mbl
ethe
MR
data
ofa
hum
anw
ech
ange
the
timin
gpa
ram
eter
sfo
rth
eac
tivat
ion
func
tion.
We
choo
seon
esp
ecifi
cset
ofM
Rm
easu
rem
entsa
ndus
eit
asth
eba
sisf
orou
rmod
ifica
tiono
fth
em
odel
.The
leng
thof
the
hear
tcy-
cle,
0.93
6m
s,is
foun
das
the
peak
tope
aktim
e(2
6in
terv
als
of36
ms
dura
tion
t t Jy¹ ! JG µ^LG
[ms]
[ms]
[ms
() ][m
s
() ]
G
atriu
m0.
362
0.46
61/
551/
751.
50
vent
ricle
0.54
50.
851
1/55
1/21
4.9
1.07
pap.
mus
c.0.
545
0.85
11/
551/
214.
90.
107
Tabl
e3.
1:T
imin
gs(o
nset
time
ofsy
stol
e
t anden
dtim
eof
syst
ole
t Jy¹ ),rat
eco
nsta
nts,l
imiti
ngva
lues
,and
initi
alva
lues
for
the
activ
atio
nfu
nctio
ns.I
nitia
ltim
efo
rm
odel
is
.
3.3
Dis
cret
efo
rmul
atio
n37
each
)inth
eM
Rre
cord
ingo
fthe
aort
icou
tflow
velo
city
ata
cent
ralre
gion
inth
eou
tflow
jet
atth
ele
velo
fth
eao
rtic
valv
e.T
heM
Rda
tado
esno
tdire
ctly
give
the
timin
gfo
rth
eco
ntra
ctio
nof
the
hear
t,bu
tby
com
parin
gvel
ocity
profi
les
atth
em
itral
ring
and
the
aort
icou
tflow
trac
tfro
mth
esi
mul
atio
nand
from
the
MR
reco
rdin
g,th
eco
nsta
ntsc
anbe
dete
rmin
edby
tria
land
erro
r.T
hese
are
aslis
ted
inTa
ble
3.1.
Fig
ure
3.6
show
sth
eac
tivat
ion
func
tions
for
the
vent
ricle
,atr
ium
and
papi
llary
mus
clef
oron
ehe
artc
ycle
.T
hedi
ffer
entia
lequ
atio
nsha
vean
alyt
icso
lutio
ns(s
eeS
ectio
n6.2
.1),
buti
nth
epr
ogra
mth
edi
ffer
entia
lequ
atio
nsar
eso
lved
with
suffi
cien
tacc
urac
yby
anim
plic
itE
uler
met
hod.
Indi
asto
lew
eha
ve
Gy1)Gy u!Gy1)
and
insy
stol
e
Gy1) Gy J G µ^LGy1)
Gy1)Gy J
Gy1) Gy1)
0.2
0.4
0.6
0.8
1
012345
i j
Fig
ure
3.6:
Act
ivat
ion
asfu
nctio
nof
time
for
the
vent
ricle
(dot
ted)
,atr
ium
(das
hed)
and
papi
llary
mus
cle(
solid
),th
ela
tteri
sin
dist
ingu
isha
blefr
omth
eve
n-tr
icul
arcu
rve
insy
stol
e.
382D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
3.3.
7To
polo
gyan
din
itial
geom
etry
ofth
ehe
art
The
geom
etric
para
met
ersd
eter
min
ethe
size
and
shap
eof
the
mod
elhe
arta
tth
ein
itial
time
ofsi
mul
atio
n.T
hein
itial
geom
etry
isso
mew
hati
deal
ized
,bei
ngco
mpo
sedo
fstr
aigh
t-lin
eseg
men
tsan
dci
rcul
arar
cs.T
his
geom
etry
isof
cour
sech
ange
ddur
ing
asi
mul
atio
nofa
card
iacc
ycle
.W
eus
efou
rkey
para
met
ersfo
rth
ege
omet
ry:th
edi
amet
erof
the
mitr
alrin
g(2
.97
cm),
the
diam
eter
ofth
eao
rtic
outfl
owtr
act(
2.58
cm),
the
angl
ebe
twee
nth
eset
wo
(45
, ),an
dthe
leng
thof
the
vent
ricle
(5.4
2cm
).T
hedi
amet
erva
lues
are
read
off
from
the
wid
ths
ofth
ein
flow
and
outfl
owje
tsin
the
MR
reco
rdin
gsan
dth
enin
crea
sedb
y10
%so
that
the
wid
thso
fthe
mea
sure
dand
sim
ulat
edin
/out
flow
jets
appe
arap
prox
imat
elyt
hesa
me
durin
gth
eir
initi
alph
ases
.Suc
hin
crea
seis
nece
ssar
yinor
dert
oco
mpe
nsat
efor
the
incr
ease
dthi
ckne
ssof
the
boun
dary
laye
r.T
hein
crea
sedth
ickn
essis
due
toth
esc
alin
goft
hevi
scos
itym
entio
nedi
nS
ectio
n3.3
.3an
dto
the
size
ofth
ere
gion
ofin
fluen
ceof
the
boun
dary
laye
ron
the
fluid
resu
lting
from
the
chos
ensp
atia
lreso
lutio
n.T
hean
gula
rval
ue,w
hich
issl
ight
lyon
the
high
side
,isch
osen
tom
ake
the
sim
ulat
edin
flow
jetd
irect
ion
fitth
em
easu
redo
neat
the
time
ofea
rlype
akin
flow
.T
hele
ngth
ofth
eve
ntric
leis
chos
enla
rge
enou
ghso
that
itca
nw
ithst
andc
ontr
actio
nunt
ilth
een
dof
syst
ole.
Fig
ure
3.7
show
sth
efin
alm
odifi
edin
itial
geom
etry.
The
boun
dary
poin
tsar
econ
nect
edby
links
repr
esen
tingm
uscl
esan
dcon
nec-
tive
tissu
es.T
his
link
stru
ctur
e,th
eto
polo
gyof
the
mod
elhe
art,i
sfix
edth
roug
h-ou
tthe
cycl
e.In
addi
tion
toth
eso
-cal
ledp
hysi
calb
ound
aryp
oint
s,th
ebo
unda
ryin
clud
eson
evi
rtua
lpoi
ntw
hich
isno
tmov
eddi
rect
lyby
the
fluid
buti
sus
edas
anin
terc
onne
ctio
npoi
ntbe
twee
nphy
sica
lpoi
nts.
Link
sco
nnec
tingt
hebo
unda
rypo
ints
may
beei
ther
activ
eor
pass
ive,s
eese
ctio
n3.3
.5.
Agi
ven
pair
ofpo
ints
may
beco
nnec
tedb
ym
ore
than
one
link.
The
boun
dary
poin
tsar
eco
nnec
teda
long
the
wal
land
inad
ditio
nby
cros
slin
ks.
Cro
sslin
ksar
ene
eded
tosi
mul
ate
circ
umfe
rent
ials
tres
sesin
the
mod
elhe
art
sinc
eth
em
odel
ison
ly2D
.The
links
them
selve
sdo
noti
nter
fere
with
the
fluid
,ex
cept
thro
ught
hefo
rces
they
exer
ton
thei
rend
poin
ts.
Inth
eor
igin
alm
odel
cros
slin
ksco
nnec
tthe
stra
ight
part
soft
heve
ntric
leto
each
othe
r.A
lso
conn
ecte
dby
cros
slin
ksar
eth
est
raig
htpa
rts
ofth
eou
tflow
trac
t.To
acco
mm
odat
ethe
mod
elto
are
alis
tichu
man
left
vent
ricle
we
incr
ease
the
stiff
ness
para
met
erso
fthe
cros
slin
ksin
the
stra
ight
part
ofth
eve
ntric
lean
din
trod
ucec
ross
links
inth
eci
rcul
arpa
rtof
the
vent
ricle
,see
Fig
ure
3.7.
Cro
sslin
ksem
anat
efrom
apo
inta
tthe
botto
mof
the
vent
ricle
toth
evi
rtua
lpo
inta
ndfr
omth
evi
rtua
lpoi
ntto
the
end
poin
tsof
the
valv
ele
aflet
s.T
helin
kfr
omth
eve
ntric
lew
allt
oth
evi
rtua
lpoi
ntre
pres
entst
hetw
opa
pilla
rym
uscl
es
3.3
Dis
cret
efo
rmul
atio
n39
Source
Sink
Atrium
Ventricle
Exteriorsink/source
Valveleaflets
Outflow
tract
Papillarymuscle
Chords
Virtualpoint
Fig
ure
3.7:
Mod
ified
initi
alhe
artg
eom
etry
with
cros
slin
ksan
dlin
ksal
ong
hear
tw
alls
how
nas
full
lines
.
(see
Fig
ure
3.7)
,whi
chin
the
real
hear
tpro
trud
efr
omth
ehe
artw
all
atpo
ints
abov
ean
dbe
low
the
plan
ein
whi
chth
em
odel
resi
des.
The
yar
ere
pres
ente
dinth
em
odel
byth
eirc
omm
onpr
ojec
tiono
nto
this
plan
e.In
the
orig
inal
mod
elth
epa
pilla
rym
uscl
ewas
conn
ecte
dto
the
apex
ofth
ehe
art(
Pes
kin,
1977
).To
im-
prov
eth
em
ovem
ento
fthe
mitr
alva
lve
we
mov
eth
epo
intt
ow
hich
the
papi
llary
mus
clei
sco
nnec
tedu
pal
ong
the
ante
riorw
all.
The
links
from
the
virt
ualp
oint
sto
the
tips
ofth
ele
aflet
repr
esen
tthe
chor
ds,c
hord
aete
ndin
eae,w
hich
arep
assiv
etis
sue.
All
activ
elin
ksar
edou
bled
bya
corr
espo
ndin
gpas
sive
link,
buts
omep
airs
ofpo
ints
are
conn
ecte
donl
yby
apa
ssive
link,
cf.
Sec
tion3
.3.5
onbo
unda
ryfo
rces
.T
helin
ksin
the
outfl
owtr
acta
ndth
eas
soci
ated
cros
slin
ksar
eal
lpas
sive.
Als
oth
elin
ksin
the
valv
ele
aflet
sand
the
links
repr
esen
tingth
ech
ords
are
pass
ive.
Inth
eor
igin
alm
odel
the
hear
twas
free
lyflo
atin
gin
the
surr
ound
ingd
omai
n,w
hich
resu
ltedi
nex
cess
ive(n
on-p
hysi
olog
ical
)mov
emen
tsdu
ring
the
ejec
tion
phas
e.D
ueto
are
coil
effe
ctfr
omth
eej
ectio
nth
ehe
artw
ould
star
taro
ta-
tion
chan
ging
itsal
ignm
entin
spac
eand
begi
na
tran
slat
ione
vent
ually
mov
ing
itth
roug
hth
ebo
unda
ry.To
over
com
ethi
spr
oble
mw
ete
ther
the
hear
tby
linea
r
402D
hear
tmod
elus
ing
the
imm
erse
dbo
unda
rym
etho
d
sprin
gs(p
assiv
elin
ks)t
oa
seto
ffixe
dpo
ints
.Afte
rtry
ing
diff
eren
tarr
ange
men
tsw
ech
oose
tote
ther
the
stra
ight
part
ofth
eou
tflow
trac
tto
fixed
poin
tssi
tuat
edat
the
initi
alpo
sitio
nof
the
outfl
owtr
act.
Thi
sim
prov
esth
em
otio
nof
the
hear
tdu
ring
ahe
artc
ycle
cons
ider
ably,
even
thou
ghit
isst
illm
oret
han
that
ofa
natu
ral
hum
anhe
art(
seeC
hapt
er4)
.
3.4
Out
line
ofal
gorit
hm
The
prev
ious
sect
ions
have
desc
ribed
diff
eren
tpar
tsof
the
2Dhe
artm
odel
.The
over
alla
imof
the
mod
elis
toco
mpu
teth
eve
loci
tyfie
ld
2y1) ,the
pres
sure
field
5y1)
and
the
hear
tbou
ndar
ycon
figur
atio
n
y1)
attim
ele
vel
:# base
don
the
sam
equa
ntiti
esat
the
prev
ious
time
leve
l
.Theal
gorit
hmis
asfo
llow
s:
1.C
ompu
teth
ebo
unda
ryfo
rce
B Q asfol
low
s:
(a)
Com
pute
the
frac
tiona
lste
pbou
ndar
ycon
figur
atio
n
byso
lvin
g
Qk Q:®B Q ) ¤¥ S #
(3.3
5)
whe
re
k Qy Q: 2y Q .
(b)
Com
pute
the
tens
ion
- inth
ebo
unda
ryba
sedo
nth
eco
nfigu
ratio
n
asde
scrib
edin
Sec
tion3
.3.5
for
the
pass
ivetis
sueu
sing
equa
tion
(3.3
6)an
dfo
rth
em
uscu
larti
ssue
usin
geq
uatio
ns(3
.37)
–(3.
41).
(c)
Com
pute
B Q using
B QB Q O © ª
« ¡¬B Q
(3.3
2)
whe
re
B Q -c ¨ Qf
¨ Q
¨ Q
(3.3
1)
2.D
istr
ibut
eth
ebo
unda
ryfo
rce
B Q tothe
Eul
eria
ngrid
,i.e
.com
pute
the
forc
efie
ld
>y usin
g
>y O QB Q 43 y Q
(3.2
4)
3.C
ompu
teth
eve
loci
tyfie
ld
2y1)
and
the
pres
sure
field
5y1)by
solv
ing
the
Nav
ier-
Sto
kese
quat
iona
ndth
eco
ntin
uity
equa
tiona
sfol
low
s:
3.4
Out
line
ofal
gorit
hm41
(a)
Suc
cess
ively
solv
efo
rth
efr
actio
nals
tep
velo
citie
s
2y1)/ ,
2y1)/)
and
2y1) . us
ing
72y1) 2y >y
(3.1
6)
78 2y1)/) 2y1)/
: y L L2y1) )= *1 L( L2y1)/)
(3.1
7)
78 2y1)/. 2y1)/)
: y 2y1)/.= *1 ( 2y1)/. (3
.18)
(b)
Sol
ve
78 2y1) 2y1) .
= : 5y1)
(3.1
9)
;2y1) sy 5y1)
(3.2
0)
for
2y1)an
d
5y1)us
ing sy 5y1
) . O Q R)Iy Q 5y1) ?y Q
(3.2
1)
?y QM Qy Q M JL y JL
(3.2
2)
Her
e
I y ) 5y1) is
give
nby
equa
tion(
3.23
)for
the
atria
lsou
rce,
whi
leth
eco
mpu
tatio
nof
I y . forth
eao
rtic
sink
isas
desc
ribed
onpp
.30–
31S
ectio
n3.3
.4.
4.C
ompu
teth
ehe
artb
ound
aryc
onfig
urat
ion
y1) Qy Q: O 2y1) 3 y Q~.
(3.2
5)
Cha
pter
4
Eva
luat
ion
ofth
e2D
mod
el
Thi
sch
apte
rpre
sent
sthe
eval
uatio
nof
the
2Dhe
artm
odel
.Firs
twe
pres
entth
ety
pica
lout
putf
rom
asi
mul
atio
nwith
the
mod
el.W
eev
alua
tetw
oge
nera
lasp
ects
ofth
eim
mer
sedb
ound
arym
etho
d(as
itis
pres
ente
dinth
isth
esis
):th
ein
fluen
ceof
the
scal
ing
fact
or
andth
ein
fluen
ceof
the
perio
dic
boun
dary
cond
ition
s.F
urth
erm
ore,
som
easp
ectso
fth
em
odel
rela
teds
peci
fical
lyto
the
sim
ulat
ion
ofth
ehe
arta
reev
alua
ted,
e.g.
the
teth
erin
gof
the
hear
t,the
Win
dkes
selm
odel
and
the
mod
ellin
gof
dist
ribut
edso
urce
sand
sink
s.
4.1
Sim
ulat
ion
outp
ut
The
mai
nou
tput
from
asi
mul
atio
ncon
sist
soft
hree
varia
bles
:the
hear
tbou
ndar
yco
nfigu
ratio
n
,the
velo
city
field
2 andth
epr
essu
refie
ld
5 .From
each
ofth
ese
seve
ralo
ther
varia
bles
can
bede
rived
.T
here
sults
pres
ente
dinth
isch
apte
rare
from
asi
mul
atio
nof
the
mod
elde
-sc
ribed
inC
hapt
er3,
with
the
follo
win
gse
tting
s:12
8
0 128
grid
,299
52tim
est
eps
perh
eart
cycl
e,t
hele
ngth
ofa
hear
tcyc
leis
0.93
6s,
# ,a
ndth
eto
tals
im-
ulat
ion
time
is2
hear
tcyc
les(
i.e.1
.872
s).W
ew
illre
fert
oth
isas
ourr
efer
ence
sim
ulat
ion.
Thi
sch
oice
ofpa
ram
eter
setti
ngsw
illbe
disc
usse
dinth
efo
llow
ing
sect
ions
.Inth
isse
ctio
nwe
only
pres
entth
ere
sults
,leav
ing
the
disc
ussi
onof
them
tola
ters
ectio
ns.
44E
valu
atio
nof
the
2Dm
odel
4.1.
1H
eart
boun
dary
confi
gura
tion
and
fluid
mar
ker
s
Ate
acht
ime
step
the
hear
tbou
ndar
ycon
figur
atio
n
iskn
own
andc
anbe
plot
ted.
We
also
com
pute
the
mot
ion
ofso
mefl
uid
mar
kers
.The
sear
epl
aced
initi
ally
atea
chgr
idpo
inti
nth
ein
terio
rof
the
hear
tand
mov
edac
cord
ingt
oth
eve
loci
tyfie
ldat
each
time
step
.No
new
mar
kers
are
intr
oduc
edas
fluid
ente
rsth
em
odel
thro
ugh
the
sour
ces,
nora
rem
arke
rsre
mov
edas
fluid
leav
esth
roug
hth
esi
nks.
The
mar
kers
dono
taffe
ctth
eflu
idm
otio
nan
dar
eon
lyus
edfo
rou
tput
.A
nan
imat
iono
fthe
hear
tbou
ndar
ycon
figur
atio
nand
fluid
mar
kers
toge
ther
isa
very
help
fult
ooli
nth
ein
vest
igat
iono
fasi
mul
atio
n.S
ince
itis
notp
ossi
ble
tosh
owth
ishe
re,w
esh
owin
stea
din
Fig
ure
4.1
apl
otof
the
hear
tbou
ndar
yco
nfigu
ratio
nand
the
fluid
mar
kers
atsi
xeq
uidi
stan
ttim
esdu
ring
the
seco
ndhe
artc
ycle
.F
rom
the
hear
tbou
ndar
ycon
figur
atio
nthe
leng
thof
the
mitr
alrin
gan
dth
edi
stan
cebe
twee
nmitr
alva
lve
leafl
ettip
sar
eco
mpu
ted.
Fur
ther
mor
e,w
eco
m-
pute
the
area
ofth
eat
rium
and
ofth
eve
ntric
le.T
hetim
eva
riatio
nof
thes
efou
rqu
antit
iesc
anbe
used
inth
ein
vest
igat
iono
fasi
mul
atio
n.
4.1.
2V
eloc
ityfie
ld
The
velo
city
field
isvi
sual
ized
thro
ugha
velo
city
vect
orfie
ldpl
ot.T
osi
mpl
ifyth
epl
ot,o
nly
asu
bset
ofth
efu
llve
loci
tyfie
ldis
plot
ted.
Mor
epr
ecis
ely,
rega
rdle
ssof
the
spat
iald
iscr
etiz
atio
nuse
dfor
asi
mul
atio
n,th
eve
loci
tyfie
ldis
plot
ted
only
ona
32
0 32gr
id.
Inth
eup
perl
eft
corn
erof
each
velo
city
field
plot
aun
itsi
zeve
ctor
corr
espo
ndin
gto
ale
ngth
of1.
25cm
and
ave
loci
tyof
1m
/sis
plot
ted.
Fur
ther
mor
e,th
ehe
artb
ound
aryc
onfig
urat
ion
ispl
otte
d.F
igur
e4.
2sh
ows
apl
otof
the
velo
city
field
and
the
hear
tbou
ndar
ycon
figur
atio
natt
hesa
mes
ixeq
uidi
stan
ttim
esas
inF
igur
e4.
1.E
ven
thou
ghve
loci
tyve
ctor
sare
seen
tocr
oss
the
hear
tbou
ndar
y,th
isdo
esno
tmea
nth
atflu
idis
leak
ing
thro
ugh
the
wal
ls.
Rat
her,t
his
isa
cons
eque
nceof
the
hear
tbou
ndar
ymov
ing
with
the
fluid
velo
city
.F
rom
the
velo
city
field
we
can
com
pute
velo
city
time
profi
les
atgi
ven
posi
-tio
ns;t
his
isth
etim
eva
riatio
nof
the
velo
city
over
agi
ven
line
segm
ento
rat
agi
ven
poin
t.W
eha
vefo
cuse
don
velo
city
time
profi
lesa
tthe
mitr
alrin
g,be
twee
nth
em
itral
leafl
ettip
s,an
dac
ross
the
aort
icou
tflow
trac
tbel
owth
eao
rtic
sink
.W
eco
mpu
teth
em
eanv
eloc
itytim
epr
ofile
sast
hem
eanv
eloc
ity(w
ithre
spec
tto
spac
e)ov
era
line
segm
entr
elat
ive
toth
ehe
artb
ound
ary,
i.e.i
na
refe
renc
efr
ame
mov
ing
with
the
hear
t.E
.g.t
heve
loci
tytim
epr
ofile
over
the
mitr
alrin
gis
the
spat
ialm
eanv
eloc
ityov
erth
em
itral
ring
rela
tive
toth
em
ovin
gm
itral
ring.
Fur
ther
mor
e,w
elo
okat
the
velo
city
time
profi
leof
the
cent
reve
loci
ty,w
hich
issi
mpl
yth
eve
loci
tyco
mpo
nent
norm
alto
the
line
segm
enta
tthe
cent
repo
into
f
4.1
Sim
ulat
ion
outp
ut45
Time0.936
Time1.092
Time1.248
Time1.404
Time1.560
Time1.716
Fig
ure
4.1:
Hea
rtbo
unda
ryan
dflu
idm
arke
rsat
six
equi
dist
antti
mes
durin
gth
ese
cond
hear
tcyc
le.
The
first
time
(0.9
36s)
isat
the
onse
tof
the
seco
ndhe
art
cycl
ein
early
dias
tole
,whi
leth
ela
sttim
e(1
.716
s)is
atth
een
dof
the
seco
ndhe
artc
ycle
durin
gve
ntric
ular
syst
ole.
46E
valu
atio
nof
the
2Dm
odel
Time0.936
Time1.092
Time1.248
Time1.404
Time1.560
Time1.716
Fig
ure4
.2:V
eloc
ityfie
ldan
dhea
rtbou
ndar
ycon
figur
atio
nats
ixeq
uidi
stan
ttimes
durin
gth
ese
cond
hear
tcyc
le.T
hefir
sttim
e(0
.936
s)is
atth
eon
seto
fthe
seco
ndhe
artc
ycle
inea
rlydi
asto
le,w
hile
the
last
time
(1.7
16s)
isat
the
endo
fthe
seco
ndhe
artc
ycle
durin
gve
ntric
ular
syst
ole.
4.1
Sim
ulat
ion
outp
ut47
the
line
segm
ent.
The
cent
reve
loci
tyis
repo
rted
asth
eve
loci
tyre
lativ
eto
the
com
puta
tiona
lgrid
,i.e
.in
are
fere
ncef
ram
efix
edin
spac
e.In
all
the
plot
sof
velo
city
time
profi
lesw
edr
awve
rtic
alda
shed
lines
atth
een
dtim
eof
each
hear
tcy
cle,
i.e.i
nou
rref
eren
cesi
mul
atio
nth
elin
esar
edr
awn
at
san
dat
# %º s.
Fig
ure
4.3
show
sth
eve
loci
tytim
epr
ofile
ofth
em
eanv
eloc
ityov
erth
em
itral
ring
andb
etw
eent
hem
itral
leafl
ettip
sdu
ring
2he
artc
ycle
s.F
igur
e4.4
show
sth
eve
loci
tytim
epr
ofile
ofth
em
eanv
eloc
ityac
ross
the
aort
icou
tflow
trac
t(be
low
the
aort
icsi
nk)d
urin
g2
hear
tcyc
les.
Fro
mth
eve
loci
tyfie
ldw
eca
nals
oobt
ainfl
owda
taov
era
give
nlin
ese
gmen
t.F
low
isco
mpu
tedb
yth
em
eanv
eloc
ityov
erth
elin
ese
gmen
tmul
tiplie
dby
the
leng
thof
the
line
segm
ent.
Thi
sgi
ves
aflo
win
the
dim
ensi
onof
area
pert
ime,
whi
chis
the
natu
rald
imen
sion
offlo
win
a2D
mod
el.
Toco
nver
tare
aflow
tovo
lum
eflow
we
mul
tiply
bya
unifo
rmth
ickn
essc
orre
spon
ding
to1/
4of
the
com
-pu
tatio
nald
omai
n.W
eco
mpu
teth
etim
eva
riatio
nof
the
volu
me
flow
over
the
mitr
alrin
g,be
twee
nthe
leafl
ettip
s,an
dac
ross
the
aort
icou
tflow
trac
t(be
low
the
aort
icsi
nk).
The
mod
elna
tura
llygi
ves
the
volu
me
flow
thro
ugh
the
sour
cesa
ndsi
nks
inth
em
odel
,see
e.g.
equa
tion(
3.9)
for
the
flow
thro
ught
heat
rials
ourc
e.T
hetim
eva
riatio
nof
the
flow
thro
ugh
the
atria
lsou
rcea
ndth
eao
rtic
sink
are
plot
ted
in
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
-40
-200
20
40
60
velocity[cm/s]
Fig
ure
4.3:
Vel
ocity
time
profi
leof
the
mea
nvel
ocity
(cm
/s)o
vert
hem
itral
ring
(sol
id)a
ndbe
twee
nthe
leafl
ettip
s(d
ashe
d).
48E
valu
atio
nof
the
2Dm
odel
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
0
20
40
60
80
velocity[cm/s]
Fig
ure
4.4:
Vel
ocity
time
profi
leof
the
mea
nve
loci
ty(c
m/s
)acr
osst
heao
rtic
outfl
owtr
act(
belo
wth
eao
rtic
sink
).
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
-600
-400
-2000
200
sourceflow[cm^3/s]
Fig
ure
4.5:
Flo
wtim
epr
ofile
for
the
flow
(cm
» /s)
thro
ugh
the
mitr
also
urce
(das
hed)
and
the
aort
icsi
nk(s
olid
).
4.1
Sim
ulat
ion
outp
ut49
Fig
ure
4.5.
Inth
efig
ure
vert
ical
dash
edlin
esat
the
end
time
ofea
chhe
artc
ycle
are
also
incl
uded
.
4.1.
3P
ress
ure
field
The
mod
elgi
vest
hepr
essu
refiel
d
543 asou
tput
,and
this
canb
epl
otte
dasa
sur-
face
plot
,aco
ntou
rplo
t,or
ade
nsity
plot
,whe
reth
eco
lour
ofa
cell
corr
espo
nds
toth
eva
lue
ofth
epr
essu
refie
ldat
inth
atce
ll.T
hepr
essu
refie
ldis
nots
moo
thdu
eto
the
num
eric
alm
etho
dsus
ed(P
eski
n,20
01).
Thu
san
aver
agin
gpro
cedu
reis
nece
ssar
ybef
orei
tca
nbe
plot
ted.
The
non-
smoo
thpr
essu
refie
lddo
esno
taf-
fect
the
com
puta
tiono
fthe
velo
city
field
,sin
ceon
lyth
egr
adie
ntof
the
pres
sure
field
isus
edin
the
com
puta
tions
and
this
issm
ooth
.P
lots
ofth
epr
essu
refie
ldis
notv
ery
usef
ul,s
ince
the
pres
sure
field
insi
deth
ehe
arti
sra
rely
know
n.It
ism
ore
usef
ulto
com
pute
the
time
varia
tion
ofth
epr
essu
reat
agi
ven
posi
tion,
e.g.
over
asm
alls
quar
ein
the
atriu
mor
the
vent
ricle
,sim
ilarly
toth
eco
mpu
tatio
noft
hepr
essu
reat
aso
urce
asde
scrib
edin
Sec
tion3
.2.1
.F
igur
e4.6
depi
ctst
hetim
eva
riatio
noft
hepr
essu
reata
squa
rein
the
vent
ricle
,in
the
atriu
m,a
ndat
the
aort
icsi
nk.T
hepr
essu
reat
the
aort
icsi
nkco
rres
pond
sto
p . ineq
uatio
n(3.
10).
Inth
efig
ure
vert
ical
dash
edlin
esar
edr
awn
atth
een
d
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
10
20
30
40
50
60
pressure[mmHg]
Fig
ure
4.6:
Pre
ssur
etim
epr
ofile
for
the
pres
sure
(mm
Hg)
inth
eve
ntric
le(s
olid
),at
rium
(dot
ted)
and
atth
eao
rtic
sink
(das
hed)
.
50E
valu
atio
nof
the
2Dm
odel
0.25
0.5
0.75
11.25
1.5
1.75
2t
[s]
Fig
ure
4.7:
Tim
eva
riatio
nof
vent
ricul
arpr
essu
re(s
olid
)and
vent
ricul
arac
tiva-
tion
(das
hed)
show
non
sam
etim
eax
isan
dsc
aled
toac
hiev
eth
esa
mem
axim
umle
vel.
time
ofea
chhe
artc
ycle
.T
heve
ntric
ular
pres
sure
varia
tion
follo
ws
clos
ely
the
vent
ricul
arac
tivat
ion
curv
e.T
his
isill
ustr
ated
inF
igur
e4.
7w
here
both
the
time
varia
tion
ofth
eve
n-tr
icul
arpr
essu
rean
dth
eve
ntric
ular
activ
atio
nar
esh
own
onth
esa
met
ime
axis
and
scal
edto
achi
eve
the
sam
emax
imum
leve
l.W
eex
ploi
tthi
sfe
atur
ein
Cha
p-te
r6.
4.2
The
scal
ing
fact
or
¼
The
scal
ing
fact
or
isuse
dto
scal
eall
times
and
leng
thst
oac
hiev
enu
mer
ical
stab
ility
ofth
eflu
idso
lver
ata
reas
onab
lecom
puta
tiona
ltime.
Pes
kini
ntro
duce
dth
esc
alin
gfac
tor
inhis
orig
inal
2Dm
odel
(Pes
kin,
1972
a).H
ela
terc
hang
edth
eflu
idso
lver
ina
subs
eque
nt3D
mod
elan
dth
usre
mov
edth
ene
edfo
rsc
alin
g(La
ian
dP
eski
n,20
00;M
cQue
enan
dP
eski
n,20
00).
But
sinc
ethe
2Dm
odel
still
uses
the
old
fluid
solv
er,t
hene
edfo
r
isstill
pres
ent,a
ndit
isim
port
antto
inve
stig
ate
the
influ
ence
of
onthe
sim
ulat
ionr
esul
ts.
We
belie
veth
atth
esc
alin
gw
ith
doesh
ave
anin
fluen
ceon
the
flow
,si
nce
scal
inga
llle
ngth
sand
times
by
( D# )c
orre
spon
dsto
scal
ingt
hevi
scos
ity
4.2
The
scal
ing
fact
or
51
#
Time
0.234
Time
0.234
Time
0.234
Time
0.234
Fig
ure
4.8:
Vel
ocity
field
and
hear
tbou
ndar
yat
s,w
ith
0.0
4,0.
1,0.
3an
d0.
5.
with
#$ ,thu
seffe
ctiv
ely
low
erin
gth
eR
eyno
ldsn
umbe
r.Pes
kin(
1977
)jus
tified
the
intr
oduc
tiono
f
bythe
com
para
tive
phys
iolo
gyof
the
hear
tofd
iffer
entm
am-
mal
s:he
arts
ofdi
ffer
entm
amm
alsa
rero
ughl
ysc
aled
mod
elso
feac
hoth
er.
Thi
sis
the
argu
men
tfor
Pes
kin’s
hypo
thes
istha
tess
entia
lfeat
ures
ofth
ebl
ood
flow
inth
ehe
arta
reno
tver
yse
nsitiv
eto
the
Rey
nold
snum
ber(
Pes
kin,
1977
).To
inve
stig
atet
hein
fluen
ceof
wemak
e6
sim
ulat
ions
vary
ing
only
the
valu
eof
.Thesi
mul
atio
nsar
eper
form
edon
a25
6gr
idw
ith29
952t
ime
step
sper
hear
tcy
cle
and
ato
talc
ompu
tatio
ntim
eof
1he
artc
ycle
.T
he
values
are:
0.04
,0.1
,0.
2,0.
3,0.
4an
d0.
5.
0.0
4w
asth
eor
igin
alva
lue
inth
eco
dew
ere
ceive
d
52E
valu
atio
nof
the
2Dm
odel
#
Time
0.546
Time0.546
Time
0.546
Time0.546
Fig
ure
4.9:
Vel
ocity
field
and
hear
tbou
ndar
yat
s,w
ith
0.0
4,0.
1,0.
3an
d0.
5.
from
Pes
kin.
Toco
mpu
teat
values
clos
erto
1it
isne
cess
aryto
com
pute
ona
finer
grid
.U
nfor
tuna
tely,
we
coul
dno
tcom
pute
afu
llhe
artc
ycle
with
# .
On
a51
2gr
idw
eca
nco
mpu
teap
prox
imat
ely2
/3of
ahe
artc
ycle
.We
belie
veth
ata
#si
mul
atio
ncan
beac
com
plis
hedo
nly
ona
1064
grid
,and
unfo
rtun
atel
ywe
did
not
have
suffi
cien
ttim
efo
rth
is.I
tis
oure
xper
ienc
etha
tit
isno
tpos
sibl
eto
com
pute
afu
llhe
artc
ycle
ona
256
grid
with
a
value
high
erth
an0.
5.N
ote
thou
gh,th
atit
isno
tpos
sibl
eto
com
plet
e2he
artc
ycle
son
a25
6gr
id.
We
plot
the
velo
city
field
from
the
6si
mul
atio
nsto
com
pare
the
over
allp
ictu
re
4.2
The
scal
ing
fact
or
53
#
Time
0.936
Time
0.936
Time
0.936
Time
0.936
Fig
ure
4.10
:Vel
ocity
field
and
hear
tbou
ndar
yat
s,
with
0.0
4,0.
1,0.
3an
d0.
5.
ofth
eve
loci
tyfie
ld.
Fig
ures
4.8–
4.10
show
the
velo
city
field
at
s,
san
d
s,re
spec
tivel
y.E
ach
figur
eco
ntai
nspl
ots
from
the
sim
ulat
ions
with
0.0
4,0.
1,0.
3an
d0.
5.T
heflo
win
Fig
ure
4.8
occu
rsdu
ring
the
fillin
gof
the
vent
ricle
,and
itis
seen
that
the
flow
patte
rnis
gene
rally
the
sam
e,th
ough
the
vort
ices
beco
mem
ore
dis-
tinct
as
increas
es.The
hear
tbou
ndar
ycon
figur
atio
nis
prac
tical
lyth
esa
mea
tall
4
values
.The
flow
inF
igur
e4.
9oc
curs
durin
gat
rials
ysto
leju
stbe
fore
vent
ric-
ular
syst
ole.
Als
ohe
reit
isse
enth
atth
evo
rtic
esar
emor
edi
stin
ctw
ithin
crea
sing
,andth
ein
trav
entr
icul
arve
loci
tyin
crea
sesw
ith
.Furth
erm
ore,
we
seet
hata
s
54E
valu
atio
nof
the
2Dm
odel
increas
es,th
reev
ortic
esar
epr
esen
tinth
eve
ntric
le.T
his
isve
rydi
fficu
ltto
see
at
0.0
4.E
ven
thou
ghth
eve
loci
tyfie
ldis
diff
eren
tinsi
deth
ehe
art,t
hehe
art
boun
dary
ispr
actic
ally
the
sam
efor
all4
values
.T
heflo
win
Fig
ure
4.10
occu
rsat
the
end
ofth
ehe
artc
ycle
.W
ese
etha
tat
0.0
4it
ispo
ssib
leto
seeo
nein
trav
entr
icul
arvo
rtex
,w
hile
at
0.1
,0.3
and
0.5
two
vort
ices
arep
rese
nt.N
ote
that
avo
rtex
isfo
rmed
atth
eao
rtic
outfl
owtr
acta
saco
nseq
uenc
eofth
esi
mpl
ege
omet
ryof
this
trac
t.T
heco
mpa
rison
ofve
loci
tyfie
ldsi
ndic
ates
that
the
orig
inal
choi
ceof
0.04
isno
topt
imal
,sin
ceat
0.0
4th
evo
rtex
stru
ctur
eis
notc
aptu
redv
ery
wel
l.To
furt
here
luci
date
the
influ
ence
of
weplo
tpre
ssur
etim
epr
ofile
s,ve
loci
tytim
epr
ofile
sand
sour
ceflo
wtim
epr
ofile
sfro
mth
esi
mul
atio
nsw
ith
0.0
4,0.
1an
d0.
5.F
igur
e4.
11sh
ows
the
pres
suret
ime
profi
lefo
rthe
vent
ricle
,atr
ium
and
aort
icsi
nk.T
heve
loci
tycu
rves
for
0.1
and
0.5
areq
uite
sim
ilar,
whi
lefo
r
0.0
4th
em
axim
umle
velf
orth
eve
ntric
ular
curv
eis
too
low
and
the
max
imum
leve
lat
the
aort
icsi
nkis
too
high
.We
belie
veth
atth
elo
wer
max
imum
leve
loft
heao
rtic
sink
pres
sure
isre
late
dto
the
form
atio
nof
the
vort
exin
the
aort
icou
tflow
trac
tas
seen
inF
igur
e4.
10.
Fig
ure
4.12
show
sth
eve
loci
tytim
epr
ofile
atth
em
itral
ring
and
betw
eent
hem
itral
leafl
ettip
s,w
hile
Fig
ure
4.13
show
sth
eve
loci
tytim
epr
ofile
acro
ssth
eao
rtic
sink
.F
rom
Fig
ure
4.12
we
seet
hatt
heve
loci
tyov
erth
em
itral
ring
isno
t
#
0.2
0.4
0.6
0.8
t[s]
10
20
30
40
50
60
pressure[mmHg]
0.2
0.4
0.6
0.8
t[s]
10
20
30
40
50
60
pressure[mmHg]
0.2
0.4
0.6
0.8
t[s]
10
20
30
40
50
60
pressure[mmHg]
Fig
ure4
.11:
Pre
ssur
etim
epr
ofile
fort
hepr
essu
re(m
mH
g)in
the
vent
ricle
(sol
id),
atriu
m(d
otte
d)an
dat
the
aort
icsi
nk(d
ashe
d),w
ith
(le
ft),
#
(mid
dle)
and
(r
ight
).
4.2
The
scal
ing
fact
or
55
#
0.2
0.4
0.6
0.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
0.2
0.4
0.6
0.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
0.2
0.4
0.6
0.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
Fig
ure
4.12
:V
eloc
itytim
epr
ofile
ofth
em
ean
velo
city
(cm
/s)o
ver
the
mitr
alrin
g(s
olid
)and
betw
eent
hele
aflet
tips
(das
hed)
,with
(le
ft),
#
(mid
dle)
and
(r
ight
).
#
0.2
0.4
0.6
0.8
t[s]
0
20
40
60
80
100
velocity[cm/s]
0.2
0.4
0.6
0.8
t[s]
0
20
40
60
80
100
velocity[cm/s]
0.2
0.4
0.6
0.8
t[s]
0
20
40
60
80
100
velocity[cm/s]
Fig
ure
4.13
:Vel
ocity
time
profi
leof
the
mea
nvel
ocity
(cm
/s)a
cros
sthe
aort
icou
tflow
trac
tbel
owth
eao
rtic
sink
,with
(le
ft),
# (m
iddl
e)an
d
(r
ight
).
56E
valu
atio
nof
the
2Dm
odel
affe
cted
very
muc
hby
incr
easi
ng
value:
peak
valu
esar
eal
mos
tthe
sam
eand
the
time
varia
tion
issi
mila
r,th
ough
mor
eos
cilla
tions
are
seen
as
increa
ses.
Not
eth
atfo
r
0.5
the
velo
city
over
the
mitr
alrin
gis
zero
atap
prox
imat
ely
º
sbe
twee
nthe
two
forw
ard
peak
s.A
sw
esh
alls
eein
the
com
paris
onbe
twee
nthe
mod
elan
dMR
mea
sure
men
tsinC
hapt
er5,
this
isa
desi
rabl
efea
ture
.Fo
rth
eve
loci
tybe
twee
nthe
leafl
ettip
sth
etim
eva
riatio
nis
sim
ilar,
whi
leth
efo
rwar
dpe
akve
loci
tiesa
rehi
gher
with
incr
easi
ng
value.
For
0.5
we
see
ane
gativ
eve
loci
tybe
twee
nthe
two
forw
ard
peak
s,an
dfo
ral
l
values
we
seea
nega
tive
velo
city
peak
follo
win
gth
ese
cond
forw
ard
peak
.The
latte
ris
rela
tedt
oth
ecl
osur
eoft
hem
itral
valv
e.F
igur
e4.1
3sho
ws
that
the
gene
rala
ortic
velo
city
profi
leis
sim
ilarf
ordi
ffer
ent
values
.As
seen
fort
hem
itral
velo
city
the
osci
llatio
nsar
emor
epro
noun
cedw
ithin
crea
sing
value
and
the
peak
valu
ein
crea
ses.
Gen
eral
ly,w
ese
etha
tthe
scal
ingf
acto
r
doesh
ave
anin
fluen
ceon
the
flow
patte
rn.It
does
notc
hang
ethe
over
allfl
owpa
ttern
,but
loca
ldiff
eren
cesa
rese
en.
The
intr
aven
tric
ular
velo
city
incr
ease
swith
incr
easi
ng
valuea
ndth
evo
rtex
for-
mat
ion
ism
ore
dist
inct
.F
urth
erm
ore,
the
max
imum
pres
sure
sach
ieved
inth
eve
ntric
lean
dat
the
aort
icsi
nkar
eal
soaf
fect
ed.
Whe
nth
em
odel
isus
edfo
rph
ysio
logi
calin
vest
igat
ions
the
user
isin
tere
sted
inas
shor
tasi
mul
atio
ntim
eas
poss
ible
whi
lest
illob
tain
ing
reas
onab
lycor
rect
resu
lts.A
shor
tsim
ulat
ion
time
requ
iresa
smal
lval
ueof
.Asim
ulat
ion
with
0.5
ispo
ssib
leon
a25
6(o
rfine
r)gr
id,w
hile
asi
mul
atio
nw
ith
0.1
can
run
ona
128
grid
with
2995
2tim
est
eppe
rhea
rtcy
cle,
and
asi
mul
atio
nw
ith
ca
nru
non
a64
grid
with
2808
time
step
sper
hear
tcyc
le.
Bas
edon
our
com
paris
onsb
etw
eens
imul
atio
nsw
ithdi
ffer
ent
values
and
cons
ider
ings
imul
atio
ntim
es,w
ebe
lieve
that
0.1
isa
good
choi
ce.A
t
0.1
the
pres
sure
time
varia
tion
and
the
velo
city
time
profi
les
are
clos
eto
thos
eat
0.5
,and
the
flow
patte
rnin
side
the
hear
tissi
mila
rto
the
flow
patte
rnse
enat
0.5
.In
Tho
rup
(199
7)it
was
conc
lude
dtha
t
0.0
4w
asa
good
choi
ce,
butt
hein
vest
igat
iont
here
was
perf
orm
edw
ithth
em
odel
befo
reth
ete
ther
ingo
fth
ehe
artw
asim
plem
ente
d.It
turn
sou
ttha
tthe
teth
erin
gdec
reas
esth
est
abili
tyof
the
mod
el.
4.3
Per
iodi
cbo
unda
ryco
nditi
ons
Inth
em
odel
the
hear
tispl
aced
ina
com
puta
tiona
ldom
ainw
ithpe
riodi
cbou
ndar
yco
nditi
ons.
Thi
sm
eans
that
ifth
ehe
artm
oves
past
one
boun
dary
itw
illap
pear
atth
eop
posi
tebo
unda
ry.A
nex
ampl
eofs
ucha
situ
atio
ncan
bese
enin
Fig
ure
4.1
4.3
Per
iodi
cbo
unda
ryco
nditi
ons
57
Time0.156
Time0.312
Time0.468
Time0.624
Time0.780
Time0.936
Fig
ure
4.14
:Hea
rtbo
unda
ryat
six
equi
dist
antti
mes
durin
ga
hear
tcyc
lefr
omth
ere
fere
nces
imul
atio
n(
½ )an
dfr
oma
sim
ulat
ion
with
ala
rge
com
puta
tiona
ldo-
mai
n(
¾ ).F
ield
ofvi
ewis
iden
tical
toth
eco
mpu
tatio
naldo
mai
noft
here
fere
nce
sim
ulat
ion.
58E
valu
atio
nof
the
2Dm
odel
at
#
s.F
urth
erm
ore,
ifth
edi
men
sion
ofth
ehe
arte
xcee
dsth
edo
mai
ndi
men
sion
the
hear
twill
over
lap
itsel
f.To
dete
rmin
ethe
influ
ence
ofth
epe
riodi
cbo
unda
ryco
nditi
onsw
em
ake
asi
mul
atio
nw
here
the
hear
tis
imm
erse
din
ala
rge
com
puta
tiona
ldom
ain
(on
a51
2gr
id),
but
whe
real
lpa
ram
eter
valu
esde
term
inin
gthe
initi
alse
tup
ofth
ehe
art(
incl
udin
gth
esi
ze)a
reid
entic
alto
ourr
efer
ence
sim
ulat
ion.
InF
igur
e4.
14th
ehe
artb
ound
aryc
onfig
urat
ionf
rom
the
sim
ulat
ion
with
the
larg
eco
mpu
tatio
naldo
mai
nand
from
the
refe
renc
esim
ulat
ioni
sde
pict
ed.W
ese
ea
smal
ldev
iatio
nbe
twee
nthe
two
hear
tbou
ndar
ycon
figur
atio
n.T
here
isha
rdly
any
diff
eren
cebe
twee
nthe
pres
sure
time
profi
less
eeni
nF
igur
e4.
15or
betw
een
the
velo
city
time
profi
les
atth
em
itral
ring
and
betw
eent
hele
aflet
sdep
icte
din
Fig
ure
4.16
.F
igur
e4.
17sh
ows
the
velo
city
time
profi
lefo
rth
eao
rtic
outfl
owtr
act,
whi
leF
igur
e4.
18sh
ows
the
flow
time
profi
lesf
orth
em
itral
sour
ceflo
wan
dth
eao
rtic
sink
flow
.In
both
ofth
esefi
gure
swe
find
smal
ldis
crep
anci
esbe
twee
nthe
resu
ltsfr
omth
ere
fere
nces
imul
atio
nand
from
the
sim
ulat
ionw
itha
larg
eco
mpu
tatio
nal
dom
ain.
Fro
mF
igur
e4.1
8w
ese
etha
tthe
aort
icsi
nkop
ensa
ndcl
oses
atth
esa
me
time
inth
etw
osi
mul
atio
nsan
dth
atth
epe
akin
flow
thro
ugh
the
sour
cean
dth
e
larg
edo
mai
nre
fere
nce
0.2
0.4
0.6
0.8
t[s]
10
20
30
40
50
60
pressure8mmHg]
0.2
0.4
0.6
0.8
t[s]
10
20
30
40
50
60
pressure[mmHg]
Fig
ure
4.15
:P
ress
uret
ime
profi
les
for
the
pres
sure
(mm
Hg)
inth
eve
ntric
le(s
olid
),at
rium
(dot
ted)
and
atth
eao
rtic
sink
(das
hed)
from
asi
mul
atio
nw
itha
larg
eco
mpu
tatio
naldo
mai
n(le
ft)an
dfr
omth
ere
fere
nces
imul
atio
n(rig
ht).
4.3
Per
iodi
cbo
unda
ryco
nditi
ons
59
larg
edo
mai
nre
fere
nce
0.2
0.4
0.6
0.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
0.2
0.4
0.6
0.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
Fig
ure4
.16:
Vel
ocity
time
profi
lefo
rthe
mea
nvel
ocity
(cm
/s)o
vert
hem
itral
ring
and
betw
eent
hele
aflet
tips
from
asi
mul
atio
nwith
ala
rge
com
puta
tiona
ldom
ain
(left)
and
from
the
refe
renc
esim
ulat
ion(
right
).
larg
edo
mai
nre
fere
nce
0.2
0.4
0.6
0.8
t[s]
0
20
40
60
80
velocity[cm/s]
0.2
0.4
0.6
0.8
t[s]
0
20
40
60
80
velocity[cm/s]
Fig
ure
4.17
:Vel
ocity
time
profi
lefo
rth
em
eanv
eloc
ity(c
m/s
)acr
osst
heao
rtic
outfl
owtr
actb
elow
the
aort
icsi
nkfr
oma
sim
ulat
ion
with
ala
rge
com
puta
tiona
ldo
mai
n(le
ft)an
dfr
omth
ere
fere
nces
imul
atio
n(rig
ht).
60E
valu
atio
nof
the
2Dm
odel
larg
edo
mai
nre
fere
nce
0.2
0.4
0.6
0.8
t[s]
-600
-400
-2000
200
sourceflow[cm^3/s]
0.2
0.4
0.6
0.8
t[s]
-600
-400
-2000
200
sourceflow[cm^3/s]
Fig
ure
4.18
:Flo
wtim
epr
ofile
sfor
the
flow
(cm
» /s)t
hrou
ghth
em
itral
sour
cean
dao
rtic
sink
from
asi
mul
atio
nw
itha
larg
eco
mpu
tatio
nald
omai
nan
dfr
omth
ere
fere
nces
imul
atio
n.
peak
outfl
owfr
omth
esi
nkis
the
sam
e.W
eco
nclu
deth
atre
sults
from
the
hear
tmod
elar
eno
tinfl
uenc
edsi
gnifi
cant
lyby
the
perio
dic
boun
dary
cond
ition
s.H
owev
er,
for
visu
alre
ason
sitw
ould
bede
sira
blei
fth
eco
mpu
tatio
nald
omai
nw
asa
little
bit
larg
erto
allo
wth
ehe
art
tost
ayin
side
the
dom
ain
durin
ga
full
sim
ulat
ion.
Thi
sw
ould
mea
nth
atth
eco
mpu
tatio
nssh
ould
bepe
rfor
med
eith
eron
the
sam
egrid
size
butw
itha
slig
htly
smal
lerh
eart
(giv
ing
aco
arse
rreso
lutio
noft
hehe
art)
oron
afin
ergr
idw
ithth
ehe
arts
ize
iden
tical
toth
esi
zeat
the
coar
segr
id(g
ivin
gap
prox
imat
ely
. times
asm
uch
com
puta
tiona
ltime,
beca
useth
enu
mbe
rofg
ridpo
ints
hast
obe
apo
wer
of2)
.S
ince
the
resu
ltsar
eno
taffe
cted
byth
epe
riodi
cbo
unda
ryco
nditi
ons,
we
have
chos
enno
tto
chan
geth
egr
idsi
zeor
the
hear
tdim
ensi
on.
4.4
Tet
herin
gof
the
hear
t
Inth
eor
igin
alm
odel
the
hear
tfloa
tedf
reel
yin
the
com
puta
tiona
ldom
ain,
andt
his
resu
ltedi
nno
n-ph
ysio
logi
calm
ovem
ents
,esp
ecia
llydu
ring
the
ejec
tion
phas
ew
hen
atr
ansl
atio
nwou
ldbe
gin
and
even
tual
lym
ove
the
hear
tthro
ught
hebo
und-
ary.
Fig
ure
4.19
show
sth
eve
loci
tyfie
ldan
dhe
artb
ound
aryc
onfig
urat
iona
t4
4.4
Teth
erin
gof
the
hear
t61
Time
0.624
Time
0.936
Time
1.248
Time
1.872
Fig
ure
4.19
:Vel
ocity
field
and
hear
tbou
ndar
ycon
figur
atio
natf
ourt
imes
durin
gtw
ohe
artc
ycle
sfro
ma
sim
ulat
ionw
ithno
teth
erin
goft
hehe
art.
times
durin
g2
hear
tcyc
leso
fasi
mul
atio
nwhe
reth
ehe
artis
nott
ethe
red.
Inth
efir
stfr
ame
inF
igur
e4.
19it
isse
enth
atth
ehe
artm
oves
thro
ugh
the
uppe
rbou
ndar
y.W
hile
from
the
seco
ndfr
ame
(at
s),w
hich
isat
the
end
ofth
efir
sthe
artc
ycle
,it
isse
enth
atth
ehe
arth
asbe
gun
tom
ove
thro
ught
helo
wer
boun
dary.
Thi
sde
velo
pmen
tisev
encl
eare
rinth
eth
irdfr
ame.
Inth
ela
stfr
ame,
atth
een
dof
the
seco
ndhe
artc
ycle
,the
hear
tisag
aini
ntac
tins
idet
hefie
ldof
view
.T
hus
durin
gtw
ohe
artc
ycle
sth
ehe
artm
oves
adi
stan
ceco
rres
pond
ing
toth
esi
zeof
the
com
puta
tiona
ldom
ain.
Tore
stra
inth
ehe
artf
rom
mov
ing,
we
teth
ered
itto
ase
tof
fixed
poin
tsas
desc
ribed
inS
ectio
n3.3
.7.I
nF
igur
e4.
20th
eve
loci
tyfie
ldan
dhe
artb
ound
ary
confi
gura
tionf
rom
asi
mul
atio
nwhe
reth
ehe
artis
teth
ered
ares
how
nfo
rthe
sam
e
62E
valu
atio
nof
the
2Dm
odel
Time
0.624
Time0.936
Time
1.248
Time1.872
Fig
ure
4.20
:Vel
ocity
field
and
hear
tbou
ndar
ycon
figur
atio
natf
ourt
imes
durin
gtw
ohe
artc
ycle
sfro
mth
ere
fere
nces
imul
atio
n.
times
asin
Fig
ure
4.19
Fig
ure
4.20
show
sth
atth
ete
ther
ing
wor
ksef
ficie
ntly
inke
epin
gth
ehe
art
inpl
ace.
The
hear
tisal
low
edto
mov
ea
little
,bu
tnot
too
muc
h.F
urth
erm
ore,
we
see
that
the
flow
field
inth
ehe
arti
sno
tver
ydi
ffer
entf
rom
that
ina
hear
tw
ithno
teth
erin
g,ex
cept
durin
gou
tflow
thro
ught
heao
rtic
sink
(the
first
fram
eat
s)
.In
the
sam
efra
me
we
seet
hatw
ithth
ete
ther
ingt
hein
trav
entr
icul
arflo
wfie
ldis
dom
inat
edby
anou
tflow
jet
from
the
vent
ricle
tow
ard
the
aort
icsi
nk,w
hile
inth
esi
mul
atio
nwith
outt
ethe
ringt
hein
trav
entr
icul
arflo
wfie
ldse
ems
calm
eran
dex
hibi
tsa
vort
exbe
twee
nthe
ante
riorw
alla
ndth
em
itral
leafl
ets.
The
teth
erin
ghas
also
intr
oduc
edos
cilla
tions
inbo
thth
eat
rium
and
the
vent
ricle
.The
osci
llatio
nsof
the
atriu
mca
nbe
seen
whe
nco
mpa
rings
ucce
ssive
plot
sof
the
4.4
Teth
erin
gof
the
hear
t63
note
ther
ing
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
10
20
30
40
50
60
pressure[mmHg]
refe
renc
e
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
10
20
30
40
50
60
pressure[mmHg]
Fig
ure4
.21:
Pre
ssur
etim
epr
ofile
fort
hepr
essu
re(m
mH
g)in
the
vent
ricle
(sol
id),
atriu
m(d
otte
d)an
dat
the
aort
icsi
nk(d
ashe
d)fr
oma
sim
ulat
ionw
ithno
teth
erin
gof
the
hear
t(to
p)an
dfr
omth
ere
fere
nces
imul
atio
n(bo
ttom
).
hear
twal
l.T
his
isno
tsee
nin
the
plot
ssh
own
here
beca
useth
eyar
eto
ofa
rapa
rtin
time.
The
osci
llatio
nsof
the
vent
ricle
are
best
seen
whe
nan
anim
atio
nof
the
pict
ures
ispe
rfor
med
.To
furt
here
xam
inet
hein
fluen
ceof
the
teth
erin
gwe
com
pare
dthe
time
varia
-tio
nof
the
pres
sure
,velo
city
and
flow
from
the
two
sim
ulat
ions
.
64E
valu
atio
nof
the
2Dm
odel
note
ther
ing
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
-40
-200
20
40
60
velocity[cm/s]
refe
renc
e
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
-40
-200
20
40
60
velocity[cm/s]
Fig
ure
4.22
:Vel
ocity
time
profi
lefo
rth
em
eanv
eloc
ity(c
m/s
)att
hem
itral
ring
(sol
id)a
ndbe
twee
nthe
leafl
ettip
s(d
ashe
d)fr
oma
sim
ulat
ion
with
note
ther
ing
ofth
ehe
art(
top)
and
from
the
refe
renc
esim
ulat
ion(
botto
m).
Fig
ure
4.21
show
sth
etim
eva
riatio
nof
the
pres
sure
.Itis
seen
that
the
max
-im
umle
velf
orth
eve
ntric
ular
pres
sure
issl
ight
lyhi
gher
with
outt
hete
ther
ing,
whi
leth
em
axim
umpr
essu
rele
velf
orth
eao
rtic
sink
ishi
gher
inth
ese
cond
cy-
cle
with
teth
erin
g.A
part
from
that
,the
time
varia
tion
ofth
epr
essu
resa
reve
rysi
mila
r.
4.4
Teth
erin
gof
the
hear
t65
note
ther
ing
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
-600
-400
-2000
200
sourceflow[cm^3/s]
refe
renc
e
0.25
0.5
0.75
11.25
1.5
1.75
2t[s]
-600
-400
-2000
200
sourceflow[cm^3/s]
Fig
ure
4.23
:F
low
time
profi
lefo
rth
eflo
w(c
m
» /s)
thro
ugh
the
mitr
also
urce
(das
hed)
and
the
aort
icsi
nk(s
olid
)fr
oma
sim
ulat
ion
with
note
ther
ing
ofth
ehe
art(
top)
and
from
the
refe
renc
esim
ulat
ion(
botto
m).
InF
igur
e4.
22th
eve
loci
tytim
epr
ofile
sfo
rth
em
itral
ring
and
betw
eent
hele
aflet
tips
ares
how
n.E
spec
ially
inth
efir
stcy
cle
diff
eren
cesb
etw
eent
heve
loci
typr
ofile
sex
ist:
the
peak
forw
ard
velo
city
inth
epr
ofile
sin
the
sim
ulat
ion
with
note
ther
ing
iscl
early
high
erth
anin
the
sim
ulat
ion
with
teth
erin
g.F
urth
erm
ore,
the
shap
eoft
heve
loci
typr
ofile
durin
gth
efir
stpe
akis
wid
erw
ithte
ther
ingt
han
66E
valu
atio
nof
the
2Dm
odel
with
out.
Inth
ese
cond
cycl
eth
eve
loci
typr
ofile
sfr
omth
etw
osi
mul
atio
nsar
em
uch
mor
esi
mila
r.F
igur
e4.
23sh
ows
the
time
varia
tiono
fthe
flow
thro
ught
hem
itral
sour
cean
dao
rtic
sink
.T
hepe
akou
tflow
issl
ight
lyla
rger
inth
esi
mul
atio
nw
ithte
ther
ing
than
inth
esi
mul
atio
nw
ithno
teth
erin
g,bu
tthe
shap
esof
the
profi
les
are
sim
i-la
r.In
the
sim
ulat
ion
with
teth
erin
gthe
flow
thro
ugh
the
mitr
also
urce
beco
mes
nega
tive,
indi
catin
gth
atbl
ood
isflo
win
gba
ckth
roug
hthe
mitr
also
urce
.Thi
sis
nota
nun
phys
iolo
gica
lfeat
ure,
sinc
ethe
pres
surei
nsid
ethe
hear
tcan
besu
chth
atbl
ood
leav
esth
roug
hthe
pulm
onar
yvei
ns(t
hepu
lmon
aryv
eins
have
nova
lves
topr
even
tthi
s).
The
intr
oduc
tiono
ftet
herin
gpre
vent
sthe
hear
tfrom
mov
ing
too
muc
hw
ith-
outs
igni
fican
tlych
angi
ngth
eflo
wpa
ttern
.The
time
varia
tion
ofth
epr
essu
reis
alm
ostth
esa
me,
whi
leso
med
iffer
ence
sare
seen
inth
eve
loci
tyan
dso
urce
flow
profi
les.
Fort
heve
loci
typr
ofile
sthe
mai
ndi
ffer
ence
sare
inth
efir
sthe
artc
ycle
.S
ince
prev
entin
gth
ehe
artf
rom
mov
ing
arou
ndin
the
com
puta
tiona
ldom
ain
isa
maj
orim
prov
emen
t,we
conc
lude
that
the
teth
erin
gof
the
hear
tissu
cces
sful
,ev
enth
ough
som
echa
nges
are
intr
oduc
edin
the
resu
ltsw
eha
vem
adet
his
part
ofth
em
odel
.
4.5
The
Win
dkes
selm
odel
The
Win
dkes
selm
odel
for
the
afte
rload
ofth
ehe
artw
asin
trod
uced
inJa
cobs
en(1
999)
and
eval
uate
din
both
Jaco
bsen
(199
9)an
dA
dele
rand
Jaco
bsen
(200
1).
We
incl
ude
here
asu
mm
aryo
fth
atev
alua
tion
toco
mpl
etet
hede
scrip
tiona
ndev
alua
tiono
four
2Dm
odel
.T
heW
indk
esse
lmod
elw
asin
trod
uced
toim
prov
eth
eou
tflow
cond
ition
sat
the
aort
icsi
nk.
Orig
inal
ly,
the
outfl
oww
asde
term
ined
bya
linea
rre
sist
ance
mod
el(s
imila
rto
the
onef
orth
eat
rials
ourc
e),bu
tthi
sga
vean
unre
alis
ticpr
essu
re
onse
tsys
t.en
dsy
st.
u! JG µ^LG
[ms]
[ms]
[ms
() ][m
s
() ]
G
atriu
m0.
347
0.42
21/
601/
542.
00
vent
ricle
0.44
70.
547
1/60
1/15
4.9
1.07
pap.
mus
c.0.
447
0.54
71/
601/
154.
90.
107
Tabl
e4.
1:T
imin
gs,r
ate
cons
tant
s,lim
iting
valu
esan
din
itial
valu
esfo
rth
eac
ti-va
tion
func
tions
for
the
dog
hear
t(M
cQue
enet
al.,
1982
).
4.5
The
Win
dkes
selm
odel
67
10
20
30
40
50
60
10
20
30
40
50
60
Fig
ure4
.24:
Initi
alge
omet
ryof
the
dog
hear
t,com
pare
with
Fig
ure3
.7.N
ote
that
the
hear
tisno
ttet
here
dto
any
fixed
poin
ts.
time
profi
leat
the
sink
,whe
reth
epr
essu
rew
asal
mos
tcla
mpe
datt
hepr
escr
ibed
cons
tant
pres
sure
inth
eao
rta.
Toin
vest
igat
ethe
influ
ence
ofth
eW
indk
esse
lm
odel
conn
ecte
dtoth
eao
rtic
sink
,we
mad
eaco
mpa
rison
betw
eena
sim
ulat
ion
with
Pes
kin’s
orig
inal
mod
elan
da
sim
ulat
ion
whe
reth
eon
lym
odifi
catio
nin
-tr
oduc
edis
the
Win
dkes
selm
odel
.P
eski
n’sor
igin
alm
odel
isba
sedo
na
dog’
she
art,
i.e.n
one
ofth
eim
prov
emen
tspr
esen
tede
arlie
rin
this
thes
isto
mak
eth
em
odel
hum
anw
ere
impl
emen
ted.
The
leng
thof
ado
g’s
hear
tcyc
lew
asta
ken
tobe
0.67
5s.
The
para
met
ersfo
rth
eac
tivat
ion
func
tion
are
liste
din
Tabl
e4.
1an
dth
ein
i-tia
lge
omet
ryfo
rth
edo
g’s
hear
tis
show
nin
Fig
ure
4.24
(com
pare
thes
ewith
Tabl
e3.
1an
dF
igur
e3.
7fo
rth
em
odifi
edm
odel
).N
ote
that
the
leng
thsc
ales
are
nott
hesa
mei
nth
em
odel
ofa
dog’
she
art(
Fig
ure
3.7)
and
inth
em
odifi
edm
odel
(Fig
ure
4.24
):fo
rth
edo
g’s
hear
tthe
leng
thof
the
com
puta
tiona
ldom
ain
corr
espo
ndsto
7.35
cm,w
hile
inou
rmod
ified
mod
elit
corr
espo
ndsto
13cm
.F
igur
e4.
25sh
ows
pres
sure
sfrom
asi
mul
atio
nof
ado
g’s
hear
twith
outt
heW
indk
esse
lmod
elov
er3
cycl
es.I
tis
appa
rent
that
the
mod
elop
erat
esat
apr
es-
sure
whi
chis
only
abou
t1/2
ofth
atof
the
phys
iolo
gica
llyco
rrec
tpre
ssur
efor
ado
g,w
hich
corr
espo
ndsto
appr
oxim
atel
y1/3
ofth
ehu
man
case
.Not
eth
atth
epr
essu
reatt
heao
rtic
sink
ishe
ldat
anal
mos
tcon
stan
tleve
lwhe
nthe
sink
isop
en.
68E
valu
atio
nof
the
2Dm
odel
0.5
11.5
2
Time [s]
0
10
20
30
40
Fig
ure
4.25
:P
ress
uret
ime
profi
ledu
ring
thre
ehe
artc
ycle
sfo
rth
epr
essu
re(m
mH
g)in
the
vent
ricle
(sol
id),
atriu
m(d
otte
d)an
dat
the
aort
icsi
nk(d
ashe
d)fr
oma
sim
ulat
ionw
ithou
tthe
Win
dkes
selm
odel
.
0.5
11.5
2
Time [s]
0
10
20
30
40
Fig
ure
4.26
:P
ress
uret
ime
profi
ledu
ring
thre
ehe
artc
ycle
sfo
rth
epr
essu
re(m
mH
g)in
the
vent
ricle
(sol
id),
atriu
m(d
otte
d),a
tthe
aort
icsi
nk(lo
ngda
shed
)an
dth
elo
adpr
essu
re
p tyQ (sho
rtda
shed
)from
asi
mul
atio
nwith
the
Win
dkes
sel
mod
el.
4.6
Dis
trib
uted
sour
cesa
ndsi
nks
69
How
ever
,the
atria
lpre
ssur
eisap
prox
imat
elya
tthe
phys
iolo
gica
lleve
l.F
igur
e4.
26sh
ows
the
pres
sure
profi
lesc
orre
spon
ding
toth
osei
nF
igur
e4.
25,
butw
ithth
eou
tflow
thro
ught
heao
rtic
sink
dete
rmin
edby
the
Win
dkes
selm
odel
.T
hefig
ure
also
incl
udes
the
time
varia
tion
ofth
elo
adpr
essu
re
p"tyQ intr
oduc
edin
equa
tion(
3.10
).T
heva
lues
ofth
eW
indk
esse
lpar
amet
ersa
read
apte
dfro
mC
appe
lloet
al.(
1995
)as
follo
ws:
Sin
ceth
em
odel
oper
ates
ata
pres
sure
of1/
3of
the
phys
iolo
gica
lleve
lsus
edin
Cap
pello
etal
.(19
95),
the
valu
esfo
rth
ere
-si
stan
cesa
redi
vide
dby
3w
hile
the
com
plia
ncei
sm
ultip
lied
by3.
Thu
sth
eW
indk
esse
lpar
amet
ersa
re:
w tyQ ) ,w tyQ .
0.8
mm
Hg/
(cm
» /s)
and
| 1.5
cm
» /mm
Hg.
The
intr
oduc
tion
ofth
eW
indk
esse
lmod
eldo
esno
tcha
nget
heve
ntric
ular
pres
sure
sign
ifica
ntly,
but
the
pres
sure
atth
eao
rtic
sink
isno
long
ercl
ampe
dto
anal
mos
tcon
stan
tval
ue.
Rat
her,
itva
ries
ina
mor
eph
ysio
logi
calr
ealis
ticw
ay.
The
time
inte
rval
durin
gw
hich
the
aort
icsi
nkis
open
isla
rger
whe
nth
eW
indk
esse
lmod
elis
intr
oduc
ed.
The
resu
ltspr
esen
tedh
ere
show
that
the
pres
sure
profi
leat
the
aort
icou
t-flo
wtr
acti
sco
nsid
erab
lyim
prov
edth
roug
hthe
useo
faW
indk
esse
lmod
elra
ther
than
alin
earr
esis
tanc
emod
elco
uple
dto
aco
nsta
ntpr
essu
rere
servo
ir.Ja
cob-
sen
(199
9)al
sore
port
sth
atit
does
not
mak
ea
sign
ifica
ntdi
ffer
ence
whe
ther
ano
n-lin
earc
ompl
ianc
e(i.e
.apr
essu
rede
pend
entco
mpl
ianc
e)is
used
.Bas
edon
thes
ein
vest
igat
ions
we
ther
efor
ech
ose
toim
plem
enta
Win
dkes
selm
odel
with
aco
nsta
ntco
mpl
ianc
ein
the
hear
tmod
el.
Afte
rim
plem
entin
gall
ofou
rm
odifi
catio
nsin
the
mod
el,w
ead
just
edth
eW
indk
esse
lpar
amet
ersb
ased
ona
com
paris
onof
sim
ulat
ions
and
MR
data
.We
ende
dwith
the
follo
win
gpa
ram
e-te
rcho
ice:
w tyQ ) #
mm
Hg/
(cm
» /s),
w tyQ . m
mH
g/(c
m
» /s)a
nd| cm
» /mm
Hg.
4.6
Dis
trib
uted
sour
cesa
ndsi
nks
The
wor
kpr
esen
tedin
this
sect
ionw
aspe
rfor
med
byJa
kobs
enan
dN
iss
(200
0)ba
sedo
nan
unpu
blis
hedn
oteb
yP
eski
n(19
98).
We
incl
udeh
erea
brie
fsum
mar
yof
Jako
bsen
and
Nis
s’s
(200
0)w
ork.
Whe
ndis
trib
uted
sour
cesa
ndsi
nksw
erei
ntro
duce
din
the
hear
tmod
el(P
eski
n,19
77)t
hest
anda
rdco
ntin
uity
equa
tione
xpre
ssin
ginc
ompr
essi
bilit
y
< ;2
was
mod
ified
to
< ;2 ?
70E
valu
atio
nof
the
2Dm
odel
whe
re
? (dep
endi
ngon
both
time
and
spac
e)ex
pres
sesth
edi
strib
utio
nof
the
sour
cesa
ndsi
nks,
asde
scrib
edin
Sec
tion3
.2.1
.Thi
sw
asdo
new
ithou
tmod
ifyin
gth
em
omen
tum
equa
tion(
the
Nav
ier-
Sto
kese
quat
ion)
78 9 2 9 : 2 ;< 2= < 5 *<.2:>
even
thou
ghth
ede
rivat
ion
ofth
eN
avie
r-S
toke
sequ
atio
nis
base
don
the
cont
inu-
ityeq
uatio
n.A
ssum
ptio
nsab
outt
hem
omen
tum
atth
eso
urce
sand
sink
sha
veto
bem
ade
toco
rrec
tlyde
rive
the
mom
entu
meq
uatio
nwith
the
mod
ified
form
ofth
eco
n-tin
uity
equa
tion.
At
asi
nkit
isas
sum
edth
atflu
idle
avin
gth
em
odel
take
sits
mom
entu
mw
ithit.
At
aso
urce
one
can
eith
eras
sum
ethat
fluid
ente
rsth
em
odel
with
nove
loci
ty(a
ndth
usw
ithno
mom
entu
mbe
caus
ethe
fluid
isin
com
pres
sibl
ean
dho
mog
eneo
us)or
that
fluid
ente
rsw
ithth
esa
mev
eloc
ityas
the
fluid
atth
eso
urce
.The
form
eras
sum
ptio
nwas
prop
osed
byP
eski
n(19
98),
whi
leJa
kobs
enan
dN
iss
(200
0)pr
opos
edth
ela
tter.
The
diff
eren
cebe
twee
nder
ivin
gth
em
o-m
entu
meq
uatio
nbas
edon
the
form
eror
the
latte
rass
umpt
ionc
once
rnin
gflui
den
terin
gthe
mod
elis
inth
ete
rmde
scrib
ingt
hech
ange
inm
omen
tum
caus
edby
the
sour
cesa
ndsi
nks.
Thi
sdi
ffer
ence
isde
scrib
edbe
low
,w
ere
fert
oJa
kobs
enan
dN
iss
(200
0)fo
rde
tails
.To
beab
lede
scrib
einfl
owan
dou
tflow
sepa
rate
lythe
follo
win
gar
ede
fined
?1 ? if
? ¶
if
? D
? ( if
? z? if
?
Thu
s
?1 desc
ribes
the
dist
ribut
ion
ofin
flow
thro
ugh
sour
ces,a
nd
? ( desc
ribes
the
dist
ribut
ion
ofou
tflow
thro
ughs
inks
.W
ithth
eas
sum
ptio
nsth
atflu
iden
terin
gth
em
odel
has
nom
omen
tum
and
fluid
leav
ing
the
mod
elta
kes
itsm
omen
tum
with
it,th
ech
ange
inm
omen
tum
rela
tedt
oth
eso
urce
sand
sink
sis
@ ¿ 7? (C·
whe
re
· isa
fixed
area
.Usi
ngth
isin
the
deriv
atio
nof
the
mom
entu
meq
uatio
ngi
ves
78 9 2 9 : 2 ;< 2= < 5 *<.2:> ?12
Thu
sthe
mom
entu
meq
uatio
nsis
nott
hest
anda
rdN
avie
r-S
toke
sequ
atio
n.
4.7
Sim
ulat
ion
over
seve
ralh
eart
cycl
es71
With
the
assu
mpt
ions
that
fluid
ente
rsth
em
odel
with
the
sam
evel
ocity
(i.e.
sam
emom
entu
m)a
sth
eflu
idat
the
sour
cean
dflu
idle
avin
gth
em
odel
take
sits
mom
entu
mw
ithit,
the
chan
gein
mom
entu
mre
late
dto
the
sour
cesa
ndsi
nks
beco
mes
@ ¿ 7?C·
Usi
ngth
iste
rmin
the
deriv
atio
nof
the
mom
entu
meq
uatio
nlea
dsto
the
follo
win
geq
uatio
n:
78 9 2 9 : 2 ;< 2= < 5 *<.2:>
Thi
sis
the
stan
dard
Nav
ier-
Sto
kese
quat
ion,
even
thou
ghth
eco
ntin
uity
equa
tion
was
noti
nst
anda
rdfo
rm.
Thi
ssh
ows
that
the
useo
fthe
stan
dard
Nav
ier-
Sto
kese
quat
iont
oget
herw
ithth
eno
n-st
anda
rdcon
tinui
tyeq
uatio
ndoe
snot
lead
toin
cons
iste
ncie
sinth
em
odel
,bu
tis
base
don
sens
ible
assu
mpt
ions
conc
erni
ngth
eflu
iden
terin
gthe
mod
el.
The
sede
rivat
ions
are
nots
peci
ficto
the
hear
tmod
el,b
utar
eva
lidfo
ran
arbi
-tr
ary
fluid
dyna
mic
alm
odel
with
dist
ribut
edso
urce
sand
sink
s.In
the
case
ofth
ehe
artm
odel
,itis
nece
ssar
ytodi
scus
swhi
chof
the
two
assu
mpt
ions
conc
erni
ngth
eflu
iden
terin
gthe
mod
elis
corr
ect.
Sin
cebl
ood
ente
rsth
eat
rium
durin
gbo
thdi
asto
lean
dsy
stol
e,w
ebe
lieve
itw
ould
beun
phys
iolo
gica
ltoas
sum
ethat
bloo
den
ters
the
hear
twith
the
sam
evel
ocity
asth
ebl
ood
alre
adyi
nth
eat
rium
has.
Itis
mor
eco
rrec
tto
assu
met
hatb
lood
ente
rsw
itha
low
erve
loci
ty.
Thu
sw
ebe
lieve
that
none
ofth
est
ated
assu
mpt
ions
abou
tflui
den
terin
gthe
mod
elar
eco
rrec
t,but
the
assu
mpt
iont
hatt
heflu
iden
terin
gthe
mod
elha
sno
velo
city
mig
htbe
clos
erto
the
phys
iolo
gica
llyco
rrec
tsitu
atio
n.Ja
kobs
enan
dN
iss
(200
0)ex
amin
eth
edi
ffer
ence
sinsi
mul
atio
nre
sults
(par
-tic
ular
lyth
eve
loci
tyfie
ld)
betw
eent
hetw
oeq
uatio
nsof
mom
entu
mde
scrib
edab
ove.
The
yfin
dth
atth
erea
reon
lym
inor
diff
eren
cesin
the
velo
city
field
,pro
b-ab
lydu
eto
the
fact
that
the
velo
city
ofth
eflu
idat
the
sour
ces,w
here
fluid
ente
rsth
em
odel
,issm
all.
We
conc
lude
that
even
thou
ghdi
ffer
enta
ssum
ptio
nsab
outth
eflu
iden
terin
gthe
mod
elle
adto
diff
eren
tmom
entu
meq
uatio
ns,th
isdo
esno
tlea
dto
maj
ordi
ffer
ence
sinth
esi
mul
atio
nres
ults
.Thu
swe
dono
tfind
itne
cess
aryto
incl
ude
the
new
form
ofth
em
omen
tum
equa
tioni
nth
em
odel
.
4.7
Sim
ulat
ion
over
seve
ralh
eart
cycl
es
Whe
nth
em
odel
isus
edfo
rph
ysio
logi
cale
xper
imen
tsan
dco
mpa
rison
sitis
im-
port
antto
dete
rmin
eatw
hich
poin
tthe
mod
elbe
com
espe
riodi
c.T
his
can
bede
-
72E
valu
atio
nof
the
2Dm
odel
term
ined
bym
akin
ga
sim
ulat
iono
verm
any
hear
tcyc
lesa
ndex
amin
ing
atw
hich
poin
tthe
mod
elbe
com
es(p
ract
ical
ly)p
erio
dicf
rom
one
hear
tcyc
leto
the
next
.W
hen
dete
rmin
ingt
heap
prop
riate
hear
tcyc
lenu
mbe
r,it
isal
sone
cess
ary
toco
nsid
erth
eco
mpu
tatio
ntim
e.Fo
rou
rref
eren
cesi
mul
atio
ntw
ohe
artc
ycle
sta
keap
prox
imat
ely3
1/2
hour
sto
com
pute
ona
700
MH
zP
entiu
mIII
proc
esso
r.W
hen
perf
orm
ing
phys
iolo
gica
lexp
erim
ents
itis
our
expe
rienc
etha
t,be
caus
eof
the
com
plex
ityof
the
mod
el,m
any
sim
ulat
ions
are
need
edbe
fore
the
corr
ect
setti
ngof
the
mod
elis
achi
eved
.Thu
sash
ortc
ompu
tatio
ntim
eis
very
impo
rtan
t.W
em
ake
aru
nw
ithth
ere
fere
ncem
odel
over
15he
artc
ycle
s.It
isev
iden
tfr
omth
ere
sults
that
the
outp
utbe
com
es(p
ract
ical
ly)p
erio
dic
with
inth
efir
st4
hear
tcyc
les.
We
dono
tsee
that
the
outp
utbe
com
espe
rfec
tlype
riodi
c,no
teve
nw
ithin
15he
artc
ycle
s,e.
g.th
eam
plitu
deso
fesp
ecia
llyth
eve
loci
tyan
dflow
time
profi
lesv
ary
alit
tle.
Fig
ure
4.27
show
sth
epr
essu
retim
epr
ofile
s,F
igur
e4.
28th
eve
loci
tytim
epr
ofile
atth
em
itral
ring,
Fig
ure
4.29
the
velo
city
time
profi
leac
ross
the
aort
icou
tflow
trac
tand
Fig
ure
4.30
the
sour
cesfl
owpr
ofile
,in
allc
ases
for
the
first
4he
artc
ycle
sofo
urre
fere
nces
imul
atio
n.F
rom
Fig
ure
4.27
itis
seen
that
the
pres
sure
inth
eat
rium
issl
ight
lydi
ffer
ent
inth
efir
stcy
cle
than
inth
efo
llow
ing,
the
vent
ricul
arpr
essu
reis
prac
tical
lyth
esa
me,
whi
leth
em
axim
umle
velo
fthe
pres
surea
tthe
aort
icsi
nkin
crea
sess
light
lydu
ring
the
four
hear
tcyc
les.
The
shap
eoft
hem
itral
velo
city
profi
le(F
igur
e4.
28)
isdi
ffer
enti
nth
efir
stcy
cle
than
inth
efo
llow
ing
ones
:the
first
peak
ishi
gher
and
the
shap
edur
ing
that
peak
isw
ider
.F
urth
erm
ore,
from
the
seco
ndcy
cle
and
onw
ard
thre
efo
rwar
dpe
aksa
repr
esen
t.T
hefir
stpe
aksl
ight
lyde
crea
ses
durin
gth
ela
stth
reec
ycle
splo
tted.
The
diff
eren
cebe
twee
nthe
first
cycl
ean
dth
efo
llow
ing
cycl
esar
ecl
osel
yrel
ated
toth
efa
ctth
atth
em
odel
begi
nsw
ithth
eflu
idbe
ing
atre
st.
InF
igur
e4.
29it
isse
enth
atth
eve
loci
typr
ofile
acro
ssth
eao
rtic
trac
tfro
mth
eth
irdan
dfo
urth
hear
tcyc
lesa
reve
rysi
mila
r.In
the
first
two
cycl
esth
epe
akve
loci
tyis
high
er,a
ndth
esh
apeo
fthe
curv
eis
slig
htly
diff
eren
tinth
efir
stcy
cle
com
pare
dto
the
follo
win
gcy
cles
.T
heflo
wth
roug
hth
eat
rials
ourc
efro
mth
ese
cond
,third
and
four
thhe
artc
ycle
isve
rysi
mila
r,w
hile
min
ordi
ffer
ence
sexi
stbe
twee
nthe
first
cycl
ean
dth
efo
llow
ing
cycl
es.F
orth
eflo
wth
roug
hth
eao
rtic
sink
we
seet
hesa
met
ende
ncie
sasf
orth
eve
loci
tyac
ross
the
aort
icou
tflow
trac
t:th
eth
irdan
dfo
urth
hear
tcyc
les
are
very
sim
ilar.
The
shap
eof
the
curv
ein
the
seco
ndcy
cle
issi
mila
rto
the
shap
ein
the
third
and
four
th,b
utth
epe
akou
tflow
isla
rger
and
for
the
first
cycl
eth
esh
apes
how
ssm
alld
iffer
ence
scom
pare
dto
the
follo
win
gcy
cles
and
the
peak
outfl
owis
larg
erth
anin
the
third
and
four
thcy
cles
.W
eco
nclu
deth
atit
isre
ason
ablet
ota
keth
ere
sults
from
the
seco
ndhe
art
4.7
Sim
ulat
ion
over
seve
ralh
eart
cycl
es73
0.5
11.5
22.5
33.5
t[s]
0
10
20
30
40
50
60
pressure[mmHg]
Fig
ure
4.27
:Pre
ssur
etim
epr
ofile
durin
g4
hear
tcyc
lesf
orth
epr
essu
re(m
mH
g)in
the
vent
ricle
(sol
id),
atriu
m(d
otte
d)an
dat
the
aort
icsi
nk(d
ashe
d).
0.5
11.5
22.5
33.5
t[s]
-40
-200
20
40
velocity[cm/s]
Fig
ure
4.28
:V
eloc
itytim
epr
ofile
durin
g4
hear
tcyc
les
for
the
mea
nve
loci
ty(c
m/s
)att
hem
itral
ring.
74E
valu
atio
nof
the
2Dm
odel
0.5
11.5
22.5
33.5
t[s]
-200
20
40
60
80
velocity[cm/s]
Fig
ure
4.29
:V
eloc
itytim
epr
ofile
durin
g4
hear
tcyc
les
for
the
mea
nve
loci
ty(c
m/s
)acr
osst
heao
rtic
outfl
owtr
actb
elow
the
aort
icsi
nk.
0.5
11.5
22.5
33.5
t[s]
-600
-400
-2000
sourceflow[cm^3/s]
Fig
ure
4.30
:Flo
wtim
epr
ofile
durin
g4
hear
tcyc
lesf
orth
eflo
w(c
m» /s
)thr
ough
the
mitr
also
urce
(das
hed)
and
the
aort
icsi
nk(s
olid
).
4.8
Con
clus
ions
75
cycl
ew
hen
phys
iolo
gica
lexp
erim
ents
are
carr
ied
outw
ithth
em
odel
.T
his
isba
sedo
nth
ego
odco
mpa
rison
betw
eent
here
sults
from
the
seco
ndhe
artc
ycle
andt
hefo
llow
ing
third
andf
ourt
hcy
cles
and
onth
ene
edto
run
man
ysi
mul
atio
nsw
hen
expe
rimen
tsar
eca
rrie
dou
t.T
hus
choo
sing
the
seco
ndan
dno
tthe
third
cycl
ere
duce
sthe
com
puta
tiona
ltime.
4.8
Con
clus
ions
We
have
seen
that
the
scal
ingf
acto
r
hasan
influ
ence
onth
eflo
wpa
ttern
insi
deth
ehe
art.
Itdo
esno
tcha
nget
heov
eral
lflow
patte
rn,b
utlo
cald
iffer
ence
sare
seen
.Com
parin
gthe
sim
ulat
ion
resu
ltsfr
omsi
mul
atio
nsw
ithdi
ffer
ent
values
and
taki
ngth
eco
mpu
tatio
nalti
me
into
acco
unt,w
eha
vech
osen
0.1
asa
good
choi
ce.T
heflo
wpa
ttern
rese
mbl
esw
ellt
heflo
wpa
ttern
seen
inth
e
0.5
sim
ulat
ion
and
the
time
varia
tion
ofpr
essu
rean
dm
eanv
eloc
ities
are
sim
ilart
oth
oseo
btai
neda
t
0.5
.W
eha
vein
vest
igat
edth
ein
fluen
ceof
the
perio
dic
boun
dary
cond
ition
sby
embe
ddin
gthe
hear
tina
larg
erco
mpu
tatio
naldo
mai
nand
com
parin
git
with
our
refe
renc
esim
ulat
ion.
Thi
sco
mpa
rison
show
edon
lym
inor
diff
eren
cesb
etw
een
the
sim
ulat
ions
.Thu
sw
eco
nclu
deth
atth
epe
riodi
cbo
unda
ryco
nditi
onsd
ono
tin
fluen
ceth
ere
sults
and
the
size
ofth
eco
mpu
tatio
nald
omai
nin
rela
tion
toth
ech
osen
hear
tdim
ensi
onis
reas
onab
le.
We
exam
ined
the
cons
eque
nceso
ftet
herin
gthe
hear
ttoa
seto
ffixe
dpo
ints
asde
scrib
edin
Sec
tion3
.3.7
byco
mpa
rings
imul
atio
nsw
ithan
dw
ithou
ttet
herin
g.W
efo
und
that
the
teth
erin
gsuc
cess
fully
prev
ente
dthe
unph
ysio
logi
calm
ovem
ent
ofth
ehe
arta
ndon
lyin
trod
uced
afe
wch
ange
sinth
ere
sults
.W
eha
vesh
owed
that
the
impl
emen
tedW
indk
esse
lmod
elfo
rth
eou
tflow
thro
ught
heao
rtic
sink
impr
oves
the
shap
eof
the
pres
sure
curv
efo
rth
epr
essu
reat
the
aort
icsi
nkw
ithou
tint
rodu
cing
sign
ifica
ntch
ange
sinth
eve
loci
tyfie
ld.
We
have
com
pare
dthe
resu
ltsfr
omth
efir
st15
hear
tcyc
les
ofou
rref
eren
cesi
mul
atio
nand
foun
dth
atit
isac
cept
ablet
ous
ethe
resu
ltsfr
omth
ese
cond
cycl
ein
expe
rimen
tsw
ithth
em
odel
.The
first
cycl
eis
cons
ider
ablyd
iffer
entf
rom
the
follo
win
gcy
cles
,whi
leth
ere
sults
from
the
seco
ndcy
cle
are
clos
eto
thos
efro
mth
efo
llow
ing
cycl
es.
The
inve
stig
atio
nspe
rfor
med
inth
isch
apte
rsho
wth
atth
ere
sults
from
the
seco
ndhe
artc
ycle
ofa
refe
renc
esim
ulat
ionc
ompu
tedo
na
128
grid
with
2995
2tim
est
epsp
erhe
artc
ycle
,with
0.1,
andw
ithal
loth
erpa
ram
eter
sasd
escr
ibed
inC
hapt
er3,
are
the
resu
ltsfr
omth
e2D
hear
tmod
elth
atsh
ould
beta
ken
asre
pres
enta
tive.
Cha
pter
5
Com
parin
gth
e2D
mod
elw
ithan
MR
data
set
The
ultim
ate
eval
uatio
noft
hehe
artm
odel
isa
com
paris
onbe
twee
nmod
elre
sults
and
mea
sure
men
tsper
form
edon
ahu
man
,whi
chin
ourc
asea
reM
Rm
easu
re-
men
ts.
Thi
sch
apte
rispa
rtly
base
don
Jaco
bsen
etal
.(2
001)
and
Ade
lera
ndJa
cobs
en(2
001)
.H
owev
er,
we
have
chan
gedt
hein
itial
geom
etry
and
the
teth
-er
ing
sinc
eth
ere
sults
pres
ente
dinth
em
entio
nedp
aper
s,an
dhe
rew
eco
mpa
reou
rnew
mod
elre
sults
with
the
sam
eMR
data
set.
The
new
initi
alge
omet
ryan
dte
ther
inga
reth
osed
escr
ibed
inS
ectio
n3.3
.7.
The
com
paris
onfa
llsin
totw
opa
rts,
each
repr
esen
tedb
yits
type
ofda
ta:
À Veloc
ityfie
lds
and
hear
twal
lbou
ndar
y.
À Veloc
itypr
ofile
satt
hem
itral
ring
and
atth
eao
rtic
outfl
owtr
act.
5.1
The
MR
data
We
useM
Rda
tafr
oma
stud
yin
Kim
etal
.(19
95)c
ondu
cted
on26
heal
thyv
ol-
unte
ers,
18m
enan
d8
wom
en.T
heir
mea
nag
ew
as25
year
s(in
ara
ngef
rom
21to
30ye
ars)
.The
volu
ntee
rsha
dno
know
nca
rdia
cor
syst
emic
dise
ases
.For
the
com
paris
onw
ithou
rsi
mul
ated
hear
tmod
el,o
neof
the
26se
tsof
MR
data
was
chos
en.S
eeA
ppen
dix
Afo
ran
expl
anat
iono
fth
ete
chni
calit
ieso
fth
eM
Rm
etho
d.
78C
ompa
ring
the
2Dm
odel
with
anM
Rda
tase
t
Inth
eM
Rda
taco
llect
ing
proc
ess,
aco
nven
tiona
lspi
nec
hose
quen
cew
asus
edfo
ra
coro
nalim
ageo
fthe
hear
tfollo
wed
bya
sing
lean
gula
tiont
hrou
ghth
eao
rtic
valv
ean
dth
eap
exof
the
hear
ttoyi
eld
alo
ngax
isvi
ewof
the
left
vent
ricle
.In
Kim
etal
.(19
95)t
helo
ngax
ispl
aneo
fthe
left
vent
ricle
was
chos
enbe
caus
eitco
ntai
nsbo
thth
eao
rtic
outfl
owan
dth
em
itral
inflo
w,a
swel
last
hele
ftve
ntric
lean
dle
ftat
rium
.Fo
rth
eco
mpa
rison
betw
eenM
Rda
taan
dm
odel
,the
long
axis
plan
eis
wel
lsui
ted
sinc
eit
isco
nsis
tent
with
the
com
puta
tiona
lpla
neof
the
2Dm
odel
.Im
agea
cqui
sitio
nwas
pros
pect
ively
trig
gere
dby
the
Rw
ave
ofth
eel
ectr
ocar-
diog
ram
,so
the
first
reco
rded
fram
ew
as8
ms
afte
rthe
Rw
ave.
Thi
sco
rres
pond
sap
prox
imat
elyt
oth
etim
eof
clos
ureo
fthe
mitr
alva
lve.
The
intr
acar
diac
flow
velo
city
mea
sure
men
tswer
eobt
aine
dusi
ngth
est
anda
rdF
LAG
(Flo
wA
djus
tedG
radi
ents
)seq
uenc
e,whi
chis
desc
ribed
inG
roen
andD
ijk(1
987)
.The
reco
rdin
gsw
erec
ondu
cted
ina
squa
re40
cmfie
ldof
view
with
anin
-pl
aner
esol
utio
nof3
.125
mm
0 3.12
5m
mgi
ving
anim
aged
slic
eof
128
0 128
voxe
ls(v
olum
eel
emen
ts).
The
follo
win
gpa
ram
eter
setti
ngsw
ere
used
for
the
flow
mea
sure
men
ts:45
degr
eesfl
ipan
gle,
9.5
ms
echo
time,
7.1
ms
acqu
isiti
ontim
e,an
d10
mm
slic
eth
ickn
ess.
The
velo
city
sens
itivity
was
sett
opr
oduc
eam
axim
umph
ases
hift
ata
velo
city
of
Á 1.25
m/s
to
Á 1.67
m/s
toav
oid
alia
sing
.V
eloc
ityen
codi
ngw
aspe
rfor
med
inal
lthr
eesp
atia
ldire
ctio
nsto
prod
ucet
hree
-di
men
sion
alve
loci
tyin
form
atio
nfr
omea
chvo
xeli
nsid
eth
efie
ldof
view
.T
hetim
ene
eded
for
the
quan
titat
iveve
loci
tyac
quis
ition
s(us
ing
EC
Gga
ting
toav
oid
mot
ion
artif
acts
)was
15to
20m
info
rea
chsu
bjec
t,dep
endi
ngon
the
hear
trat
e.T
hepr
oced
urer
esul
tedi
nth
ere
cord
edM
Rda
tabe
ing
aver
aged
over
768
hear
tbe
ats.
For
the
26ex
amin
edpe
rson
sthe
tem
pora
lreso
lutio
nwas
28m
s
Á 3.8
ms
(mea
n
Á stan
dard
devi
atio
n(S
D))
with
25-3
2fr
ames
perc
ardi
accy
cle.
For
the
spec
ificM
Rse
twith
whi
chth
eco
mpa
rison
was
carr
iedo
ut,t
hete
mpo
ralre
solu
-tio
nw
as36
ms
with
26fr
ames
perc
ardi
accy
cle
givi
nga
tota
llen
gth
ofth
ehe
art
cycl
eof
936
ms.
The
tota
lrec
ordi
ngtim
ew
as11
52m
s,i.e
.app
roxi
mat
ely
#) Âhe
artc
ycle
s.T
heac
cura
cyof
quan
titat
iveflo
wm
easu
rem
entsb
yth
eM
Rph
aset
echn
ique
depe
ndso
nth
esu
ppre
ssio
nof
artif
acts
.T
heve
loci
tym
apsw
ere
corr
ecte
dfor
poss
ible
linea
rpha
seer
rors
resu
lting
from
eddy
curr
ents
inth
em
agne
ticfie
ld.I
nad
ditio
n,m
aski
ngof
rand
omno
isei
nth
est
atio
nary
tissu
ewas
done
.Bot
hpr
oce-
dure
swer
eper
form
edby
ase
mi-a
utom
ateds
oftw
arep
rogr
am,w
hich
isde
scrib
edin
deta
ilin
Wal
kere
tal.
(199
3).
An
MR
data
setc
onsi
stso
fthe
thre
e-di
men
sion
alvel
ocity
info
rmat
ioni
nea
chvo
xela
ndth
em
odul
usim
age
atea
chtim
efr
ame.
The
mod
ulus
imag
eco
ntai
nsin
form
atio
non
the
MR
sign
alam
plitu
de,w
hich
depe
ndse
.g.o
nth
esp
ecifi
ctis
sue
5.2
Vel
ocity
field
s79
and
the
velo
city
ofth
eflo
win
the
voxe
l.T
hem
odul
usim
age
isdi
spla
yeda
sa
grey
-sca
lepi
ctur
ean
dca
n,as
such
,be
used
toro
ughl
yde
term
inet
hein
tern
alan
atom
yand
also
the
mag
nitu
deof
the
flow
.For
the
com
paris
onw
ithth
e2D
hear
tm
odel
we
wer
eonl
yin
tere
sted
inth
ein
-pla
neco
mpo
nent
soft
heve
loci
ty.T
hein
-pl
anev
eloc
ityco
mpo
nent
scan
bedi
spla
yeda
svel
ocity
field
vect
ors.
Pre
sent
ing
the
velo
city
field
plot
supe
rimpo
sedo
nth
eco
rres
pond
ingm
odul
usim
ages
how
sbo
thth
em
agni
tude
and
the
dire
ctio
nof
the
bloo
d.It
isal
sopo
ssib
leat
each
time
fram
eto
extr
actv
eloc
ityin
form
atio
nfr
omon
eor
seve
ralv
oxel
s,an
dth
isca
nbe
used
topl
ota
velo
city
profi
le(v
eloc
ityas
func
tion
oftim
e)at
asp
ecifi
cpl
ace
inth
eim
aged
slic
e.In
the
velo
city
field
plot
we
show
belo
ww
eha
vein
dica
ted
the
voxe
lsfr
omw
hich
velo
city
info
rmat
ion
isex
trac
tedt
oca
lcul
atev
eloc
itytim
epr
ofile
s.
5.2
Vel
ocity
field
s
The
star
tingt
ime
ofth
eM
Rda
tais
diff
eren
tfro
mth
atof
the
sim
ulat
ion.
The
nat-
ural
star
toft
heM
Rda
tais
atth
ecl
osur
eoft
hem
itral
valv
e,w
hile
the
sim
ulat
ion
star
tsin
early
vent
ricul
ardi
asto
le.W
eha
vech
osen
tosh
owal
lres
ults
acco
rdin
gto
the
time
cour
seof
the
sim
ulat
ion,
i.e.s
tart
ing
with
early
vent
ricul
ardi
asto
le,
and
have
shift
edth
eve
loci
tieso
fth
eM
Rm
easu
rem
entsfr
omth
efir
stcy
cle,
ac-
cord
ingl
y.T
hem
atch
ingw
asdo
neby
sync
hron
izin
gthe
time
ofth
eea
rlype
akve
loci
ty(a
lso
calle
dth
eE
peak
)inth
efir
sthe
artc
ycle
inth
eve
loci
tytim
epr
ofile
ofth
ece
ntre
velo
city
atth
em
itral
ring
inth
em
odel
with
the
time
ofth
eE
peak
inth
eve
loci
tytim
epr
ofile
atth
em
itral
ring
inth
eM
Rda
ta.
InF
igur
es5.
1–5.
3th
eve
loci
tyfie
ldis
show
nat
8di
ffer
entt
imes
durin
ga
hear
tcyc
leas
velo
city
vect
orsa
long
with
the
hear
twal
lbou
ndar
y.W
eha
veus
edth
ere
sults
from
the
seco
ndhe
artc
ycle
ofth
em
odel
,so
the
sim
ulat
ion
time
runs
from
sto
# %º s.
Inth
em
odel
,the
hear
twal
lbo
unda
ryis
com
pute
dand
isth
eref
oree
asily
plot
ted.
Inth
eM
Rda
ta,th
ein
terio
roft
hehe
art
boun
dary
(not
the
valv
ele
aflet
s)w
asm
anua
llytr
aced
from
plot
sof
the
velo
city
field
supe
rimpo
sedo
nth
em
odul
usim
age.
The
boun
dary
isgi
ven
aw
idth
equa
ltoth
esp
atia
lreso
lutio
noft
heM
Rm
easu
rem
ents
,i.e.3
.125
mm
.We
emph
asiz
ethat
the
dete
rmin
atio
nof
the
inte
riorb
ound
aryf
rom
the
MR
data
isra
ther
unce
rtai
nan
dth
atit
shou
ldbe
only
take
nas
anin
dica
tion
ofits
posi
tion.
We
begi
nw
itha
desc
riptio
nof
the
resu
ltsin
Fig
ures
5.1–
5.3.
Thi
sw
illgi
vean
intr
oduc
tiont
oth
etim
ese
quen
ceof
ahe
artc
ycle
and
prov
ide
the
basi
sfor
our
com
paris
on.
Ea
rly
mitr
ali
nflo
w.
The
resu
ltsst
artw
ithth
eea
rlym
itral
inflo
w.
The
mitr
al
80C
ompa
ring
the
2Dm
odel
with
anM
Rda
tase
t
MR
sim
ulat
ion
Time R+512 ms
Time1.0660
à ÄÅÇÆÈÉÉ s
Time R+620 ms
Time1.1700
à ÄÅÇÆÅÊ È s
Time R+728 ms
Time1.2740
à ÄÅÇÆËÊÌ s
Fig
ure
5.1:
Vel
ocity
field
and
hear
tbou
ndar
yfro
mM
Rm
easu
rem
ents(l
eft)
and
refe
renc
esim
ulat
ion
(rig
ht)
atth
ree
times
durin
ga
hear
tcyc
lech
arac
teriz
edby
:ea
rlype
akm
itral
inflo
w(t
op),
tem
pora
rym
axim
alan
terio
rvor
tex
inth
eM
Rda
ta(m
iddl
e)an
da
decl
inei
nin
flow
(bot
tom
).
5.2
Vel
ocity
field
s81
MR
sim
ulat
ion
Time R+836 ms
Time1.3910
à ÄÅÇÆÍÎÅ s
Time R+8 ms
Time1.4950
à ÄÅÇÆÌÎÏ s
Time R+80 ms
Time1.5600
ÅÇÆÏÉÈ
s
Fig
ure
5.2:
Vel
ocity
field
and
hear
tbou
ndar
yfro
mM
Rm
easu
rem
ents(l
eft)
and
refe
renc
esim
ulat
ion
(rig
ht)
atth
ree
times
durin
ga
hear
tcyc
lech
arac
teriz
edby
:at
rials
ysto
lein
sim
ulat
ion(
top)
,clo
sure
ofm
itral
valv
e(m
iddl
e)an
dea
rlyao
rtic
outfl
ow(b
otto
m).
82C
ompa
ring
the
2Dm
odel
with
anM
Rda
tase
t
MR
sim
ulat
ion
Time R+152 ms
Time1.6380
à ÄÅÇÆÉÍÐ s
Time R+332 ms
Time1.8200
à ÄÅÇÆÐËÈ s
Fig
ure
5.3:
Vel
ocity
field
and
hear
tbou
ndar
yfro
mM
Rm
easu
rem
ents(l
eft)
and
refe
renc
esim
ulat
ion
(rig
ht)
attw
otim
esdu
ring
ahe
artc
ycle
char
acte
rized
by:
peak
aort
icou
tflow
inM
Rda
ta(t
op)
and
the
time
just
befo
recl
osur
eof
aort
icva
lve
inM
Rda
taan
dju
staf
tera
ortic
sink
clos
urei
nsi
mul
atio
n(bo
ttom
).
inflo
wty
pica
llyha
stw
ope
aksd
urin
ga
hear
tcyc
le,t
heea
rlype
ak,a
lso
refe
rred
toas
the
Epe
ak,a
ndth
ela
teat
rialp
eak,
refe
rred
toas
the
Ape
ak.(
Inth
ese
cond
hear
tcyc
leof
the
sim
ulat
ion,
thou
gh,th
reep
eaks
ares
een.
We
disc
ussth
isbe
low
).In
Fig
ure
5.1
atth
etim
eof
the
Epe
ak(1
.066
s),a
wid
ein
flow
jeti
sse
enin
both
MR
data
and
sim
ulat
ion.
Inth
eM
Rda
tath
eje
tis
wid
eran
dre
ache
sdee
per
into
the
vent
ricle
com
pare
dto
that
ofth
esi
mul
atio
n.In
both
the
MR
data
and
sim
ulat
ion,
the
begi
nnin
gof
form
atio
nof
avo
rtex
isse
enat
the
leafl
ettip
s,th
ean
terio
rvor
tex
iscl
early
seen
inth
esi
mul
atio
n.(T
hele
aflet
sare
nots
how
nin
the
MR
data
).In
the
sim
ulat
ion
avo
rtex
isal
read
ypre
sent
deep
erin
the
vent
ricle
,
5.2
Vel
ocity
field
s83
this
vort
exw
asfo
rmed
inth
efir
sthe
artc
ycle
and
notd
estr
oyed
.M
idd
iast
ole.
Inth
eM
Rda
taan
dsim
ulat
iona
t1.1
70s
(Fig
ure5
.1),
the
inflo
wha
sdec
reas
edan
din
the
sim
ulat
ions
emi-c
losu
reoft
hem
itral
valv
eis
seen
.Als
o,th
eflo
wpa
ttern
inth
eM
Rda
tasu
gges
tstha
tthe
mitr
alva
lve
isin
ase
mic
lose
dpo
sitio
n.In
the
MR
data
ala
rge
vort
exha
sdev
elop
edin
the
ante
riorp
arto
fth
eve
ntric
le,w
hile
ther
eis
only
ahi
ntof
apo
ster
iorv
orte
x.T
heflo
wpa
ttern
issi
mila
rin
the
uppe
rpar
toft
heve
ntric
lein
the
sim
ulat
ion:
two
vort
ices
arec
lear
lyse
en,a
larg
ean
terio
rvor
tex
and
asm
alle
rpos
terio
rone
.B
utin
the
low
erpa
rtof
the
vent
ricle
,clo
seto
the
apex
,a
third
vent
ricle
isal
sopr
esen
t.A
t1.
274
s(F
igur
e5.
1)th
ein
flow
hasf
urth
erde
clin
edin
both
MR
data
and
sim
ulat
ion,
but
the
vort
ices
ares
tillp
rese
ntat
low
eran
gula
rvel
ocity
.In
the
MR
data
the
post
erio
rvo
rtex
isha
rdly
dete
ctab
le.
La
tefil
ling
.A
t1.
391
s(F
igur
e5.
2)th
ein
flow
thro
ugh
the
mitr
alrin
gin
the
sim
ulat
ion
hasi
ncre
ased
,and
the
third
peak
(the
Ape
ak)o
fth
em
itral
inflo
wis
onits
way
.T
his,
ofco
urse
,sho
uld
have
been
the
seco
ndpe
ak,a
poin
ttha
tis
disc
usse
dinS
ectio
n5.3
.In
cont
rast
,onl
ya
hint
ofth
eA
peak
isse
enin
the
MR
data
.The
pres
ence
ofth
ean
terio
rvor
tex
inth
eM
Rda
tais
rath
erva
guea
tthi
stim
e,w
hile
itis
still
clea
rin
the
sim
ulat
ion,
alon
gw
ithbo
thth
epo
ster
iora
ndap
ical
vort
ices
.No
post
erio
rvor
tex
isde
tect
ablei
nth
eM
Rda
ta.
Mitr
alv
alv
ecl
osu
re.
Inbo
thth
eM
Rda
taan
dsi
mul
atio
n,th
em
itral
valv
eis
clos
ing
at1.
495
s(F
igur
e5.
2).
The
mitr
alva
lve
isno
tsho
wn
expl
icitl
yin
the
MR
data
,but
the
flow
patte
rnre
veal
stha
tthe
valv
eis
clos
ing.
Inth
eM
Rda
taa
rota
tiona
lflow
patte
rnfil
ling
upth
eve
ntric
leis
disc
erni
ble,
buti
tis
nota
clea
rvo
rtex
,w
hile
inth
esi
mul
atio
nbo
thth
ean
terio
rand
post
erio
rvor
tices
are
still
pres
ent.T
heap
ical
vort
exis
hard
lyde
tect
able
.Ve
ntr
icu
lars
ysto
le.T
heea
rlyao
rtic
outfl
owpa
ttern
near
the
outfl
owtr
acta
t1.
560
s(F
igur
e5.
2)is
sim
ilari
nth
eM
Rda
taan
dth
esi
mul
atio
n.D
eepe
rinth
eve
ntric
leth
eve
loci
tiesa
rehi
gher
inth
esi
mul
atio
ntha
nis
appa
rent
from
the
MR
data
.Fur
ther
mor
e,the
flow
from
mos
toft
heve
ntric
lein
the
sim
ulat
ionc
onve
rges
tow
ard
the
outfl
owtr
act.
Bas
edon
the
posi
tion
ofth
ehe
artw
allo
fth
eve
ntric
lein
the
sim
ulat
ion
(whe
nco
mpa
redt
oth
epr
evio
ustim
efr
ame)
,itis
seen
that
the
vent
ricul
arsy
stol
eha
sse
tin.
Thi
sis
notc
lear
from
the
hear
twal
lpo
sitio
nin
the
MR
data
,but
sinc
eaor
ticou
tflow
ispr
esen
tthe
vent
ricul
arsy
stol
emus
thav
est
arte
d.In
Fig
ure
5.3
at1.
638
sth
eao
rtic
outfl
owve
loci
tyha
sinc
reas
edin
both
MR
data
ands
imul
atio
n.F
rom
the
velo
city
profi
less
how
nin
the
follo
win
gse
ctio
nit
can
bese
enth
atth
isis
the
time
ofpe
akao
rtic
outfl
owve
loci
tyac
cord
ingt
oth
eM
Rda
taan
dju
staf
terp
eakv
eloc
ityac
cord
ingt
osi
mul
atio
n.T
heco
nver
ging
ofth
eve
ntric
ular
flow
tow
ard
the
outfl
owtr
actf
rom
mos
tpar
tsof
the
vent
ricle
isst
illm
ostc
lear
lyse
enin
the
sim
ulat
ion.
Inth
esi
mul
ated
flow
field
the
artifi
cial
84C
ompa
ring
the
2Dm
odel
with
anM
Rda
tase
t
term
inat
iono
fth
eou
tflow
trac
tcau
sesa
back
flow
alon
gth
ean
terio
rsid
eof
the
trac
tand
the
form
atio
nof
avo
rtex
.E
nd
ofs
ysto
le.T
heflo
wpa
ttern
at1.
820
s(F
igur
e5.
3)sh
ows
that
the
aort
icva
lve
isst
illop
enin
the
MR
data
,but
the
aort
icsi
nkis
clos
edin
the
sim
ulat
ion.
Fro
mth
eve
loci
typr
ofile
sin
the
follo
win
gse
ctio
nit
isse
enth
atth
isis
just
befo
reva
lve
clos
urea
ccor
ding
toth
eM
Rda
taan
dju
staf
terv
alve
clos
urei
nth
esi
mul
a-tio
n.T
hevo
rtex
seen
inth
eou
tflow
trac
toft
hesi
mul
atio
nis
ofco
urse
anar
tifac
tof
the
mod
el,a
ndth
eve
ntric
ular
vort
exse
enin
the
sim
ulat
ionw
illaf
fect
the
flow
patte
rnof
the
follo
win
ghe
artc
ycle
,asw
esa
win
the
first
time
fram
es(F
igur
e5.1
)of
this
hear
tcyc
le.
The
abov
eco
mpa
rison
betw
eent
heve
loci
tyfie
ldsf
rom
the
MR
mea
sure
men
tsan
dth
esi
mul
atio
nsh
ows
are
ason
ablea
gree
men
tinth
ege
nera
lflow
patte
rn.A
diff
eren
cein
velo
city
field
sis
seen
inth
ew
idth
ofth
ein
flow
jet(
from
atriu
mto
vent
ricle
).T
heflo
wpa
ttern
sals
odiff
erw
ithre
spec
ttoth
esi
zeof
the
vent
ricle
,the
vent
ricle
inth
esi
mul
atio
nbei
ngla
rger
than
inth
eM
Rda
ta.T
his
isa
cons
eque
nce
ofth
em
odel
bein
g2D
.The
entir
eex
pans
iont
akes
plac
ein
the
cons
ider
edpl
ane,
and
topr
oduc
ethe
right
outfl
owth
eve
ntric
leha
sto
expa
ndm
ore.
The
larg
erve
ntric
lem
ight
bean
expl
anat
ionf
orth
ecl
eare
rfor
mat
ion
ofa
post
erio
rvor
tex
and
the
mor
epr
omin
enta
nter
iorv
orte
xin
the
sim
ulat
ion.
The
flow
field
sfr
omth
esi
mul
atio
nar
eaf
fect
edby
the
sim
ple
aort
icva
lve
(sin
kco
uple
dthr
ough
adi
ode
toa
Win
dkes
selm
odel
).To
geth
erw
ithth
ete
rmi-
natio
noft
heou
tflow
trac
t,th
eao
rtic
sink
resu
ltsin
the
form
atio
nofa
vort
exat
the
sink
,whi
chof
cour
seis
nots
eeni
nth
eM
Rda
ta.F
urth
erm
ore,t
hete
ther
ingo
fthe
mod
elhe
arth
asel
imin
ated
the
exce
ssive
mov
emen
tof
the
hear
t,bu
tit
hasa
lso
intr
oduc
edos
cilla
tions
inbo
thth
eat
rium
and
the
vent
ricle
(see
Sec
tion4
.4)a
ndth
isaf
fect
sthe
flow
field
inan
unph
ysio
logi
calw
ay.T
heos
cilla
tions
ofth
eat
rium
can
bese
enw
hen
com
parin
gsuc
cess
ivepl
ots
ofth
ehe
artw
all.
Thi
sis
nots
een
inth
epl
ots
show
nhe
rebe
caus
ethey
are
too
fara
part
intim
e.T
heos
cilla
tions
ofth
eve
ntric
lear
ebe
stse
enw
hen
anan
imat
iono
fthe
pict
ures
ispe
rfor
med
.Inad
ditio
n,th
ete
ther
ingr
estr
icts
the
mod
elto
mov
emen
tsalo
ngth
eax
isof
the
out-
flow
trac
t.To
geth
erw
ithth
edy
nam
icso
fthe
mitr
alrin
g(v
aria
tion
inle
ngth
),th
ere
stric
tion
due
toth
ete
ther
ingr
esul
tsin
the
vent
ricle
axis
bein
gal
mos
talig
ned
with
the
axis
ofth
eou
tflow
trac
tins
tead
ofha
ving
the
angl
ewe
intr
oduc
edin
the
initi
alge
omet
ry.T
hepa
pilla
rym
uscl
e(m
odel
ledb
ya
link
betw
eena
virt
ualp
oint
insi
deth
ele
ftve
ntric
lean
da
poin
ton
the
post
erio
rhea
rtw
all)
resu
ltsin
anun
phys
iolo
gica
lsh
apeo
fth
eap
exof
the
left
vent
ricle
(see
Fig
ure
5.2
at
#
s).
Thi
sdi
stor
tion
affe
ctst
heflo
wpa
ttern
and
mak
esth
eag
reem
entle
ssgo
od.
Inou
rear
lierr
epor
tedc
ompa
rison
sbet
wee
nmod
elan
dM
Rda
ta(A
dele
rand
5.3
Vel
ocity
time
profi
les
85
Jaco
bsen
,2001
;Jac
obse
neta
l.,20
01)w
em
oved
the
teth
erin
gpoi
ntof
the
pap-
illar
ym
uscl
eto
the
vent
ricul
arw
all
upal
ong
the
post
erio
rwal
lto
prev
entt
hefo
rmat
ion
ofth
epo
ster
iorv
orte
x.B
utth
isre
sulte
din
ave
ryst
rong
(and
unph
ys-
iolo
gica
l)co
ntra
ctio
nof
the
post
erio
rwal
land
ina
very
asym
met
ricm
ovem
ent
ofth
em
itral
valv
e.R
egar
ding
the
wor
kde
scrib
edin
this
thes
is,it
was
deci
ded
that
the
mitr
alva
lve
mov
emen
tand
the
cont
ract
iono
fth
eve
ntric
lew
ere
mor
eim
port
antt
han
the
form
atio
nof
the
post
erio
rvor
tex.
The
refo
rew
eno
wte
ther
the
papi
llary
mus
clet
oa
poin
talit
tlebi
tup
alon
gth
ean
terio
rwal
l,as
desc
ribed
inS
ectio
n3.
3.7.
Thi
sde
cisi
onw
aspa
rtic
ular
lym
ade
with
the
sim
ulat
ions
for
Cha
pter
6in
min
d.G
ener
ally,
we
have
obta
ined
age
omet
ryw
hich
ism
ore
corr
ectc
ompa
redt
oa
hum
anle
ftve
ntric
leth
anth
atin
the
orig
inal
mod
el,b
utth
ege
omet
ryis
still
sim
-pl
ified
.W
eha
veac
hiev
eda
corr
ectle
ngth
scal
eoft
hele
ftve
ntric
leby
chan
ging
the
diam
eter
ofth
em
itral
ring
and
the
diam
eter
ofth
eou
tflow
trac
t.T
hech
ange
sof
the
cros
slin
ksim
prov
edth
eco
ntro
lof
the
mov
emen
tof
the
vent
ricle
wal
ldur
ing
the
hear
tcyc
le,t
husc
hang
ingt
heal
mos
tcirc
ular
form
ofth
eve
ntric
lein
the
orig
inal
mod
elto
the
mor
ere
alis
ticel
onga
tedf
orm
inth
epr
esen
tm
odel
.W
ithth
isse
tof
mod
ifica
tions
,whi
chm
ustb
eco
nsid
ered
asm
inim
al,w
eof
cour
sedi
dno
tobt
ain
(nor
did
we
expe
ctto
)acl
osem
atch
betw
eent
hesi
mul
ated
and
mea
sure
dhea
rtw
allg
eom
etrie
sdur
ing
the
vario
usph
ases
ofth
ehe
artc
ycle
.
5.3
Vel
ocity
time
profi
les
Fro
mth
eve
loci
tyfie
lds
ofbo
thth
eM
Rda
taan
dsi
mul
atio
nwe
cane
xtra
ctve
loc-
ities
atch
osen
spat
ialp
ositi
ons.
For
the
MR
data
we
only
have
26tim
esov
era
hear
tcyc
lew
here
we
can
extr
actv
eloc
ityin
form
atio
n,w
here
asfo
rth
esi
mul
atio
nw
eha
ve29
952t
imes
.A
part
from
the
purp
oseo
fcom
parin
gthe
velo
city
atch
osen
spat
ialp
ositi
ons,
the
velo
city
profi
les
can
also
beus
edto
eval
uate
the
timin
gof
the
sim
ulat
ion
com
pare
dto
MR
mea
sure
men
ts.W
eha
veno
exac
tmea
sure
men
tsof
the
time
ofon
seta
ndth
edu
ratio
nof
vent
ricul
aran
dat
rials
ysto
le(s
eeS
ectio
n3.3
.6)d
ueto
the
limite
dqu
ality
ofth
eE
CG
durin
gth
eM
Rsc
an,b
utby
com
parin
gthe
velo
city
profi
lest
hetim
ing
para
met
ersc
anbe
eval
uate
d.T
hein
-an
dou
tflow
velo
city
profi
lesf
rom
the
MR
data
are
obta
ined
byav
er-
agin
gth
eve
loci
tyov
er2
fixed
voxe
lsat
the
cent
reof
the
left
vent
ricul
arou
tflow
trac
tatt
hele
velo
fth
eao
rtic
valv
e,an
dov
era
squa
reof
4fix
edvo
xels
near
the
cent
reof
the
mitr
alrin
g.T
hese
voxe
lare
indi
cate
din
the
velo
city
field
plot
sin
86C
ompa
ring
the
2Dm
odel
with
anM
Rda
tase
t
Fig
ures
5.1–
5.3.
At
the
outfl
owtr
actt
heve
rtic
alco
mpo
nent
isre
cord
ed(s
eere
f-er
ence
fram
ein
Fig
ure5
.1),
whi
leat
the
mitr
alrin
git
isth
eco
mpo
nentm
akin
gan
angl
eof3
0
, with
the
vert
ical
.Bec
ause
ofth
esm
allm
ovem
ento
fth
ehe
artw
all,
noat
tem
ptis
mad
eto
trac
eth
em
ovem
ents
ofth
ehe
artin
choo
sing
thes
evox
els
and
thus
the
velo
citie
sar
eco
mpu
tedw
ithre
spec
ttoa
refe
renc
efra
me
fixed
insp
ace.
The
extr
acte
dvel
ociti
esar
e’m
ean
cent
re’v
eloc
ities
,whi
char
ea
com
bi-
natio
nof
am
eanv
eloc
ityov
eran
da
cent
reve
loci
tyat
the
rele
vant
posi
tion.
Itw
ould
bedi
fficu
ltto
com
pute
the
mea
nvel
ocity
over
the
entir
eou
tflow
trac
tor
over
the
entir
em
itral
ring
beca
useth
epo
ster
iorw
alli
sno
tkno
wn
for
the
aort
icou
tflow
trac
tand
the
ante
riorb
orde
ris
notk
now
nfo
rth
em
itral
ring.
Itis
also
diffi
cult
tore
port
the
cent
reve
loci
ty(a
tone
voxe
l)be
caus
ethe
velo
city
osci
llate
sfr
omon
evo
xelt
oth
ene
xtov
erth
eao
rtic
outfl
owtr
act.
The
sear
eou
rrea
sons
for
repo
rtin
gam
eanc
entr
evel
ocity
for
the
MR
data
.Fo
rth
esi
mul
atio
nw
ere
port
both
am
eana
nda
cent
reve
loci
ty(a
sdes
crib
edin
Sec
tion4
.1)t
obe
able
toco
mpa
reth
em
ean
cent
reve
loci
tyof
the
MR
data
with
both
type
sofv
eloc
ities
.F
igur
e5.
4sh
ows
the
time
cour
seof
the
inflo
wve
loci
tyat
the
mitr
alrin
gex
-tr
acte
dfro
mbo
thth
eM
Rm
easu
rem
entsa
ndth
esi
mul
atio
n.O
utflo
wve
loci
tiesa
tth
eao
rtic
outfl
owtr
acta
resh
own
inF
igur
e5.
5.T
hem
atch
ingo
fth
esi
mul
atio
nan
dM
Rda
tapl
aced
the
R-p
eaki
nth
eE
CG
ofth
eM
Rda
taat
1.48
6s
and
this
isin
dica
tedb
ya
vert
ical
dash
edlin
ein
Fig
ures
5.4
and
5.5.
Fig
ure
5.4
show
sth
atat
the
mitr
alrin
gth
ere
isa
reas
onab
leagr
eem
entb
e-tw
een
MR
data
and
sim
ulat
ion
resu
ltsfo
rbo
thth
ece
ntre
and
the
mea
nvel
ocity
inth
ebe
ginn
ing
ofth
ehe
artc
ycle
durin
gea
rlyfil
ling,
but
the
early
max
imum
(pea
kE)v
eloc
ityis
high
erin
the
MR
data
than
inth
esi
mul
atio
n(f
orbo
thm
ean
and
cent
reva
lues
).It
isob
tain
edat
the
sam
etim
ebe
caus
eof
the
mat
chin
gof
sim
ulat
ion
and
MR
data
.The
upst
roke
ofth
eE
peak
ofth
esi
mul
atio
nre
sults
isve
rysi
mila
r,w
hile
the
dow
nstr
oke
isfa
ster
for
the
mea
ntha
nth
ece
ntre
velo
city
,bu
tbot
hth
eup
stro
kean
dth
edo
wns
trok
em
atch
the
MR
data
fairl
yw
ell.
The
MR
data
show
sne
gativ
eflo
wve
loci
tybe
twee
nthe
Epe
akan
dth
eA
peak
,and
this
isno
tsee
nin
the
sim
ulat
ion.
On
the
cont
rary,
the
sim
ulat
ion
show
sa
forw
ard
peak
.Thi
sco
uld
bedu
eto
anin
corr
ectm
ovem
ento
fth
em
itral
valv
ele
aflet
sin
the
sim
ulat
ion,
whi
chm
ight
beca
used
byth
eth
ree
vent
ricul
arvo
rtic
es,b
ecau
seth
evo
rtic
esaf
fect
the
mitr
alva
lve
mov
emen
t.B
utth
em
itral
valv
em
ovem
entis
diffi
cult
toju
dge
sinc
eit
isno
tdet
ecte
dinth
eM
Rm
easu
rem
ents
.Alte
rnat
ivel
y,th
epe
akco
uld
beca
used
byan
inco
rrec
tact
ivat
ion
leve
loft
hem
uscl
emod
el(s
eeS
ectio
n3.3
.6).
The
Ape
akin
the
mitr
alin
flow
isca
used
byth
eon
seto
fatr
ials
ysto
le.A
tria
lsy
stol
ecan
beha
rdto
dete
ctin
the
MR
mea
sure
men
ts,be
caus
eitoc
curs
atth
een
d
5.3
Vel
ocity
time
profi
les
87
200
400
600
800
Time [msec]
-0.4
-0.20
0.2
0.4
0.6
Velocity [m/s]
Ventricular inflow - mitral valve
11.2
1.4
1.6
1.8
t[s]
-0.4
-0.20
0.2
0.4
0.6
velocity[m/s]
Fig
ure
5.4:
Vel
ocity
time
profi
lefo
rve
loci
ty(m
/s)
over
the
mitr
alrin
gfr
omM
Rda
ta(t
op)
and
from
sim
ulat
ion
(bot
tom
).F
rom
the
sim
ulat
ion
both
cent
re(d
ashe
d)an
dm
ean(
solid
)vel
ocity
isin
clud
ed.
ofea
chM
Rre
cord
ing.
The
last
part
ofan
MR
reco
rdin
gwill
besm
eare
dout
be-
caus
ethe
final
mea
sure
men
tisan
aver
ageo
ver7
68he
artb
eats
whi
chea
chm
ight
vary
inhe
artc
ycle
leng
th.T
his
cane
xpla
inw
hyth
epe
akA
velo
city
(thi
rdpe
akin
the
sim
ulat
ion
curv
es)i
sm
uch
high
erin
the
sim
ulat
ion
than
inth
eM
Rda
ta.A
sse
enin
Fig
ure
5.4,
the
sim
ulat
ions
how
sa
clea
ratr
ials
ysto
le.A
lthou
ghth
em
ea-
sure
men
tssho
won
lya
very
vagu
eatr
ials
ysto
le,th
isoc
curs
atap
prox
imat
elyt
hesa
met
ime
inth
esi
mul
atio
nand
MR
mea
sure
men
ts,co
nfirm
ing
the
timin
gof
the
atria
lsys
tole
inth
em
odel
.The
MR
data
show
noba
ckflo
wat
the
time
ofcl
osur
eof
the
mitr
alva
lve
(bet
wee
n1.4
94an
d1.5
30s)
,in
cont
rast
tobo
thve
loci
tiesf
rom
the
sim
ulat
ion.
Thi
sco
uld
bea
cons
eque
nceof
the
aver
agin
goft
heM
Rda
ta,o
rit
coul
dbe
caus
edby
the
coar
setim
ere
solu
tioni
nth
eM
Rm
easu
rem
ents
.Inot
her
88C
ompa
ring
the
2Dm
odel
with
anM
Rda
tase
t
MR
mea
sure
men
ts(not
show
n)th
isdi
spla
cem
entba
ckflo
wha
sbee
ncle
arly
seen
.F
igur
e5.
5sh
ows
the
velo
city
time
profi
leac
ross
the
aort
icou
tflow
trac
tfro
mbo
thth
eM
Rda
taan
dth
esi
mul
atio
n.It
isse
enth
atth
eou
tflow
thro
ught
heao
rtic
trac
tsta
rtsa
tapp
roxi
mat
elyt
hesa
met
ime
inth
eM
Rda
taan
dth
esi
mul
atio
n(f
orbo
thve
loci
tiesr
epor
ted)
.Thu
swe
can
conc
lude
that
the
onse
tofv
entr
icul
arsy
stol
ein
the
sim
ulat
ion
isco
rrec
t.T
hefo
rmof
the
curv
eis
muc
hw
ider
for
the
MR
data
,but
the
outfl
owth
roug
hth
etr
acte
ndsa
tapp
roxi
mat
elyt
hesa
me
time
inbo
thth
eM
Rda
taan
dth
esi
mul
atio
n.T
hem
axim
umou
tflow
velo
city
isob
tain
edea
rlier
inth
esi
mul
atio
nth
anin
the
MR
data
,sug
gest
ingt
hatt
heon
set
ofve
ntric
ular
dias
tole
coul
dbe
furt
heri
mpr
oved
.
200
400
600
800
Time [msec]
-0.20
0.2
0.4
0.6
0.81
Velocity [m/s]
Ventricular outflow - aortic valve
11.2
1.4
1.6
1.8
t[s]
0
0.2
0.4
0.6
0.81
velocity[m/s]
Fig
ure
5.5:
Vel
ocity
time
profi
lefo
rve
loci
ty(m
/s)a
cros
sthe
aort
icou
tflow
trac
tfr
omM
Rda
ta(t
op)
and
from
sim
ulat
ion
(bot
tom
).F
rom
the
sim
ulat
ion
both
cent
re(d
ashe
d)an
dm
ean(
solid
)vel
ocity
isin
clud
ed.
5.4
Dis
cuss
iona
ndco
nclu
sion
s89
Apr
imar
yef
fect
ofth
eus
eoft
heW
indk
esse
lmod
elis
that
we
obta
ina
sim
u-la
ted
aort
icou
tflow
profi
lew
hich
repr
oduc
esa
char
acte
ristic
feat
ureo
fth
em
ea-
sure
done
,nam
elyt
hatt
heup
stro
keha
sadu
ratio
nofa
bout
one
half
ofth
atof
the
dow
nstr
oke.
As
men
tione
dabo
ve,t
heac
tual
shap
eoft
hesi
mul
ated
outfl
owcu
rve
does
note
xact
lym
atch
the
mea
sure
done
.Fur
ther
adju
stm
ento
fthe
load
para
m-
eter
smig
htac
hiev
ea
bette
ragr
eem
ent,b
utth
em
uscu
larm
odel
behi
ndth
ehe
art
mod
elis
also
invo
lved
inth
esi
mul
ated
profi
lean
dthe
refo
resh
ould
bein
volv
edin
the
tuni
ngpr
oces
sasw
ell.
Itw
asal
sofo
und
that
the
shap
eoft
heou
tflow
profi
lew
asse
nsitiv
eto
the
stre
ngth
and
geom
etry
ofth
ete
ther
ing.
Itsh
ould
beno
tedt
hati
nth
epr
evio
usly
repo
rted
com
paris
onsb
etw
eenm
odel
and
MR
data
(Jac
obse
neta
l.,20
01;A
dele
rand
Jaco
bsen
,2001
)res
ults
from
the
first
hear
tcyc
lew
ere
com
pare
dwith
the
MR
data
and
seve
ralo
fth
epa
ram
eter
sw
ere
adju
sted
toac
hiev
ea
good
com
paris
onbe
twee
nthe
first
hear
tcyc
lean
dth
eM
Rda
ta.W
hen
we
late
rinv
estig
ated
whi
chhe
artc
ycle
best
repr
esen
tsape
riodi
cre
sultf
rom
the
mod
el(s
eeS
ectio
n4.7
),w
edi
dno
tcha
nget
hepa
ram
eter
setti
ngs
fort
hem
odel
,and
this
expl
ains
why
the
com
paris
onpr
esen
tedh
erei
sno
tasg
ood
asth
eon
ew
eha
vepr
evio
usly
repo
rted
.The
mod
elco
uld
befu
rthe
radj
uste
dto
mak
eth
ese
cond
hear
tcyc
lebe
tterc
ompa
redto
MR
data
.Unf
ortu
nate
ly,tim
eha
sno
tallo
wed
usto
doth
is.
5.4
Dis
cuss
iona
ndco
nclu
sion
s
The
com
paris
onof
the
velo
city
field
sfro
mth
eM
Rm
easu
rem
entsa
ndth
esi
mul
a-tio
nsh
ows
afa
irqu
alita
tive
agre
emen
tinth
ege
nera
lflow
patte
rn:a
wid
ein
flow
jet,
the
form
atio
nof
ado
min
anta
nter
iorv
orte
xfil
ling
the
vent
ricle
,and
anou
t-flo
wje
tin
the
aort
icou
tflow
trac
t.A
diff
eren
cein
velo
city
field
sis
seen
inth
ew
idth
ofth
ein
flow
jet(
from
atriu
mto
vent
ricle
)and
inth
esi
zeof
the
vent
ricle
.C
ompa
rison
ofth
evo
rtex
patte
rnsin
the
mid
dias
tolic
phas
ecou
ldin
dica
teth
ata
mor
eco
mpl
ex3D
flow
patte
rnis
pres
entin
the
mea
sure
dven
tric
leth
anth
atpr
e-di
cted
from
the
sim
ple
2Dm
odel
.Thi
sis
supp
orte
dby
resu
ltsfr
omth
e3D
mod
elpr
esen
tedin
Cha
pter
7.In
the
3Dm
odel
we
seet
hata
vort
exrin
gis
form
eddu
r-in
gfil
ling
(see
Fig
ure
7.6)
,and
whe
nlo
okin
gat
this
vort
exrin
gin
acu
t-aw
ayvi
ew(F
igur
e7.
7)it
hast
hesh
apeo
ftw
ovo
rtic
es,s
imila
rto
thos
esee
nin
the
2Dm
odel
.Fur
ther
,the
sim
ulat
ion
show
sth
ree
vort
ices
inth
eve
ntric
le,w
here
asth
eM
Rda
tash
ows
atm
ostt
wo.
The
third
vort
exis
are
sult
ofth
evo
rtex
inth
epr
e-vi
ous
hear
tcyc
leno
tbei
ngde
stroy
edbe
fore
the
onse
tofd
iast
ole.
Inth
eM
Rda
tano
vort
exis
pres
enta
tthe
end
ofa
hear
tcyc
le.T
heth
irdvo
rtex
inth
esi
mul
atio
nis
unph
ysio
logi
cal,a
ndit
wou
ldim
prov
eth
em
odel
cons
ider
ably
ifit
coul
dbe
90C
ompa
ring
the
2Dm
odel
with
anM
Rda
tase
t
rem
oved
.Rem
ovin
gth
isvo
rtex
mig
htim
prov
eth
em
ovem
ento
fthe
mitr
alva
lve
and
cons
eque
ntlya
lso
rem
ove
the
third
peak
inm
itral
velo
city
profi
le.
Gen
eral
ly,w
ese
eth
atth
evo
rtex
patte
rnis
mor
edi
stin
ctin
the
sim
ulat
ion
and
the
velo
citie
sin
the
vort
ices
are
larg
er.
Thi
sis
part
lyre
late
dto
the
form
ofth
eve
ntric
lean
dth
em
itral
valv
em
otio
nin
the
sim
ulat
ion.
Bec
ause
the
mod
elis
2D,v
entr
icul
arex
pans
ionh
asto
take
plac
ein
the
cons
ider
edpl
ane,
resu
lting
inan
over
size
vent
ricle
.F
urth
erm
ore,
the
vent
ricle
isst
illto
oci
rcul
arin
form
,ev
enth
ough
we
have
impr
oved
this
byad
just
ing
the
cros
slin
ks.
The
mot
ion
ofth
em
itral
valv
eis
notc
ompl
etel
yco
rrec
t;it
shou
ldop
enup
mor
ean
dth
epo
ster
iorl
eafle
tsho
uld
actu
ally
touc
hth
epo
ster
iorv
entr
icul
arw
all.
The
latte
rw
ould
prob
ably
prev
entt
hefo
rmat
ion
ofth
epo
ster
iorv
orte
x.T
heflo
wfie
lds
from
the
sim
ulat
ion
are
affe
cted
byth
esi
mpl
eao
rtic
valv
e(d
iode
coup
ledt
osi
nk),
resu
lting
inth
efo
rmat
ion
ofa
vort
exat
the
sink
,whi
chof
cour
seis
nots
eeni
nth
eM
Rda
ta.
Fur
ther
mor
e,th
ete
ther
ing
ofth
em
odel
hear
tintr
oduc
esos
cilla
tions
inbo
thth
eat
rium
and
the
vent
ricle
and
this
affe
cts
the
flow
field
inan
unph
ysio
logi
calw
ay.
The
time
ofat
rials
ysto
lese
emsto
bese
trea
sona
blyw
elli
nth
em
odel
,but
this
isdi
fficu
ltto
judg
ebe
caus
eof
the
very
vagu
eatr
ials
ysto
lein
the
MR
data
.The
onse
tof
vent
ricul
arsy
stol
esee
mst
obe
right
,whi
leit
wou
ldpr
obab
lyim
prov
eth
em
odel
ifth
edu
ratio
nof
vent
ricul
arsy
stol
ewas
exte
nded
,thus
lead
ing
toa
late
rons
etof
vent
ricul
ardi
asto
le.
The
velo
city
profi
lesa
tthe
mitr
alrin
gan
daor
ticou
tflow
trac
tfro
msi
mul
atio
nan
dMR
mea
sure
men
tsmat
chre
ason
ablyw
ell.
The
timin
gof
the
velo
city
profi
les
isin
good
agre
emen
t,whe
reas
the
shap
eand
max
imum
valu
esof
the
velo
city
curv
essh
owso
med
iffer
ence
.The
best
agre
emen
tfor
the
mitr
alve
loci
typr
ofile
isin
the
begi
nnin
gof
the
hear
tcyc
le,w
hile
the
best
agre
emen
tfor
the
aort
icpr
ofile
isfr
omth
ebe
ginn
ing
until
max
imum
outfl
owve
loci
ty.
The
depe
nden
ceof
the
sim
ulat
edre
sults
onth
eva
rious
para
met
ersin
the
mod
elis
very
com
plex
,and
we
ares
uret
hata
furt
hera
djus
tmen
tofth
epa
ram
eter
sco
uld
lead
toan
even
bette
ragr
eem
entbe
twee
nsim
ulat
edan
dm
easu
redv
eloc
-ity
data
.Pa
ram
eter
sof
part
icul
arim
port
ance
inth
isre
gard
are
the
dura
tion
ofve
ntric
ular
syst
ole,
the
exci
tatio
nan
dre
laxa
tionp
aram
eter
s,and
the
Win
dkes
sel
and
mus
clep
aram
eter
s.How
ever
,the
com
paris
onsh
ows
that
desp
iteth
ege
omet
-ric
alco
nstr
aint
spos
edby
the
2Dm
odel
ling
ofa
3Dph
enom
enon
,are
ason
able
agre
emen
tisob
tain
edbe
twee
nsim
ulat
edan
dm
easu
redre
sults
.T
heco
mpa
rison
supp
ortst
heco
ncep
tofc
ombi
neds
imul
atio
nand
MR
stud
ies
toin
vest
igat
ethe
flow
inth
ehe
arta
ndits
inte
ract
ionw
ithth
est
ruct
urea
ndfu
nc-
tion
ofhe
artm
uscl
esan
dhe
artv
alve
s,pr
ovid
edth
atth
elim
itsof
the
mod
elar
eke
ptin
min
d.T
hede
taile
dflow
insi
deth
ehe
arts
houl
dnot
beus
edfo
rin
vest
iga-
5.4
Dis
cuss
iona
ndco
nclu
sion
s91
tions
befo
reth
em
odel
isim
prov
ed,e
spec
ially
with
resp
ectto
the
vort
expa
ttern
.B
ased
onth
eco
mpa
rison
betw
eent
hem
itral
velo
city
profi
les,
itis
just
ified
tous
eth
em
odel
toin
vest
igat
ethe
early
fillin
gof
the
vent
ricle
,par
ticul
arly
the
influ
ence
onth
em
itral
velo
city
profi
le.I
nves
tigat
ions
invo
lvin
gth
eao
rtic
outfl
owve
loci
typr
ofile
sw
ould
also
beju
stifi
ed.
Exp
erim
ents
invo
lvin
gth
eov
eral
lres
pons
eof
the
hear
tto
diff
eren
tsce
nario
sisye
tano
ther
poss
ibili
ty,e
.g.h
owdo
esth
ehe
art
reac
tto
anin
crea
sedr
esis
tanc
einth
eou
tflow
,w
hath
appe
nsif
part
ofth
eve
n-tr
icul
arw
allg
row
sst
iffer
orw
hata
reth
eco
nseq
uenc
esofa
decr
ease
inpa
pilla
rym
uscl
ete
nsio
n?In
answ
erin
gthe
sequ
estio
nson
esh
ould
note
xpec
tto
obta
inth
eco
mpl
etefl
owfie
ldin
side
the
hear
t,but
rath
ersu
chqu
antit
iesa
sth
eca
rdia
cou
tput
,the
form
ofth
eou
tflow
velo
city
profi
le,o
rth
ein
fluen
ceon
the
fillin
gof
the
vent
ricle
.
Cha
pter
6
Sim
ulat
ing
path
olog
ical
hear
tco
nditi
ons
An
impo
rtan
tpur
pose
ofou
rw
ork
with
the
2Dm
odel
isto
use
itto
stud
yth
ein
fluen
ceof
path
olog
ical
cond
ition
sin
the
hear
t,sp
ecifi
cally
cond
ition
swhi
chca
nbe
rela
tedt
oth
em
echa
nica
lactiv
atio
nof
the
hear
tmus
cle.
The
bloo
dflo
wpa
ttern
chan
gesw
hen
apa
tient
suffe
rsfr
oma
hear
tdis
ease
,re
sulti
ngin
e.g.
ade
crea
sein
the
ejec
tion
frac
tion
ora
redu
cedfi
lling
velo
city
.M
echa
nism
sinvo
lved
inhe
artd
isea
sesh
ave
been
inve
stig
ated
byth
eus
eof
dif-
fere
ntm
easu
ringte
chni
ques
(one
ofth
embe
ingM
R).
Thi
sha
sled
toth
ede
finiti
onof
inde
xeso
fthe
bloo
dve
loci
tyto
char
acte
rizeth
ele
ftve
ntric
ular
dias
tolic
func
-tio
n.S
ever
alst
udie
shav
efo
cuse
don
the
early
dias
tolic
fillin
g,an
dsi
ncew
eha
vefo
und
are
ason
ablyg
ood
com
paris
onbe
twee
nthe
mod
elan
dM
Rda
tain
exac
tlyth
eea
rlyfil
ling
phas
e,th
em
echa
nism
sresp
onsi
blefo
rth
ech
ange
inflo
wpa
ttern
and
the
velo
city
inde
xes
can
best
udie
din
the
com
pute
rmod
el.H
owev
er,t
obe
able
tosi
mul
atep
atho
logi
calc
ondi
tions
,itis
nece
ssar
yfirs
tto
refin
eth
em
odel
for
the
activ
atio
nof
the
hear
tmus
cle.
The
impr
oved
mod
elca
nth
enbe
used
tost
udys
omeg
ener
alm
echa
nism
sinvo
lved
inhe
artd
isea
ses.
Thi
sch
apte
rbeg
ins
with
the
med
ical
back
grou
ndfo
rim
prov
ing
the
mod
elfo
rth
eac
tivat
ion
ofth
ehe
artm
uscl
ean
dde
scrib
esth
eim
plem
enta
tiono
fth
eim
prov
edac
tivat
ion
func
tion.
The
mod
elpa
ram
eter
sare
then
setu
pto
sim
ulat
ea
heal
thy
hum
an.
Usi
ngth
esem
odel
para
met
ersw
em
ake
anin
vest
igat
iono
fso
meo
fthe
mec
hani
smsin
volv
edin
hear
tdis
ease
sand
com
pare
sim
ulat
ions
ofa
dise
ased
hear
twith
MR
mea
sure
men
ts.
94S
imul
atin
gpa
thol
ogic
alhe
artc
ondi
tions
6.1
Med
ical
back
grou
nd
Inm
osth
eart
dise
ases
dias
tolic
dysf
unct
iono
ften
coex
ists
with
syst
olic
dysf
unc-
tion.
The
asse
ssm
entof
dias
tolic
func
tion,
or,m
ore
prec
isel
y,th
eas
sess
men
tofle
ftve
ntric
ular
dias
tolic
func
tion,
hasb
eeng
iven
muc
hatte
ntio
n(H
oulin
d,20
01b;
Nis
hiha
raet
al.,
2000
;Oht
eeta
l.,19
99;S
tein
eeta
l.,19
99;S
tuga
arde
tal.,
1994
a;S
tuga
arde
tal.,
1994
b;S
tuga
arde
tal.,
1993
;Yam
amot
oeta
l.,19
95;S
t.G
oar
etal
.,19
91).
As
are
sults
ofre
cent
prog
ress
the
intr
aven
tric
ular
bloo
dflo
wpa
t-te
rnan
dde
rived
bloo
dve
loci
tyin
dexe
scan
now
beus
edto
eval
uate
the
stat
eof
the
dias
tolic
func
tion.
The
intr
aven
tric
ular
bloo
dflo
wpa
ttern
chan
gesin
adi
s-ea
sedh
eart
,but
atpr
esen
tthe
mec
hani
smsre
spon
sibl
efor
this
chan
gear
eno
tco
mpl
etel
yund
erst
ood.
The
com
pute
rmod
elis
ato
olfo
rst
udyi
ngth
esem
echa
-ni
sms.
The
dias
tolic
func
tion
can
beev
alua
tede
.g.b
ym
easu
ringt
hepr
essu
re,th
ew
allt
hick
ness
,the
wal
lmot
ion
orth
ebl
ood
flow
/vel
ocity
.Nat
ural
ly,w
eha
vefo
-cu
sedo
nco
mpa
ringt
hem
odel
with
stud
ieso
fthe
dias
tolic
func
tion
that
areb
ased
onth
ebl
ood
flow
patte
rnor
bloo
dflo
wve
loci
ty.T
hese
stud
iesa
repe
rfor
med
us-
ing
MR
Ior
echo
card
iogr
aphy(
e.g.
colo
urM
-mod
eD
oppl
erte
chni
que)
.We
look
atth
ege
nera
lflow
patte
rndu
ring
early
fillin
gan
dw
eco
mpa
reco
mpu
teda
ndob
serve
dve
loci
tyin
dexe
sder
ived
from
the
intr
aven
tric
ular
velo
city
field
.
6.1.
1M
echa
nism
sdur
ing
early
fillin
g
Inth
eno
rmal
stat
eofe
arly
left
vent
ricul
arfil
ling,
the
rapi
dle
ftve
ntric
ular
rela
x-at
ion
gene
rate
sapr
essu
regr
adie
ntin
the
left
vent
ricle
with
the
high
estp
ress
ure
atth
eba
sean
dth
elo
wes
tpre
ssur
eat
the
apex
.T
hepr
essu
regr
adie
ntdr
ives
the
bloo
den
terin
gatt
hem
itral
valv
eto
war
dth
eap
exof
the
vent
ricle
(Hou
lind
etal
.,20
01).
Inth
edi
seas
edhe
artw
ithdi
asto
licdy
sfun
ctio
nthe
left
vent
ricul
arre
laxa
tioni
ssl
owed
and
the
intr
aven
tric
ular
pres
sure
grad
ient
isre
duce
d(se
ee.g
.S
tein
eeta
l.,19
99).
So,
regi
onal
diff
eren
cesin
rela
xatio
nrat
eaf
fect
the
early
di-
asto
licfu
nctio
n.T
here
duce
dint
rave
ntric
ular
pres
sure
grad
ient
tend
sto
caus
easl
ower
prop
agat
iono
fthe
bloo
dflo
win
toth
eve
ntric
le(H
oulin
det
al.,
2001
).
6.1.
2P
atho
logi
calh
eart
cond
ition
s
The
reex
ists
ever
alst
udie
soft
hedi
asto
licfu
nctio
n,co
verin
ga
rang
eofh
eart
dis-
ease
s,tha
tpro
vide
data
that
canb
eus
edto
dete
rmin
eapp
ropr
iatei
nput
para
met
ers
toth
em
odel
and
toco
mpa
rew
ithsi
mul
atio
nre
sults
.Thi
sse
ctio
ngiv
esa
brie
fde
scrip
tiono
fso
mei
mpo
rtan
thea
rtdi
seas
esan
dth
eiri
nflue
nceo
nth
edi
asto
lic
6.1
Med
ical
back
grou
nd95
func
tion.
Isch
emic
hear
tdis
ease
isa
com
mon
term
fori
nade
quat
eblo
odsu
pply
toan
area
ofth
ehe
artw
all.
Ifth
isis
due
tobl
ocka
geof
the
coro
nary
arte
riesi
tis
calle
dco
rona
ryar
tery
dise
ase,
seeb
elow
.Is
chem
iaca
nbe
both
loca
land
glob
al.I
nth
eca
seof
glob
alis
chem
iath
efil
ling
ofth
eve
ntric
leis
dela
yed
(Stu
gaar
deta
l.,19
94b)
.Loc
alis
chem
iais
ofte
nse
enin
anac
utei
nfar
ctio
n(K
im,
2001
),se
e’A
cute
myo
card
ialin
farc
tion’
belo
w.
Cor
onar
yar
tery
dise
ase(
CA
D)
isth
ena
rrow
ing
ofth
eco
rona
ryar
terie
sca
used
byat
hero
scle
rosi
s.Aco
rona
ryar
tery
isan
arte
ryth
atsu
pplie
sbl
ood
toth
ehe
artm
uscl
e.R
elax
atio
nis
ofte
npr
olon
gedd
urin
gth
isill
ness
(Wei
sset
al.,
1976
;Oht
eet
al.,
1999
)and
the
fillin
gof
the
vent
ricle
isde
laye
d(S
tuga
arde
tal.,
1994
a;S
tuga
arde
tal.,
1993
).
Acu
tem
yoca
rdia
linf
arct
ion
(AM
I),o
ften
refe
rred
toas
ahe
arta
ttack
,isth
ede
ath
ofpa
rtof
the
hear
tmus
cle
due
toa
sudd
enlo
ssof
bloo
dsu
pply.
The
loss
ofbl
ood
supp
lyis
caus
edby
the
com
plet
eblo
ckag
eof
aco
rona
ryar
tery
bya
bloo
dcl
ot.
AM
Iaf
fect
sthe
mot
ion
and
the
rela
xatio
noft
hedi
seas
edpa
rtof
the
hear
twal
l.Ty
pica
lly,t
hem
otio
nof
the
hear
twal
lis
abno
rmal
and
both
the
rela
xatio
nand
the
fillin
gof
the
vent
ricle
isde
laye
d(S
tein
eeta
l.,19
99).
Fur
ther
mor
e,lo
wer
fillin
gpe
akve
loci
ties
are
seen
(Ste
ine
etal
.,19
99;H
oulin
det
al.,
2001
).
Car
diom
yopa
thy
isa
gene
ralte
rmfo
ra
dise
aseo
fth
ehe
artm
uscl
e(t
hem
y-oc
ardi
um).
Hyp
ertr
ophi
cca
rdio
myo
path
y(H
CM
)is
char
acte
rized
byin
crea
sed
grow
th(h
yper
trop
hy)in
the
thic
knes
sof
the
left
vent
ricul
arw
all.
Inth
isca
seth
eej
ectio
nfr
actio
n(s
eeS
ectio
n2.2
)is
ofte
nno
taffe
cted
,bu
tthe
rela
xatio
nis
prol
onge
d(N
ishi
hara
etal
.,20
00).
Dila
ted
card
iom
yopa
thy
(DC
M)
isch
arac
teriz
edby
adi
late
dve
ntric
le.
Thi
sis
ofte
nse
enin
chro
nic
hear
tfai
lure
,whe
rea
glob
aldi
latio
nof
the
vent
ricle
ispr
esen
tand
the
rela
xatio
nis
abno
rmal
(Kim
,200
1).I
nth
eca
seof
DC
Mth
eej
ectio
nfr
actio
nis
decr
ease
dand
the
fillin
gof
the
vent
ricle
isab
norm
al(H
oulin
d,20
01a;
Yam
amot
oeta
l.,19
95;S
t.G
oare
tal.,
1991
).It
isno
tcle
arw
heth
erth
eab
norm
alfil
ling
isca
used
bydi
asto
licor
syst
olic
dysf
unct
iono
fthe
vent
ricle
(Yam
amot
oeta
l.,19
95).
96S
imul
atin
gpa
thol
ogic
alhe
artc
ondi
tions
6.1.
3V
eloc
ityin
dexe
s
The
dias
tolic
func
tion
can
bede
term
ined
byin
vest
igat
ing(
spat
ial)
velo
city
pro-
files
alon
glin
esac
ross
the
left
vent
ricle
,i.e.
velo
city
profi
lesa
long
lines
perp
en-
dicu
lart
oth
elo
ngax
isof
the
vent
ricle
,atd
iffer
entt
imes
durin
gth
eea
rlyin
flow
.T
his
has
been
done
usin
gbo
thM
RI
(Hou
lind
etal
.,20
01)a
ndpu
lsed
Dop
pler
echo
card
iogr
aphy
(Ste
ine
etal
.,19
99;Y
amam
otoe
tal.,
1995
).Ty
pica
lly3-
6lin
esar
eex
amin
ed.
We
pres
enth
ere
the
met
hodo
fH
oulin
det
al.(
2001
),w
here
six
lines
are
ex-
amin
eddu
ring
anM
Rsc
an.O
nelin
eis
plac
edat
the
leve
lof
the
mitr
alle
aflet
tips
(ML
T)
perp
endi
cula
rtoth
ein
flow
jet.
The
othe
rlin
esar
epl
aced
para
llelt
oth
efir
stlin
ebu
tdis
plac
edei
ther
upst
ream
ordo
wns
trea
mof
the
first
:on
elin
eis
plac
ed2
cmup
stre
am(a
ppro
xim
atel
yatt
hele
velo
fthe
mitr
alrin
g),a
ndlin
esar
epl
aced
1,2,
3an
d4
cmdo
wns
trea
m.T
hese
lines
are
fixed
insp
acea
nddo
not
mov
ew
ithth
ehe
art.
Thi
sis
acce
ptab
lebec
ause
the
mot
ion
ofth
ehe
artis
smal
ldu
ring
dias
tole
.Alo
ngea
chof
thes
elin
esth
eab
solu
teve
loci
tyis
com
pute
dbas
edon
allt
hree
velo
city
com
pone
nts,a
ndth
esp
atia
lvel
ocity
profi
leis
plot
ted.
The
inflo
wje
tis
man
ually
trac
edan
dal
lvel
ociti
esou
tsid
ethe
jeta
rese
tto
zero
.The
velo
city
profi
lesa
reex
amin
edat
5tim
es:t
hetim
eof
Epe
akve
loci
tyat
the
mitr
alle
aflet
tips,
the
time
fram
eprio
rto
this
,and
the
thre
esub
sequ
enttim
efr
ames
.The
time
diff
eren
cebe
twee
nsuc
ceed
ingti
me
fram
esar
eof
the
orde
rof2
5m
s.F
rom
the
velo
city
profi
lest
hree
quan
titie
scan
bede
rived
that
char
acte
rizeth
eve
ntric
ular
dias
tolic
func
tion
(Hou
lind
etal
.,20
01).
The
sear
eth
epe
akve
loc-
ity(P
V),
the
velo
city
inde
x(V
I),
and
the
time
dela
y(T
D).
PV
isth
em
axim
umve
loci
ty(in
both
spac
eand
time)
ata
part
icul
arlin
e.V
Ifo
ra
part
icul
arlin
eis
com
pute
das
the
PV
atth
atlin
ere
lativ
eto
the
PV
atth
elin
eat
the
mitr
alle
aflet
tips.
Fin
ally
,TD
isca
lcul
ated
asth
etim
edi
ffer
ence
betw
eent
ime
ofoc
curr
ence
ofP
Vin
apa
rtic
ular
line
and
time
ofoc
curr
ence
ofP
Vat
the
line
atth
em
itral
leafl
ettip
s.F
igur
e6.1
show
sth
eve
loci
typr
ofile
sfro
ma
heal
thyp
erso
nalo
ngth
esi
xlin
esat
five
times
:27
ms
befo
reP
Vat
the
mitr
alle
aflet
tips,
atth
etim
eof
PV
atth
em
itral
leafl
ettip
s,an
d27
,54
and
81m
saf
tert
hetim
eof
PV
atth
em
itral
leafl
ettip
s.It
isse
enth
atth
eve
loci
typr
ofile
sar
ew
ider
and
low
erfo
rth
elin
escl
oses
tto
the
apex
(the
2,3
and
4cm
lines
).F
urth
erm
ore,
the
figur
esh
ows
that
PV
isob
tain
edal
mos
tatt
hesa
met
ime
atal
lthe
lines
.In
Hou
lind
etal
.(20
01)P
V,V
Ian
dTD
areg
iven
fora
grou
pofh
ealth
yper
sons
and
for
agr
oup
ofA
MI
patie
nts.
Fig
ure
6.2
show
sth
eP
Vat
the
six
lines
from
the
heal
thyg
roup
and
the
AM
Igr
oup.
The
data
are
plot
ted
asth
em
eanv
alue
sw
ithth
est
anda
rdde
viat
ion
show
nas
erro
rbar
s.F
rom
this
figur
eit
iscl
eart
hat
6.1
Med
ical
back
grou
nd97
+4
+3
+2
+1
ML
T -2
-27
ms
PV
ÑÒÓ
27m
s54
ms
81m
s
Fig
ure6
.1:V
eloc
itypr
ofile
sofa
bsol
utev
eloc
ity(m
/s)a
long
six
lines
atfiv
etim
esfr
oma
heal
thyp
erso
nobt
aine
dby
MR
I.P
eakv
eloc
ityin
each
mea
surin
gpos
ition
ism
arke
dby
a
Ô .From
Hou
lind
etal
.(20
01).
the
peak
velo
city
atal
llin
esar
elo
wer
inan
AM
Ipa
tient
than
ina
heal
thyp
erso
n.P
Vis
only
sign
ifica
ntly
low
er(f
rom
ast
atis
tical
poin
tofv
iew
)at
the
3an
d4
cmlin
es(H
oulin
det
al.,
2001
).F
igur
e6.
3sh
ows
the
velo
city
inde
x,V
I,at
the
six
lines
from
the
heal
thya
ndth
eA
MI
grou
p.T
hefig
ure
show
sth
atcl
oset
oth
eap
ex(a
tthe
3an
d4
cmlin
es)
VI
islo
wer
inth
eA
MI
patie
ntst
han
for
the
heal
thyg
roup
.In
Fig
ure
6.4
the
time
dela
y,T
D,
atth
esi
xlin
esis
show
n.T
Dis
mar
kedl
yhi
gher
clos
erto
the
apex
(att
he3
and
4cm
lines
)for
the
AM
Igr
oup
than
for
the
heal
thyg
roup
.All
thes
efig
ures
(Fig
ures
6.1–
6.4)
show
that
inA
MI
patie
ntst
hefil
ling
patte
rndu
ring
early
dias
tole
inth
elo
wer
part
ofth
eve
ntric
leis
chan
gedc
ompa
redto
heal
thyp
erso
ns:
the
fillin
gis
dela
yed,
peak
velo
citie
sare
low
eran
dth
epe
akve
loci
tyat
ML
Tis
notm
aint
aine
ddow
nstr
eam
ofM
LT.
98S
imul
atin
gpa
thol
ogic
alhe
artc
ondi
tions
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
PV[m/s]
MR
healthy
MR
AMI
Fig
ure
6.2:
Pea
kve
loci
ty(P
V)
atsi
xlin
esfr
omM
Rm
easu
rem
entso
fhe
alth
yper
sons
(
Õ )an
dA
MI
patie
nts
(
Ö ).D
ataa
resh
own
asth
em
eanv
alue
with
the
stan
dard
devi
atio
n(S
D)a
ser-
rorb
ars.
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
1.2
1.4
1.6
VI
MR
healthy
MR
AMI
Fig
ure
6.3:
Vel
ocity
inde
x(V
I)at
six
lines
from
MR
mea
sure
men
tsof
heal
thyp
erso
ns(
Õ )an
dA
MI
patie
nts
(
Ö ).D
ataa
resh
own
asth
em
eanv
alue
with
the
stan
dard
devi
atio
n(S
D)a
ser-
rorb
ars.
-2
01
23
4Distance
from
MLT
[cm]
-500
50
100
TD[ms]
MR
healthy
MR
AMI
Fig
ure
6.4:
Tim
ede
lay(
TD
)at
six
lines
from
MR
mea
sure
men
tsof
heal
thyp
er-
sons
(
Õ )an
dA
MI
patie
nts(
Ö ).D
ata
are
show
nas
the
mea
nval
uew
ithth
est
an-
dard
devi
atio
n(S
D)a
serr
orba
rs.
6.1
Med
ical
back
grou
nd99
6.1.
4T
ime
cons
tant
ofre
laxa
tion
The
rela
xatio
nof
the
left
vent
ricle
iscl
osel
yre
late
dto
the
pres
sure
ther
e.W
eiss
etal
.(19
76)s
how
edth
atdu
ring
isov
olum
icre
laxa
tiont
hepr
essu
re-t
imev
aria
tion
follo
ws
anex
pone
ntia
ldec
ay(
×ØÙÚ Û )w
itha
cert
ain
time
cons
tant
Ü ,and
Ü hassu
bseq
uent
lybee
nuse
das
anin
dex
for
the
rela
xatio
n.T
hem
easu
redp
ress
ure-
time
varia
tion
durin
gis
ovol
umic
rela
xatio
nis
fitte
dto
anex
pone
ntia
ldec
ayto
dete
rmin
ethe
time
cons
tant
Ü .P
ress
ureis
mea
sure
dby
inva
sive
met
hods
(not
MR
I).
The
pres
sure
mea
sure
-m
ents
can
bedo
neat
seve
ralp
oint
sin
the
vent
ricle
,thu
sm
akin
git
poss
ible
toob
tain
Ü fordiff
eren
treg
ions
inth
eve
ntric
le.T
ypic
ally
,
Ü isdete
rmin
edfo
rth
eba
sala
ndap
ical
part
s,se
ee.g
.Ste
inee
tal.
(199
9).
6.1.
5R
egio
nsof
the
vent
ricle
Inst
udie
sof
the
left
vent
ricle
the
vent
ricul
arw
alli
sdi
vide
din
tofiv
ere
gion
sof
appr
oxim
atel
ythe
sam
esiz
e(S
chro
eder,
1999
):th
eba
salh
alfo
fthe
ante
riorw
all
regi
on(A
B),
the
apic
alha
lfof
the
ante
riorw
allr
egio
n(A
A),
the
apic
alre
gion
(A),
the
apic
alha
lfof
the
post
erio
rwal
lreg
ion
(PA
),an
dth
eba
salh
alfo
fthe
post
erio
rw
allr
egio
n(P
B).
The
five
regi
onsa
resh
own
onan
MR
imag
ein
Fig
ure
6.5.
A
AoV MiV
Ao LA
RV
AB
AA
PA
PBLV
Fig
ure
6.5:
MR
imag
ewith
the
five
regi
onso
fthe
vent
ricul
arw
alli
ndic
ated
:AB
:ba
salh
alfo
fan
terio
rwal
l,A
A:
apic
alha
lfof
ante
riorw
all,
A:
apex
,PA
:ap
ical
half
ofpo
ster
iorw
all,
PB
:bas
alha
lfof
post
erio
rwal
l.F
urth
erm
ore,
the
posi
tion
ofth
ele
ftve
ntric
le(L
V),
the
left
atriu
m(L
A),
the
right
vent
ricle
(RV
),th
em
itral
valv
e(M
iV),
the
aort
icva
lve
AoV
and
the
aort
a(A
o)ar
ein
dica
ted.
100
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
6.2
The
new
activ
atio
nfu
nctio
n
The
prev
ious
sect
ions
how
edth
atin
adi
seas
edhe
artt
here
laxa
tiono
fth
ehe
art
mus
clei
sof
ten
affe
cted
.The
rela
xatio
noft
hehe
artw
allc
anea
sily
beco
ntro
lled
and
chan
gedi
nth
em
odel
thro
ugh
man
ipul
atio
nof
the
para
met
ersd
eter
min
ing
the
activ
atio
nfu
nctio
n.In
the
orig
inal
mod
el,t
hepa
ram
eter
sthat
dete
rmin
ethe
rela
xatio
nand
con-
trac
tion
ofth
ehe
artw
all(
the
para
met
ersin
the
activ
atio
nfu
nctio
n
Ý )arede
fined
sepa
rate
lyfor
the
atriu
man
dve
ntric
le.
Tobe
able
tosi
mul
ate
regi
onal
diff
er-
ence
s,whi
chis
nece
ssar
yinth
epa
thol
ogic
alca
sesa
bove
,the
para
met
ersh
ave
tobe
regi
onal
lygi
ven
rath
erth
angi
ven
for
each
hear
tcha
mbe
ras
aw
hole
.Thi
sis
achi
eved
bydi
vidi
ngth
ehe
artw
alli
nto
five
regi
ons,
sim
ilart
oth
eon
esde
scrib
edin
Sec
tion6
.1.5
.The
para
met
ersd
eter
min
ingt
heac
tivat
ion
func
tion
are
then
de-
fined
with
inea
chre
gion
.T
hese
para
met
ersa
reth
ere
laxa
tiona
ndex
cita
tion,
orco
ntra
ctio
n,tim
eco
nsta
nts
Þuß and
Þuà ,thelim
iting
and
initi
alva
lues
ofth
eac
-tiv
atio
nfu
nctio
n
Ýâáãä
and
Ýå ,th
eon
setti
me
ofsy
stol
e
æèç andth
een
dtim
eof
syst
ole
æèç éàêë
(whi
chis
also
the
onse
ttim
eof
dias
tole
).We
call
this
refin
emen
tof
the
activ
atio
nfu
nctio
na
regi
onal
activ
atio
n.In
addi
tion
tom
akin
git
poss
ible
tosi
mul
atep
atho
logi
calc
ondi
tions
,we
wis
hto
impr
ove
the
activ
atio
nfu
nctio
nfr
oma
gene
ralp
hysi
olog
ical
poin
tof
view
.T
heac
tivat
ion
func
tion
Ýì æí give
sth
etim
eco
urse
ofco
ntra
ctio
nand
rela
xatio
nof
the
hear
tmus
cle.
Ýì æí isa
mod
elfo
rth
ere
leas
eof
Ca
î ï (Pes
kin,
2001
),w
hich
initi
ates
the
mec
hani
calco
ntra
ctio
nofh
eart
mus
cle(
seeS
ectio
n2.4
).T
here
leas
eofC
a
î ï istr
igge
redb
yth
eac
tion
pote
ntia
l,whi
chis
prop
agat
edth
roug
hth
eco
nduc
tions
yste
mof
the
hear
t.S
ince
the
cont
ract
ilere
spon
seof
aca
rdia
cm
uscl
ebe
gins
just
afte
rth
eac
tion
pote
ntia
lhas
reac
hedt
hem
uscl
efib
er,
the
cont
ract
iona
lso
prop
agat
esas
aw
ave
over
the
hear
twal
lwith
the
sam
evel
ocity
field
asth
eel
ectr
ical
impu
lses
.But
this
isno
thow
the
cont
ract
iona
ndre
laxa
tion
are
impl
emen
tedin
the
orig
inal
mod
el.
Inth
eor
igin
alm
odel
the
cont
ract
iono
fthe
hear
twal
locc
urss
imul
tane
ousl
yinth
een
tire
atriu
man
d,af
tera
dela
y,si
mul
tane
ousl
yinth
een
tire
vent
ricle
,bec
ause
æèç isthe
sam
efor
the
entir
eatr
ium
andf
orth
een
tire
vent
ricle
.The
sam
ehol
dsfo
rth
ere
laxa
tioni
nth
eor
igin
alm
odel
,bec
ause
also
æèç éàêë isth
esa
mef
orth
een
tire
atriu
man
dfo
rth
een
tire
vent
ricle
.Cha
ngin
gto
are
gion
alac
tivat
ion
wou
ldst
illno
tbe
phys
iolo
gica
llyco
rrec
t,be
caus
e
æðç and
æðç éàêë wou
ldbe
the
sam
ewith
inea
chre
gion
.T
heph
ysio
logi
cally
corr
ectw
ayto
mod
elth
etim
eco
urse
ofco
n-tr
actio
nan
dre
laxa
tioni
sto
lett
heco
ntra
ctio
nand
rela
xatio
npro
paga
teove
rthe
hear
twal
lwith
afin
iteve
loci
ty.
With
resp
ectto
the
mod
el,t
his
mea
nsth
atth
eon
setti
me
ofsy
stol
e
æðç andth
een
dtim
eof
syst
ole
æðç éàêëar
egi
ven
indi
vidu
ally
6.2
The
new
activ
atio
nfu
nctio
n10
1
for
each
link
inth
ehe
arta
ndno
tfor
agr
oup
oflin
ksto
geth
er.T
his
impr
ovem
ent
ofth
eac
tivat
ion
func
tion
we
call
agr
adua
lact
ivat
ion.
The
grad
uala
ctiv
atio
naf
fect
sth
etim
ing
ofth
eco
ntra
ctio
nand
rela
xatio
n,w
hile
the
regi
onal
activ
atio
naf
fect
sbot
hth
etim
ing
andt
hera
teof
cont
ract
iona
ndre
laxa
tion.
The
grad
uala
ndre
gion
alac
tivat
ions
aren
otco
mpl
emen
taryp
roce
sses
but
can
beus
edto
geth
eror
sepa
rate
ly.If
they
are
used
toge
ther,
the
timin
gis
dete
rmin
edby
the
grad
uala
ctiv
atio
n,an
dth
era
teof
cont
ract
iona
ndre
laxa
tion
are
dete
rmin
edby
the
regi
onal
activ
atio
n.
6.2.
1T
heor
igin
alm
odel
Inth
isse
ctio
nwe
brie
flyre
peat
how
the
activ
atio
nfu
nctio
nis
defin
edin
the
orig
-in
alm
odel
.In
dias
tole
the
gove
rnin
gequ
atio
nfor
the
activ
atio
nfu
nctio
n
Ýì æí is(s
eeS
ec-
tion
3.3.
6)
ñ Ý ñ æò óÞußÝ
with
the
anal
ytic
solu
tion
Ýì æíòÝì æëí ×Ø
ôõö Ù ØÙ ÷ø
whe
reÞ ß isth
ere
laxa
tionr
ate
cons
tant
and
æ ë iseith
erth
eon
setti
me
ofdi
asto
leor
the
star
tingt
ime
ofth
esi
mul
atio
n.In
syst
olet
hego
vern
ing
equa
tions
are
(see
Sec
tion3
.3.6
)
ñ Ýúù ñ æòÞ"àì ÝáãäóÝ ùíüû
ñ Ý ñ æòÞ"àì Ý ùóÝí
with
the
anal
ytic
solu
tion
Ý ùì æíòÝ áãäóìÝ áãäóÝ ùì æèçíí ×Øôýö Ù ØÙÿþø
Ýì æíòÝ áãäóìÝ áãäóÝì æðçííì Þ àì æ óæðçíí ×Øôýö ÙØÙÿþø
whe
re
Þuà isthe
exci
tatio
nra
teco
nsta
nt,
Ý áãä
isth
elim
iting
valu
eof
the
activ
a-tio
nfu
nctio
nan
d
æèç isthe
time
ofon
seto
fsys
tole
.The
equa
tions
are
solv
edw
itha
com
mon
initi
alva
lue
for
the
auxi
liary
func
tion
Ý ù and
the
activ
atio
nfu
nctio
n
Ý :Ý ùì æèçíòÝì æèçí .T
hesi
xpa
ram
eter
sthat
dete
rmin
ethe
time
cour
seof
the
activ
atio
nfu
nctio
nar
e:
Þ ß ,Þ"à ,Ýáãä
,
Ýå òÝì æ òí ,
æðç time
ofon
seto
fsy
stol
eand
æðç éàêë time
ofen
dof
syst
ole(
whi
chis
equa
lto
æ ë thetim
eof
onse
tofd
iast
ole)
.The
reis
one
seto
fpar
amet
ersfo
rth
elin
ksin
the
atriu
m,o
nefo
rth
elin
ksin
the
vent
ricle
and
102
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
one
for
the
papi
llary
mus
cle
link.
The
only
dist
inct
ion
betw
eent
hese
tfor
the
vent
ricul
arlin
ksan
dth
atfo
rth
epa
pilla
rym
uscl
elin
kis
the
initi
alva
lue
Ýå .T
heva
lues
ofth
epa
ram
eter
sare
liste
din
Tabl
e3.
1in
Sec
tion3
.3.6
.
6.2.
2T
hene
wim
plem
enta
tion
The
core
ofth
ene
wim
plem
enta
tiono
fthe
activ
atio
nfu
nctio
n
Ý isthat
this
func
-tio
nin
clud
ing
the
six
para
met
ers
Þ"ß ,
Þ"à ,Ý áãä
,
Ýå ,
æðç and
æðç éàêë are
defin
edse
para
telyf
orea
chlin
kin
the
hear
tbou
ndar
y.
Gra
dual
activ
atio
n
Inth
eco
deof
the
orig
inal
mod
el,th
eac
tivat
ion
func
tion
was
give
nby
thre
evar
i-ab
les:
,
and
,re
pres
entin
gthe
activ
atio
nfu
nctio
nin
the
atriu
m,v
entr
icle
and
papi
llary
mus
cle
(cho
rd),
resp
ectiv
ely.
The
seva
riabl
esar
eno
wde
fined
asar
rays
with
inde
x
:
ì í ,
ì í an
d
ì í ,w
here
isan
inde
xto
the
link.
Eac
hof
thes
evar
iabl
esho
lds
only
valu
esof
links
inits
own
dom
ain,
e.g.
for
ì í w
ede
alon
lyw
ithin
dexe
s
tolin
ksth
atar
epa
rtof
the
atria
lhea
rtbo
unda
ry.T
hetw
opa
ram
eter
saffe
cted
byth
egr
adua
lact
ivat
ion
æèç and
æèç éàêë
wer
eorig
-in
ally
defin
edas
cons
tant
varia
bles
for
the
thre
egro
upsa
triu
m,v
entr
icle
and
pap-
illar
ych
ord,
resp
ectiv
ely.
The
yar
eno
w,
sim
ilarly
to
,
and
,de
fined
asar
rays
with
inde
x
,e.g
.the
time
æèç ofons
etof
atria
lsys
tole
,whi
chus
edto
beth
eva
riabl
e
,is
now
ì í .
We
setu
ptw
och
ains
oflin
ks,o
nefo
rth
eat
rium
,
,
and
one
for
the
vent
ricle
,
.
The
seva
riabl
esho
lds
the
inde
x,in
age
omet
rical
lyna
tura
lord
er,
toal
lth
elin
ksin
the
atriu
man
dve
ntric
le,r
espe
ctive
ly.
By
ge-
omet
rical
lyna
tura
lord
erw
em
eant
hat
ì í
and
ì í
hold
inde
xest
one
ighb
ourin
glin
kson
the
hear
tbou
ndar
y.F
igur
e6.
6sh
ows
how
the
two
chai
nsar
ese
tup.
Eac
hcr
ossl
ink
inth
eve
ntric
leis
num
bere
ddire
ctly
afte
rthe
link
alon
gth
erig
ht(p
oste
rior)
boun
dary
with
whi
chit
has
aco
mm
onbo
unda
rypo
int.
We
defin
east
artin
glin
k(
an
d
)
for
each
ofth
ech
ains
,i.e.
and
ar
eth
enu
mbe
rsof
the
links
whe
reth
ew
ave
ofco
ntra
ctio
nor
rela
x-at
ion
star
tsin
each
chai
n,re
spec
tivel
y.Fo
rphy
siol
ogic
alre
ason
sthe
star
tingl
ink
inth
eat
rium
shou
ldbe
atth
epo
sitio
nof
the
SA
node
and
the
star
tingl
ink
inth
eve
ntric
lesh
ould
beat
the
posi
tion
ofth
eA
Vno
de(s
eeS
ectio
n2.3
).N
ext,
we
pres
crib
eave
loci
ty
with
whi
chth
eac
tivat
ion
shou
ldpr
opag
ate
over
the
hear
tbou
ndar
y.T
heve
loci
tydo
esno
thav
eto
beth
esa
mef
orth
eat
rium
6.2
The
new
activ
atio
nfu
nctio
n10
3
aN
Nv
i+1
i+3
i+5
i+4
i+2
12
3
mitr
alrin
g
atriu
mve
ntric
le
12
mitr
alrin
g
i
vN
−v
N − v
N − v
4 2
N − 3
1
a1
N −
N − a
N − a
23
Fig
ure
6.6:
Illus
trat
ion
ofth
enu
mbe
ringo
fth
elin
ksin
the
two
chai
nsin
the
atriu
m(le
ft)an
dth
eve
ntric
le(r
ight
).
ã and
are
the
tota
lnum
bero
flin
ksin
the
chai
ns,re
spec
tivel
y.N
ote,
the
mitr
alrin
gis
mod
elle
dby
two
links
,one
inea
chof
the
chai
ns.
and
the
vent
ricle
,nor
does
itha
sto
beth
esa
mef
oral
llin
ksin
eith
erth
eat
rium
orth
eve
ntric
le.I
nth
efo
llow
ing
we
intr
oduc
eatp
ropa
gatio
nvel
ocity
for
the
atriu
m
ò ã a
nda
prop
agat
ionv
eloc
ityfo
rth
eve
ntric
le
ò .T
heor
etic
ally
the
prop
agat
ionv
eloc
itysh
ould
beeq
ualto
the
velo
city
ofco
nduc
tiono
fthe
elec
tric
alim
puls
esin
the
hear
t(se
eSec
tion2
.3).
Toim
plem
entt
hegr
adua
lact
ivat
ion
func
tion,
the
para
met
ersæðç a
nd
æðç éàêë
shou
ldbe
setf
orea
chlin
kac
cord
ingt
oth
elin
k’s
spat
ialp
ositi
onat
time
æ òæðç
and
æ òæðç éàêë .
How
ever
,to
sim
plify
the
calc
ulat
ions
we
com
pute
æðç and
æðç éàêë
acco
rdin
gto
the
initi
alpo
sitio
nof
the
link.
Tha
tis,
first
we
set
æðç and
æðç éàêë
for
the
star
ting
link
(
an
d
)
inea
chch
ain.
The
n,ba
sedo
nth
epr
opag
a-tio
nve
loci
tyan
dth
edi
stan
cebe
twee
nlin
ksin
the
initi
alhe
artc
onfig
urat
ion,
we
com
pute
æðç and
æðç éàêë
for
the
rest
ofth
elin
ksin
each
chai
n.T
hege
nera
lform
ula
for
the
onse
ttim
eof
syst
ole
æðç fora
link
is
æèçì í òæèçì çÙ ãßÙí
ì í
whe
re
çÙ ãßÙ ise
ither
or
,
isth
epr
opag
atio
nvel
ocity
and
ì í
isth
edi
stan
cebe
twee
nlin
k
and
the
star
tingl
ink
(eith
er
or
).
Typi
cally
,we
have
that
the
dist
ance
ì í
isof
the
form
óçÙ ãßÙ
å ,w
here
å isth
ein
itial
dist
ance
betw
eenn
eigh
bour
ingb
ound
aryp
oint
s.
104
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
Thu
s
æðçì í òæðçì çÙ ãßÙí ó çÙ ãßÙ
å
We
assu
met
hatt
hedu
ratio
nof
syst
ole
æèç éë ß
isth
esa
mef
oral
llin
ksin
the
atriu
man
dfo
ral
llin
ksin
the
vent
ricle
(Gan
ong,
1993
).T
hust
hege
nera
lform
ula
for
the
end
time
ofsy
stol
e
æðç éàêë
is
æðç éàêëì íòæðçìíæðç éë ß
Inth
ein
itial
hear
tcon
figur
atio
nthe
dist
ance
betw
eenn
eigh
bour
ingp
oint
sis
å .By
com
putin
gthe
valu
esfo
rthe
links
acco
rdin
gto
thei
rini
tialp
ositi
ons,
we
gett
herig
hton
setti
me
ofsy
stol
e
æðçì í on
lyif
the
dist
ance
betw
eenn
eigh
bour-
ing
poin
tsat
æðçì í is
å .T
his
cond
ition
does
noth
old,
and
we
take
that
into
acco
untw
hen
we
com
pute
æðçì í .
Fort
heat
rium
we
know
that
atth
eon
seto
fsys
tole
æ òæðç
the
dist
ance
betw
een
neig
hbou
ringp
oint
sis
ÙFþò ! å ,
i.e.
åò"ÙFþ .
So
toas
sure
the
corr
ectv
alue
of
æðçì í w
em
ultip
lyth
epr
opag
atio
nvel
ocity
inth
eat
rium
ò
ã by
2.T
hust
heeq
uatio
nsfo
rth
eon
setti
me
ofsy
stol
eand
end
time
ofsy
stol
ein
the
atriu
mar
e
æðçì í òæðçì í ó
å " ã
æðç éàêëì íòæðçì íæðç éë ß
Inth
eve
ntric
leth
edi
stan
cebe
twee
nnei
ghbo
urin
gpoi
ntsa
tthe
onse
tofs
ys-
tole
æ òæðç isa
ppro
xim
atel
y
Ù #ò $$å ,
i.e.
åò $Ùÿþ .
Thu
sw
ene
edto
mul
tiply
the
prop
agat
ionv
eloc
ityin
the
vent
ricle
ò b
y
$ toas
-su
reth
eco
rrec
tval
ueof
æðçì í .
The
onse
tand
end
time
ofsy
stol
efo
ral
lth
elin
ksal
ong
the
left
(ant
erio
r)ve
ntric
ular
wal
l(c
alle
dsh
ortl
inks
)ar
eco
mpu
ted
sim
ilarly
toth
elin
ksin
the
atriu
m,i.
e.
æèçì í òæèçì í ó
å $
æèç éàêëì í òæèçì í æèç éë ß
Her
e
æðçì í ha
sbee
nset
toa
valu
egre
ater
than
æðç forth
ela
stlin
kin
the
atria
lch
ain,
the
diff
eren
cebe
ing
equa
ltoth
ede
layi
nth
eA
Vno
de(s
eeS
ectio
n2.3
).Fo
rth
esh
ortl
inks
alon
gth
erig
htve
ntric
ular
wal
lthe
prin
cipl
eis
the
sam
e,pr
ovid
edw
eta
kein
toac
coun
tthat
ever
yse
cond
inde
xis
anin
dex
toa
cros
slin
k.
6.2
The
new
activ
atio
nfu
nctio
n10
5
Thu
s
æèçì í òæèçì í " ó
å $
æèç éàêëì í òæèçì í æèç éë ß
Ons
etan
dend
time
ofsy
stol
efor
each
cros
slin
kin
the
vent
ricle
area
ssoc
iate
dw
ithth
esh
ortl
inks
alon
gth
erig
htve
ntric
ular
wal
l.T
heva
lues
of
æðç and
æðç éàêë
for
acr
ossl
ink
are
iden
tical
toth
eva
lues
for
the
next
shor
tlink
inth
eve
ntric
ular
chai
n,e.
g.th
esh
ortl
ink
with
inde
x
" an
dth
ecr
ossl
ink
with
inde
x
in
Fig
ure
6.6
have
iden
tical
valu
esof
onse
tand
end
time
ofsy
stol
e.In
sum
mar
ywe
have
that
with
the
grad
uala
ctiv
atio
nth
etim
eco
urse
ofth
eac
tivat
ion
func
tion
isde
term
ined
byth
efo
llow
ing
para
met
ers:t
hest
artin
glin
ks
an
d
,t
heon
setti
mes
ofsy
stol
efor
the
star
ting
links
æèçì í an
d
æèçì í ,
the
velo
citie
s ã a
nd
,and
the
dura
tion
times
ofsy
stol
e
æèç éë ß
inth
eat
rium
and
vent
ricle
.
Reg
iona
lact
ivat
ion
The
regi
onal
activ
atio
nfu
nctio
nis
intr
oduc
edba
sedo
nth
ew
ish
tosi
mul
ater
e-gi
onal
diff
eren
cesin
the
vent
ricul
arac
tivat
ion,
part
icul
arly
inth
eve
ntric
ular
re-
laxa
tion.
Hen
ce,th
isim
prov
emen
toft
heac
tivat
ion
func
tion
affe
ctso
nly
the
acti-
vatio
nfu
nctio
nfo
rth
eve
ntric
ular
links
.Whi
leth
egr
adua
lact
ivat
ion
affe
ctso
nly
æèç andæèç éàêë ,the
regi
onal
activ
atio
naf
fect
sall
six
para
met
ersth
atde
term
inet
heac
tivat
ion
func
tion,
æèç ,æèç éàêë ,
Þ à ,Þuß ,Ýâáãä
and
Ýå .W
ede
finefi
vere
gion
soft
heve
ntric
ular
hear
twal
lofa
ppro
xim
atel
ythe
sam
esi
ze:
regi
on1
alon
gth
eup
pera
nter
iorw
all,
regi
on2
alon
gth
elo
wer
ante
rior
wal
l,re
gion
3al
ong
the
apex
,reg
ion
4al
ong
the
low
erpo
ster
iorw
alla
ndre
gion
5al
ong
the
uppe
rpos
terio
rwal
l.F
igur
e6.
7sh
ows
the
five
regi
ons
atth
ein
itial
time.
The
regi
onsc
orre
spon
dto
the
five
regi
onsp
rese
nted
inS
ectio
n6.1
.4.T
hepa
pilla
rym
uscl
elin
kbe
long
sto
regi
on3.
Inea
chof
the
five
regi
ons
all
the
para
met
ers(
æðç ,æðç éàêë ,
Þ à ,
Þ"ß ,Ýå an
d
Ýáãä )de
term
inin
gthe
activ
atio
nfu
nctio
nar
efix
ed.T
hus
ifw
ech
oose
regi
onal
and
notg
radu
alac
tivat
ion,
the
valu
esof
æðç and
æðç éàêë are
the
sam
ewith
inea
chre
gion
.It
isal
sopo
ssib
leto
have
both
regi
onal
and
grad
uala
ctiv
atio
nat
the
sam
etim
e.In
this
case
æèç and
æèç éàêë areg
iven
byth
egr
adua
lact
ivat
ion
proc
essa
ndth
ere
mai
ning
four
para
met
ersa
rede
term
ined
byth
eva
lue
for
the
give
nre
gion
.We
give
the
valu
esfo
rth
epa
pilla
rym
uscl
elin
kin
depe
nden
tlyof
the
regi
onva
lues
,as
sum
ingc
ondi
tions
whe
reth
epa
pilla
rym
uscl
eact
ivat
ion
isno
taffe
cted
byth
ehe
artd
isea
se.
106
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
2
4
1
5
aort
a
valv
em
itral
atriu
m
vent
ricle
3
Fig
ure
6.7:
The
five
regi
onso
fthe
vent
ricul
arhe
artw
alla
tini
tialt
ime.
6.2.
3T
ime
cons
tant
ofre
laxa
tion
InS
ectio
n4.
1.3
we
foun
da
clos
ere
latio
nbe
twee
nthe
time
cour
seof
the
ac-
tivat
ion
func
tion
Ý andth
etim
eco
urse
ofth
eve
ntric
ular
pres
sure
,par
ticul
arly
from
the
begi
nnin
gof
rela
xatio
nun
tilth
em
itral
valv
eop
ens(
see
Fig
ure
4.7)
.F
urth
erm
ore,
the
prev
ious
sect
ions
show
that
durin
gre
laxa
tion
Ý isdesc
ribed
byan
expo
nent
iald
ecay
ofth
efo
rm
×Øô õÙ .
Thi
ssu
ppor
tsth
eus
eof
apr
essu
re-
dete
rmin
edin
dex
(the
time
cons
tant
Ü ofan
expo
nent
iald
ecay
durin
gre
laxa
tion)
for
the
eval
uatio
noft
here
laxa
tion,
asde
scrib
edin
Sec
tion6
.1.4
.Hen
ce,a
ssum
-in
gth
at
Ü ò% Þuß ,w
eca
nus
emea
sure
men
tsoft
hetim
eco
nsta
nt
Ü todete
rmin
e
Þuß inthe
activ
atio
nfu
nctio
n.
6.3
Cal
ibra
ting
the
mod
el
The
para
met
ersd
eter
min
ingt
hetw
oim
prov
emen
tsoft
heac
tivat
ion
func
tion
(gra
d-ua
land
regi
onal
activ
atio
n)ar
ese
tup
tom
atch
ahe
alth
yhum
anby
com
paris
ons
with
MR
data
and
calib
ratio
nsba
sedo
nph
ysio
logi
calk
now
ledg
e.
6.3.
1V
eloc
itypr
ofile
sal
ong
tran
sver
selin
es
Inad
ditio
nto
the
typi
calo
utpu
tfro
mth
em
odel
(see
Sec
tion4
.1)w
eal
solo
okat
spat
ialv
eloc
itypr
ofile
s(of
abso
lute
velo
city
)alo
ngtr
ansv
erse
lines
inth
eve
ntri-
cle
bya
met
hods
imila
rto
that
used
byH
oulin
det
al.(
2001
)for
MR
data
durin
gth
eea
rlyfil
ling
phas
e.W
ede
scrib
edth
atm
etho
din
Sec
tion
6.1.
3.S
ince
we
are
only
inte
rest
edin
the
dias
tolic
fillin
gph
ase,
we
only
inve
stig
atet
heve
loci
typr
ofile
sdu
ring
dias
tole
,but
we
inve
stig
atet
hem
for
alo
nger
time
perio
dan
dat
6.3
Cal
ibra
ting
the
mod
el10
7
smal
lert
ime
inte
rval
s(a
ppro
xim
atel
y13
ms)
than
done
inH
oulin
det
al.(
2001
).W
hen
we
inve
stig
atet
here
sults
from
the
six
lines
,we
look
atth
esp
atia
lve-
loci
typr
ofile
alon
gea
chof
the
lines
atdi
ffer
entt
imes
and
com
pute
PV,
VI
and
TD
for
each
ofth
elin
es(s
eeS
ectio
n6.1
.3fo
ra
defin
ition
ofth
eseq
uant
ities
).B
ecau
sew
ein
vest
igat
ethe
lines
for
alo
nger
time
perio
dth
ando
nein
Hou
lind
etal
.(20
01)w
ear
eab
leto
find
high
erva
lues
ofT
D.
InH
oulin
det
al.
(200
1)th
elin
esar
efix
edin
spac
e,an
dth
isis
reas
onab
lebe
caus
ethe
mot
ion
ofth
ehe
arti
sre
lativ
ely
smal
l.H
owev
er,
inth
em
odel
the
hear
tmot
ion
ism
ore
pron
ounc
ed.T
here
fore
we
exam
inet
hedi
ffer
ence
betw
een
resu
ltsfr
omlin
esfix
edin
spac
eand
resu
ltsfr
omlin
esfo
llow
ing
the
hear
tmot
ion.
The
lines
fixed
insp
acea
rese
tup
sim
ilarly
toth
em
etho
dfor
the
MR
data
:at
the
time
ofpe
akve
loci
tybe
twee
nthe
leafl
ettip
sa
line
ispl
aced
atth
ele
velo
fth
em
itral
leafl
ettip
s(M
LT
)bu
tper
pend
icul
arto
the
inflo
wje
t.A
noth
erlin
eis
plac
edpa
ralle
ltoth
efir
ston
ebu
tdis
plac
ed2
cmup
stre
am,a
ppro
xim
atel
yatt
hele
velo
fth
em
itral
ring.
Fur
ther
mor
e,4
lines
are
plac
edpa
ralle
lto
the
first
line
butd
ispl
aced
1,2,
3an
d4
cmdo
wns
trea
m.T
heto
pro
wof
Fig
ure
6.8
show
sth
eve
loci
tyfie
ld,t
hehe
artb
ound
arya
ndth
esi
xlin
esat
two
times
durin
gth
efir
stcy
cle
ofa
sim
ulat
ionw
here
the
lines
are
fixed
insp
ace.
Fort
helin
esfo
llow
ing
the
hear
tmot
ion
we
take
,ate
acht
ime
step
,the
mitr
alrin
gas
the
basi
clin
e,be
caus
ethis
isa
wel
ldefi
nedl
ine
inth
em
odel
and
the
flow
ispr
actic
ally
perp
endi
cula
rtoth
islin
e.T
hen
we
plac
e5lin
espa
ralle
ltoth
islin
ebu
tdis
plac
ed2,
3,4,
5an
d6
cmdo
wns
trea
m,se
ebot
tom
row
ofF
igur
e6.
8.T
helin
e2
cmdo
wns
trea
mof
the
mitr
alrin
gis
appr
oxim
atel
yatt
hele
velo
fthe
mitr
alle
aflet
tips
and
we
thus
deno
teit
ML
T.T
heco
mpa
rison
betw
eent
here
sults
obta
ined
atlin
esfix
edin
spac
eand
lines
follo
win
gth
ehe
artm
otio
nre
veal
sth
atev
enth
ough
we
only
look
atdi
asto
le(w
here
the
mot
ion
isle
ssth
anin
syst
ole)
,the
hear
tmot
ion
isto
ogr
eatt
oob
-ta
inre
ason
abler
esul
tsfr
omth
elin
esfix
edin
spac
e.E
.g.t
helin
epo
sitio
ned1
cmdo
wns
trea
mof
the
mitr
alle
aflet
tips
mov
esab
ove
the
leve
lof
the
mitr
alle
aflet
tips,
and
the
line
atth
em
itral
ring
ispo
sitio
nedi
nth
em
iddl
eof
the
atriu
m,s
eeto
prig
htpi
ctur
ein
Fig
ure
6.8.
Thu
sw
eco
nclu
deth
atw
esh
ould
choo
selin
esfo
llow
ing
the
hear
tmot
ion.
Sin
ceth
elin
esth
atfo
llow
the
hear
tmot
ion
are
base
don
the
line
atth
em
itral
ring
and
noto
na
line
posi
tione
dper
pend
icul
arto
the
inflo
wje
t,w
eca
nnot
bece
rtai
ntha
tthe
lines
are
perp
endi
cula
rtoth
ein
flow
jet.
Thi
sis
nota
prob
lem
for
the
velo
city
inde
xess
ince
they
areb
ased
onth
em
axim
umof
anab
solu
teve
loci
ty,
butt
hesp
atia
lvar
iatio
noft
heve
loci
tyal
ongt
helin
esm
ight
bedi
ffer
entth
anse
enin
MR
data
obta
ined
byth
em
etho
ddes
crib
edin
Hou
lind
etal
.(20
01).
108
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
Line
sfix
edin
spac
e
Time
0.164
Time
0.244
Line
sfo
llow
ing
the
hear
tmot
ion
Time
0.164
Time
0.244
Fig
ure
6.8:
The
six
lines
whe
resp
atia
lvel
ocity
profi
lesa
reco
mpu
tedd
urin
gdi
-as
tole
inth
eca
seof
lines
fixed
insp
ace(
top)
and
lines
follo
win
gth
ehe
artm
otio
n(b
otto
m).
The
pict
ures
inth
ele
ftsi
deis
atth
etim
eof
peak
velo
city
atth
em
itral
leafl
ettip
s.
6.3.
2G
radu
alac
tivat
ion
The
para
met
ersd
eter
min
ingt
hegr
adua
lact
ivat
ion
are:
the
star
tingl
ink
inat
rium
,t
hest
artin
glin
kin
vent
ricle
,t
heon
setti
me
ofsy
stol
efor
the
star
ting
links
æðçì í an
d
æðçì í ,
the
velo
citie
s
and
ã ,the
dura
tion
times
ofsy
stol
e
æðç éë ß
inth
eat
rium
and
inth
eve
ntric
le,a
ndth
eon
set/e
ndtim
esof
syst
ole
for
the
papi
llary
mus
clel
ink.
The
para
met
ersfo
rth
epa
pilla
rym
uscl
elin
kar
ese
tse
para
telyf
rom
the
vent
ricul
arpa
ram
eter
s,bec
ause
ittu
rns
outt
hatt
hem
odel
isve
ryse
nsitiv
eto
the
activ
atio
nof
the
papi
llary
mus
cle.
We
perf
orm
eda
serie
sof
sim
ulat
ions
tode
cide
the
best
setti
ngof
the
pa-
6.3
Cal
ibra
ting
the
mod
el10
9
ram
eter
sfor
the
grad
uala
ctiv
atio
n.W
ede
fine
the
best
setti
ngas
one
that
best
impr
oves
the
com
paris
onw
ithM
Rda
ta(s
eeC
hapt
er5)
with
outi
ntro
duci
ngno
n-ph
ysio
logi
calb
ehav
iour
ofth
ehe
art.
E.g
.it
isim
port
antt
hatt
hem
ovem
ento
fth
em
itral
valv
eis
sym
met
rical
and
nota
sym
met
rical
(see
Ade
lera
ndJa
cobs
en(2
001)
and
Jaco
bsen
etal
.(20
01))
.For
allo
fthe
sim
ulat
ions
we
chos
eto
lett
hedu
ratio
nofs
ysto
lebe
the
sam
easi
nou
rref
eren
cem
odel
,i.e.
æðç éë ßis
0.10
4san
d0.
306
sfo
rth
eat
rium
and
vent
ricle
,res
pect
ively
.Fur
ther
mor
e,
æðç forth
est
artin
glin
ksw
erec
hose
nto
beth
esa
mea
s
æðç inour
refe
renc
emod
el,i.
e.in
the
first
hear
tcy
cle
æðçì íò 0.36
2s
and
æðçì íò 0.54
5s.
The
sim
ulat
ions
indi
cate
that
the
mod
elis
notv
ery
sens
itive
toth
eac
tivat
ion
ofth
eat
rium
,sho
win
gon
lym
inor
diff
eren
cesw
hen
we
chan
geth
eve
loci
ty
ã and
the
star
tingl
ink
.
We
chos
eto
lett
heat
riala
ctiv
atio
npr
opag
ates
ymm
etri-
cally
over
the
hear
twal
lfro
mth
em
idpo
into
nth
eat
rialw
alla
ndw
itha
velo
city
of
ãò 0.5m
/s,w
hich
isph
ysio
logi
cally
corr
ect(
seeS
ectio
n2.3
).
is
not
plac
edat
the
posi
tion
ofth
eS
Ano
de,b
ecau
sea
sym
met
rical
activ
atio
nof
the
atriu
mgi
vesa
mov
emen
toft
hem
itral
valv
eth
atis
slig
htly
mor
esy
mm
etric
al.
The
phys
iolo
gica
llyco
rrec
tvel
ocity
inth
eve
ntric
leis
appr
oxim
atel
y5.0
m/s
,so
ò 5.0m
/s.T
hest
artin
glin
kin
the
vent
ricle
isth
eto
plin
kon
the
ante
rior
wal
l,si
nce
this
isap
prox
imat
elyw
here
the
AV
node
islo
cate
din
the
hear
t,se
eS
ectio
n2.3
.B
ecau
seth
em
odel
star
tsup
from
anun
phys
iolo
gica
lsitu
atio
n(a
llth
ebl
ood
isat
rest
at
æ ò ),
we
chos
eto
lett
here
laxa
tionb
esi
mul
tane
ousin
the
entir
eve
ntric
lein
the
first
hear
tcyc
le.T
hepr
opag
atio
noft
heac
tivat
ion
over
the
vent
ricul
arw
alld
oesn
otch
ange
the
sim
ulat
ionr
esul
tsve
rym
uch,
the
bigg
est
diff
eren
cebe
ing
seen
inth
eou
tflow
thro
ught
heao
rtic
sink
.How
ever
,the
mod
elis
very
sens
itive
toth
eac
tivat
ion
time
ofth
epa
pilla
rym
uscl
elin
k.W
hen
the
papi
llary
mus
clel
ink
isde
fined
aspa
rtof
the
vent
ricul
arlin
kch
ain
and
thus
isac
tivat
edac
cord
ingt
oth
epr
opag
atio
noft
heac
tivat
ion
wav
e,th
ean
-te
riorv
entr
icul
arw
allm
otio
nis
very
abno
rmal
beca
useo
favi
goro
usco
ntra
ctio
nof
this
part
ofth
ew
all.
Thi
sis
beca
uset
hepa
pilla
rym
uscl
eco
ntra
ctsb
efor
eth
epo
ster
iorw
all.
Eve
nif
we
lett
hepa
pilla
rym
uscl
econ
trac
t20
ms
late
r,th
eco
ntra
ctio
noft
hean
terio
rwal
lis
dist
orte
dand
the
mitr
alva
lve
clos
esve
ryas
ym-
met
rical
ly.T
his
can
bese
enin
Fig
ure
6.9
whe
reth
ehe
artb
ound
aryc
onfig
urat
ion
issh
own
attw
otim
esfr
oma
sim
ulat
ionw
ithth
egr
adua
lact
ivat
ion
and
from
the
refe
renc
esim
ulat
iond
escr
ibed
inC
hapt
er4.
The
dist
orte
dcon
trac
tiono
fthe
ante
riorw
alla
ndth
eas
ymm
etric
alclo
sing
ofth
em
itral
valv
eis
seen
beca
useth
epa
pilla
rym
uscl
eis
teth
ered
toa
poin
talo
ngth
ean
terio
rwal
l.In
ourp
revi
ousl
yre
port
edw
ork
(Jac
obse
neta
l.,20
01;A
dele
rand
Jaco
bsen
,2001
)the
papi
llary
mus
clew
aste
ther
edto
apo
inta
long
the
post
erio
rw
all,
and
here
the
cont
ract
iono
fth
epo
ster
iorw
allw
asdi
stor
teda
ndth
em
itral
110
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
Time1.5600
Time1.6380
Fig
ure
6.9:
Hea
rtbo
unda
ryat
two
times
durin
ga
hear
tcyc
lefr
oma
sim
ulat
ion
with
grad
uala
ctiv
atio
nw
ith
ò 5.0m
/san
d20
ms
late
ract
ivat
ion
ofpa
pilla
rym
uscl
e(
Ö )an
dfr
omth
ere
fere
nces
imul
atio
n(
& ).
valv
ecl
osed
asym
met
rical
lyto
the
othe
rsid
e.T
hebe
stw
ayto
min
imiz
eth
edi
stor
tion
ofth
ean
terio
rwal
lis
toin
crea
seth
eve
loci
ty
inth
eve
ntric
le.I
fw
ese
t
ò 10.0
m/s
and
lett
hepa
pilla
rym
uscl
eco
ntra
ct10
ms
late
rtha
nit
shou
ldac
cord
ingt
oth
epr
opag
atio
nof
the
activ
atio
nw
ave,
we
obta
inre
ason
ables
imul
atio
nre
sults
.How
ever
,
ò 10.0
m/s
isto
ohi
ghco
mpa
redt
oty
pica
lvel
ociti
esin
the
Pur
kinj
efib
ers
ofap
prox
imat
ely3
.0-
5.0
m/s
(see
Sec
tion2
.3).
Fig
ure
6.10
show
sth
ehe
artb
ound
arya
tsix
times
durin
gth
ese
cond
hear
tcy
cle
from
asi
mul
atio
nwith
ò 10.0
m/s
and
10m
sla
tera
ctiv
atio
nof
the
pap-
illar
ym
uscl
eand
from
the
refe
renc
esim
ulat
ion.
We
sees
omed
iffer
ence
inth
ehe
artb
ound
aryc
onfig
urat
iona
ndna
tura
llyth
ebi
gges
tdiff
eren
ceis
durin
gsy
s-to
le.B
utth
efig
ure
also
show
sa
bette
rcon
trac
tiono
fthe
ante
riorw
allc
ompa
red
toth
ere
sults
inF
igur
e6.
9an
dan
impr
oved
mov
emen
tof
the
mitr
alva
lve
dur-
ing
clos
ing,
thou
ghth
eva
lve
does
notc
lose
assy
mm
etric
ally
asin
the
refe
renc
esi
mul
atio
n.To
furt
herj
udge
the
chan
gesin
sim
ulat
ionr
esul
tsdu
eto
the
grad
uala
ctiv
atio
nw
eco
mpa
repr
essu
retim
epr
ofile
s,ve
loci
tytim
epr
ofile
san
dflo
wtim
epr
ofile
s.T
hepr
essu
retim
epr
ofile
s(n
otsh
own)
show
hard
lyan
ydi
ffer
ence
com
pare
dto
the
refe
renc
esim
ulat
ion.
The
velo
city
time
profi
les
over
the
mitr
alrin
gan
dbe
-tw
een
the
leafl
ettip
sar
esh
own
inF
igur
e6.
11.
The
figur
esh
ows
that
the
peak
velo
citie
sfor
the
mitr
alrin
gpr
ofile
are
low
erin
the
sim
ulat
ion
with
the
grad
ual
6.3
Cal
ibra
ting
the
mod
el11
1
Time1.0920
Time1.2480
Time1.4040
Time1.5600
Time1.7160
Time1.8720
Fig
ure
6.10
:Hea
rtbo
unda
ryat
six
times
durin
ga
hear
tcyc
le.F
rom
asi
mul
atio
nw
ithgr
adua
lact
ivat
ion
with
ò 10.0
m/s
and
10m
sla
tera
ctiv
atio
nof
papi
llary
mus
cle(
Ö )an
dfr
omth
ere
fere
nces
imul
atio
n(
& ).
112
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
11.2
1.4
1.6
1.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
11.2
1.4
1.6
1.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
Fig
ure
6.11
:Vel
ocity
time
profi
leof
the
mea
nvel
ocity
(cm
/s)a
tthe
mitr
alrin
g(s
olid
)and
betw
eent
hele
aflet
tips
(das
hed)
from
asi
mul
atio
nw
ithgr
adua
lact
i-va
tion
(left)
and
from
the
refe
renc
esim
ulat
ion(
right
).
11.2
1.4
1.6
1.8
t[s]
0
20
40
60
80
velocity[cm/s]
11.2
1.4
1.6
1.8
t[s]
0
20
40
60
80
velocity[cm/s]
Fig
ure
6.12
:Vel
ocity
time
profi
leof
the
mea
nvel
ocity
(cm
/s)a
cros
sthe
aort
icou
tflow
trac
tbel
owth
eao
rtic
sink
from
asi
mul
atio
nwith
grad
uala
ctiv
atio
n(le
ft)an
dfr
omth
ere
fere
nces
imul
atio
n(rig
ht).
6.3
Cal
ibra
ting
the
mod
el11
3
11.2
1.4
1.6
1.8
t[s]
-600
-400
-2000
200
sourceflow[cm^3/s]
11.2
1.4
1.6
1.8
t[s]
-600
-400
-2000
200
sourceflow[cm^3/s]
Fig
ure
6.13
:F
low
time
profi
lefo
rth
eflo
w(c
m
' /s)
thro
ugh
the
mitr
also
urce
(das
hed)
and
the
aort
icsi
nk(f
ull)
from
asi
mul
atio
nwith
grad
uala
ctiv
atio
n(le
ft)an
dfr
omth
ere
fere
nce(
right
).
activ
atio
nim
plem
ente
d.T
hepe
akin
the
leafl
ettip
profi
leca
used
byth
eat
rialc
on-
trac
tion
ishi
gher
inth
esi
mul
atio
nw
ithth
egr
adua
lact
ivat
ion
and,
asex
pect
ed,
occu
rsla
ter.
Bas
edon
the
com
paris
onbe
twee
nthe
refe
renc
esim
ulat
iona
ndM
Rda
tapr
esen
tedin
Cha
pter
5,w
edo
notw
antt
heat
rials
ysto
leto
bede
laye
d.A
part
from
the
diff
eren
cesa
tthe
peak
s,th
epr
ofile
sar
eco
mpa
rabl
eand
have
sim
ilar
shap
es.
Fig
ure
6.12
show
sth
eve
loci
typr
ofile
acro
ssth
eao
rtic
outfl
owtr
act,
whi
leF
igur
e6.1
3sh
ows
the
flow
thro
ught
hem
itral
sour
cean
daor
ticsi
nk.T
heve
loci
typr
ofile
acro
ssth
eao
rtic
outfl
owtr
acts
how
son
lym
inor
diff
eren
cesc
ompa
redt
oth
ere
fere
nces
imul
atio
n,e.
g.a
slig
htly
low
erpe
akva
lue.
Fro
mth
eso
urce
/sin
kflo
wpr
ofile
sw
ese
eth
atth
eao
rtic
sink
open
sand
clos
essl
ight
lyla
ter
inth
esi
mul
atio
nw
ithth
egr
adua
lact
ivat
ion.
Thi
sis
expe
cted
beca
useo
nset
ofsy
stol
eat
the
first
link
inth
eve
ntric
leis
atth
esa
met
ime
ason
seto
fsys
tole
inth
een
tire
vent
ricle
inth
ere
fere
nces
imul
atio
n.T
hus
the
over
allv
entr
icul
arco
ntra
ctio
nis
dela
yedi
nth
esi
mul
atio
nwith
grad
uala
ctiv
atio
n.F
urth
erm
ore,
the
peak
outfl
owra
teis
less
inth
egr
adua
lact
ivat
ion
sim
ulat
ion
than
inth
ere
fere
nces
imul
atio
n.T
heflo
wth
roug
hthe
mitr
also
urce
show
son
lysm
alld
iscr
epan
cies
,e.g.
the
flow
isne
vern
egat
ive
inth
egr
adua
lact
ivat
ion
sim
ulat
iona
sin
the
refe
renc
esim
ulat
ion.
The
resu
ltspr
esen
tedin
this
sect
ion
show
that
we
obta
inre
ason
abler
esul
ts
114
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
only
with
anun
phys
iolo
gica
lpara
met
erset
ting(
ò 10.0
m/s
)and
the
resu
ltsar
eve
ryse
nsitiv
eto
the
activ
atio
nof
the
papi
llary
mus
cle.
Fur
ther
mor
e,th
egr
adua
lac
tivat
ion
only
slig
htly
influ
ence
sthe
sim
ulat
ion
resu
lts,th
usw
ech
oose
nott
ous
egra
dual
activ
atio
n.
6.3.
3R
egio
nala
ctiv
atio
n
Thi
sse
ctio
ndes
crib
esth
em
odel
setu
pfo
rahe
alth
yhum
anw
henw
eus
eonl
yth
ere
gion
alac
tivat
ion.
Thi
sse
tup
invo
lves
the
para
met
ers
Þ"à ,Þ"ß ,Ýå an
d
Ý áãä ,w
hile
æðç and
æðç éàêë
are
unch
ange
dfrom
the
orig
inal
mod
else
tup
(see
Tabl
e3.
1fo
rth
eor
igin
alm
odel
setu
p).
InS
ectio
n6.1
we
saw
that
path
olog
ical
hear
tcon
ditio
nsaf
fect
the
rela
xatio
nof
the
vent
ricul
arhe
artm
uscl
e.T
hust
hepr
imar
ypa
ram
eter
toad
just
isth
ere
lax-
atio
nra
teco
nsta
nt
Þuß forth
eve
ntric
le,w
hich
corr
espo
ndsto
the
reci
proc
alof
the
para
met
er
Ü inphy
siol
ogic
alm
easu
rem
ents
.Adj
ustin
g
Þ ß chang
esth
edu
ratio
ntim
eof
dias
tole
and
cons
eque
ntlyt
hepe
riod
ofth
eac
tivat
ion
func
tion,
whi
chin
turn
mea
nsth
atth
ele
ngth
ofa
hear
tcyc
leis
alte
red.
We
choo
seno
tto
adju
stan
yof
the
othe
rpar
amet
ersfo
rth
eac
tivat
ion
func
tion,
sinc
ew
ew
illlim
itou
rin
vest
igat
ions
toth
eea
rlyfil
ling
phas
ewhi
chis
durin
gre
laxa
tion.
For
ahe
alth
yhu
man
Ü isthe
sam
efor
the
entir
eve
ntric
le,s
oth
ere
are
nore
gion
aldi
ffer
ence
sinth
ere
laxa
tion.
Thu
s
Þ"ß shoul
dbe
the
sam
efor
the
entir
eve
ntric
lein
the
mod
el.
Whe
nco
mpa
ringt
heva
lue
of
Þ ß (% !!(Ø) )
inou
rre
fere
ncem
odel
with
the
reci
proc
alof
mea
sure
dval
ueso
f
Ü forthe
vent
ricle
asa
who
le(i.
e.no
tfor
any
part
icul
arre
gion
inth
eve
ntric
le)it
turn
sout
that
this
valu
eco
rres
pond
stoa
dise
ased
hear
t.T
heva
lues
ofth
eac
tivat
ion
func
tion
para
met
ers
inou
rre
fere
ncem
odel
(see
Tabl
e3.
1)w
ere
notc
hose
nbas
edon
phys
iolo
gica
lkn
owle
dge,
butw
ere
sett
oac
hiev
eth
esa
met
imin
gin
the
mod
elas
inth
eM
Rda
taw
eco
mpa
redth
em
odel
with
.Nis
hiha
raet
al.(
2000
)find
that
inpa
tient
swith
HC
M
Ü is54
* 12m
s(m
ean
* SD
),w
hile
inth
eco
ntro
lgro
upof
heal
thyp
erso
ns
Üis
32
* 7m
s.S
imila
rly,O
hte
etal
.(19
99)fi
ndth
atin
patie
ntsw
ithe.
g.A
MI
Ü is48
.5
* 6m
s,w
hile
Ü forahe
alth
ycon
trol
grou
pis
37.6
* 2.6
ms.
Not
eth
atal
lof
the
stud
iesm
entio
neda
reof
pers
onsw
ithan
agea
roun
d60
year
s.B
ased
onth
ese
mea
sure
men
tsof
Ü wecha
nge
Þ"ß from
%!!(Ø) to
% +,(Ø) .T
hepe
riod
ofth
eac
tivat
ion
func
tion
(and
cons
eque
ntlyt
hele
ngth
ofa
hear
tcyc
le)t
henc
hang
esfr
om0.
936
sto
0.90
9s.
Inre
duci
ngth
edu
ratio
nofd
iast
olet
hele
ngth
ofa
hear
tcy
cle
hasb
eenr
educ
ed,a
ndth
isis
phys
iolo
gica
llyco
rrec
t.To
dete
rmin
ethe
affe
ctof
the
chan
gedr
elax
atio
nrat
e
Þ"ß andth
esh
orte
rhea
rtcy
cle
leng
th,w
eco
mpa
rea
sim
ulat
ion
with
thes
enew
para
met
ersim
plem
ente
dw
ithth
ere
fere
nces
imul
atio
nde
scrib
edin
Cha
pter
4.G
ener
ally
the
resu
ltsar
e
6.3
Cal
ibra
ting
the
mod
el11
5
very
sim
ilar.
We
find
that
the
pres
sure
leve
l(no
tsho
wn)
isra
ised
alit
tlew
ithth
ene
wpa
ram
eter
choi
cean
dth
eve
loci
tytim
epr
ofile
atth
eao
rtic
outfl
ow(n
otsh
own)
issl
ight
lych
ange
d.T
heve
loci
tytim
epr
ofile
satt
hem
itral
valv
ear
eal
soch
ange
d,w
hich
isse
enin
Fig
ure
6.14
.The
wid
thof
the
first
peak
inth
em
itral
ring
profi
leis
wid
eran
dth
em
eanv
eloc
ityov
erth
em
itral
ring
ispr
actic
ally
zero
attw
opo
ints
intim
e.F
rom
the
com
paris
onbe
twee
nthe
refe
renc
esim
ulat
ion
and
MR
data
inC
hapt
er5
we
seet
hatb
oth
ofth
esef
eatu
resa
reim
prov
emen
tsof
the
sim
ulat
ion
resu
lts.I
nth
eve
loci
typr
ofile
betw
eent
hele
aflet
tips
from
the
sim
ulat
ionw
ith
Þ"ßò% +,(Ø) w
ese
eane
gativ
eve
loci
tyju
stbe
fore
æ ò 1.2
s,w
hich
isre
late
dto
the
sem
iclo
sure
ofth
em
itral
valv
e.T
hem
odel
with
the
abov
ede
scrib
edpa
ram
eter
adju
stm
ents(
Þ ß andle
ngth
ofhe
artc
ycle
)w
ere
fert
oas
a’h
ealth
ype
rson
’.In
the
follo
win
gw
eus
eth
ism
odel
setu
pto
stud
ydiff
eren
tmec
hani
smsin
volv
edin
the
early
fillin
gof
the
left
vent
ricle
.The
sem
echa
nism
sare
exam
ined
onth
eba
siso
fvel
ocity
profi
lesa
long
six
tran
sver
selin
esin
the
vent
ricle
asde
scrib
edin
Sec
tion6
.3.1
ando
nth
eba
siso
fth
epr
essu
redi
ffer
ence
betw
eena
triu
man
dve
ntric
le,s
ince
the
pres
sure
grad
ient
isas
sum
edto
bew
hatd
rives
the
bloo
din
toth
eve
ntric
le(s
eeS
ectio
n6.1
.1).
Þ ßò% +,(Ø)
refe
renc
e,
Þ"ßò% !!(Ø)
11.2
1.4
1.6
1.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
11.2
1.4
1.6
1.8
t[s]
-40
-200
20
40
60
velocity[cm/s]
Fig
ure
6.14
:Vel
ocity
time
profi
leof
the
mea
nvel
ocity
(cm
/s)a
tthe
mitr
alrin
g(s
olid
)an
dbe
twee
nthe
leafl
ettip
s(d
ashe
d)fr
oma
sim
ulat
ion
whe
re
Þ"ßò
% +,(Ø) (le
ft)an
dfr
omth
ere
fere
nces
imul
atio
nw
here
Þ ßò%!!(Ø)
(rig
ht).
116
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
+4
+3
+2
+1
ML
T
-2
-25
ms
PV
ÑÒÓ
+25
ms
+51
ms
+76
ms
1.01
0s1.
035
s1.
060
s1.
086s
1.11
1s
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Fig
ure
6.15
:V
eloc
itypr
ofile
sof
abso
lute
velo
city
(m/s
)alo
ngsi
xlin
esat
five
times
from
asi
mul
atio
nof
ahe
alth
yhum
an.I
fth
eve
loci
typr
ofile
ispl
otte
dat
the
time
ofP
Vat
alin
e,P
Vis
mar
ked
bya
Ô .F
igur
e6.
15sh
ows
the
spat
ialv
eloc
itypr
ofile
s(of
abso
lute
velo
city
)alo
ngth
esi
xtr
ansv
erse
lines
inth
eve
ntric
leat
five
times
:25
ms
befo
reP
Vat
the
mitr
alle
aflet
tips,
atth
etim
eof
PV
atth
em
itral
leafl
ettip
s,an
d25
,51
and
76m
saf
ter
the
time
ofP
Vat
the
mitr
alle
aflet
tips.
The
velo
city
profi
les
inF
igur
e6.
15ar
esi
mila
rto
thos
eof
ahe
alth
yper
son
from
MR
mea
sure
men
tssho
wn
inF
igur
e6.
1.F
igur
e6.
1sh
ows
that
the
velo
city
profi
les
beco
mew
ider
and
decr
ease
inhe
ight
ata
line
astim
ein
crea
ses.
Inth
esi
mul
atio
nw
edo
nots
eea
wid
erpr
ofile
ata
line
astim
ein
crea
ses,b
utth
ere
isa
decr
ease
inhe
ight
.The
MR
data
isav
erag
edov
erm
any
hear
tcyc
les
and
this
coul
dre
sulti
nw
ider
and
low
erpr
ofile
stha
nw
hatt
ypic
ally
occu
rsin
one
spec
ific
hear
tcyc
le.
The
peak
velo
city
isob
tain
edat
the
sam
etim
eat
four
ofth
elin
esin
the
MR
6.3
Cal
ibra
ting
the
mod
el11
7
data
,inth
esi
mul
atio
nthi
sis
true
for
thre
eoft
helin
es.T
hepr
ofile
sfro
mth
eM
Rda
tash
owth
atth
eje
tget
sslig
htly
wid
erfu
rthe
rdow
nin
the
vent
ricle
ata
fixed
time.
The
sim
ulat
ion
show
sth
eop
posi
te:a
slig
htly
narr
ower
jetf
urth
erdo
wn
inth
eve
ntric
leat
afix
edtim
e.F
urth
erm
ore,
inth
esi
mul
atio
nth
eve
loci
typr
ofile
sat
the
2,3
and
4cm
lines
are
tria
ngul
arin
shap
e,th
isis
nott
rue
for
the
MR
data
,ag
aint
his
coul
dbe
aco
nseq
uenc
eofth
eav
erag
ingi
nth
eM
Rda
ta.
Fig
ure
6.16
show
sth
epe
akve
loci
tyP
Vat
the
six
lines
from
the
sim
ulat
ion
alon
gw
ithth
esa
med
ata
from
MR
mea
sure
men
tsof
heal
thyp
erso
ns.T
heM
Rda
taar
eth
esa
mea
sin
Fig
ure
6.2,
soth
eM
Rda
taar
esh
own
asm
ean
valu
ew
ithth
est
anda
rdde
viat
ion
show
nas
erro
rbar
s.T
hefig
ure
show
sth
atP
Vis
gene
rally
high
erin
the
sim
ulat
ion,
exce
ptat
the
ML
Tlin
e.A
tth
elin
esju
stup
-an
ddo
wns
trea
mof
the
ML
Tlin
eP
Vis
with
inth
est
anda
rdde
viat
ion
ofth
eM
Rda
ta,b
utfu
rthe
rdow
nstr
eam
PV
ism
uch
high
erin
the
sim
ulat
ion.
Fur
ther
mor
e,P
Vin
crea
sesd
owns
trea
mof
the
ML
Tlin
ein
the
sim
ulat
ion,
whi
leit
decr
ease
sinth
eM
Rm
easu
rem
ents
.The
latte
ris
prob
ably
rela
tedt
oth
eje
tbec
omin
gnar
row
erfu
rthe
rdow
nin
the
vent
ricle
seen
inth
eve
loci
typr
ofile
sfr
omth
esi
mul
atio
nab
ove,
seeF
igur
e6.
15.I
nth
eM
Rda
tath
eje
tbec
omes
wid
erfu
rthe
rdow
nin
the
vent
ricle
,whi
chle
adst
olo
wer
PV.
The
narr
ower
jeti
nth
esi
mul
atio
nco
uld
bea
resu
ltof
the
vort
exm
otio
n,w
hich
ism
uch
mor
epr
onou
nced
inth
esi
mul
atio
nth
anse
enin
MR
mea
sure
men
ts,se
eSec
tion5
.2.M
ore
pron
ounc
edvo
rtic
esw
ould
leav
ele
ssro
omfo
rth
eje
t,he
reby
mak
ing
itna
rrow
er.
Thi
sag
ain
mig
htbe
are
sulto
fthe
mod
elbe
ing
only
2D,w
hile
the
MR
mea
sure
men
tsare
ofth
efu
ll3D
phen
omen
on.
Fig
ure
6.17
show
sth
eve
loci
tyin
dex
VI
atth
esi
xlin
esfr
omth
esi
mul
atio
nan
dth
eM
Rda
taof
heal
thyp
erso
ns.T
hela
ttera
reid
entic
alto
thos
esh
own
inF
igur
e6.
3.T
heve
loci
tyin
dex
VI
isco
nsid
erab
lyhig
heri
nth
esi
mul
atio
n,be
ing
only
iden
tical
,nat
ural
ly,at
the
ML
Tlin
ew
ithth
eM
Rda
ta.F
urth
erm
ore,
VI
isgr
eate
rthan
1fo
rlin
esdo
wns
trea
mof
ML
T,w
hich
isa
resu
ltofP
Vbe
ing
high
erat
lines
dow
nstr
eam
inth
eve
ntric
leth
anat
the
ML
Tlin
e.F
igur
e6.
18sh
ows
the
time
dela
yTD
atth
esi
xlin
esfr
omth
esi
mul
atio
nand
from
the
MR
data
ofhe
alth
yper
sons
,whi
char
eth
esa
mea
stho
sesh
own
inF
ig-
ure
6.4.
The
figur
esh
ows
that
TD
inth
esi
mul
atio
nis
com
para
blet
oT
Dfr
omth
eM
Rda
ta;o
nly
atth
elin
e2
cmup
stre
amof
the
ML
Tlin
eis
TD
notw
ithin
the
stan
dard
devi
atio
n.T
heva
lue
ofT
Dat
the
3cm
line
from
the
sim
ulat
ion
islo
w(0
ms)
and
resu
ltsin
ale
sssm
ooth
lyin
crea
sing
TD
curv
eas
fort
heM
Rda
ta.T
helo
wva
lue
ofT
Dat
the
3cm
line
isa
resu
ltsof
the
vort
expr
esen
tinth
eve
ntric
lefr
omth
epr
evio
ushe
artc
ycle
(des
crib
edin
Sec
tion5
.2).
Thi
svo
rtex
exte
ndst
hein
flow
jet(
spat
ially
)and
cons
eque
ntlyt
heve
loci
tybe
com
eshi
ghat
anea
rlier
time
low
erin
the
vent
ricle
.Thi
sis
seen
inth
ele
ftpl
otin
Fig
ure
6.19
that
show
sth
e
118
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
PV[m/s]
MR
healthy
Healthy
Fig
ure
6.16
:Pea
kvel
ocity
(PV
)att
hesi
xlin
esfr
omM
Rm
easu
rem
entso
fhe
alth
ype
rson
s(
Õ )an
dfr
oma
sim
-ul
atio
nof
ahe
alth
yper
son(
& ).
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
1.2
1.4
1.6
VI
MR
healthy
Healthy
Fig
ure6
.17:
Vel
ocity
inde
x(V
I)at
the
six
lines
from
MR
mea
sure
men
tsof
heal
thy
pers
ons(
Õ )an
dfr
oma
sim
-ul
atio
nof
ahe
alth
yper
son(
& ).
-2
01
23
4Distance
from
MLT
[cm]
-500
50
100
TD[ms]
MR
healthy
Healthy
Fig
ure
6.18
:Tim
ede
lay(
TD
)at
the
six
lines
from
MR
mea
sure
men
tsofh
ealth
ype
rson
s(
Õ )an
dfr
oma
sim
ulat
iono
fahe
alth
yper
son(
& ).
6.3
Cal
ibra
ting
the
mod
el11
9
2.he
artc
ycle
Time
1.035
1.he
artc
ycle
Time
0.151
Fig
ure
6.19
:Vel
ocity
field
atth
etim
eof
PV
atth
eM
LT
line
inth
ese
cond
hear
tcy
cle
(left)
and
inth
efir
sthe
artc
ycle
(rig
ht).
velo
city
field
atth
etim
eof
PV
atth
eM
LT
line.
Ifth
evo
rtex
was
notp
rese
ntP
Vw
ould
belo
wer
and
TD
wou
ldbe
high
erat
the
3an
d4
cmlin
es.W
eca
nsu
ppor
tth
isby
look
ing
atth
eve
loci
tyfie
ldfr
omth
efir
sthe
artc
ycle
ofth
esi
mul
atio
n,w
here
novo
rtex
ispr
esen
tlow
inth
eve
ntric
le;s
eeth
erig
htpl
otin
Fig
ure
6.19
.F
rom
this
figur
eit
iscl
eart
hatt
heje
tdoe
snot
reac
hasf
arin
toth
eve
ntric
leas
inth
ese
cond
cycl
e.F
igur
e6.2
0sh
ows
the
time
varia
tiono
fthe
pres
sured
iffer
ence
betw
eena
triu
man
dven
tric
le.T
heve
ntric
ular
pres
surei
sm
easu
reda
ppro
xim
atel
y2/3
dow
nin
the
vent
ricle
from
the
mitr
alrin
g,w
hile
the
atria
lpre
ssur
eism
easu
redin
the
mid
dle
ofth
eat
rium
.In
the
sim
ulat
ion
we
find
ape
akpr
essu
redi
ffer
ence
durin
gea
rlyfil
ling
ofap
prox
imat
ely2
mm
Hg
(the
first
peak
inF
igur
e6.
20is
rela
tedt
oea
rlyfil
ling)
.T
hela
rges
tpea
kin
Fig
ure
6.20
isca
used
byat
rials
ysto
le.S
tein
eeta
l.(1
999)
mea
sure
dusi
ngco
lour
M-m
ode
Dop
pler
echo
card
iogr
aphy
the
pres
sure
diff
eren
cebe
twee
nthe
mitr
altip
and
the
apex
and
betw
eent
heat
rium
and
the
mi-
tral
tipdu
ring
early
dias
tole
indo
gs.T
hey
foun
dth
atth
epe
akpr
essu
rediff
eren
cebe
twee
nmitr
altip
and
apex
was
1.9
* 0.9
mm
Hg
and
betw
eena
triu
man
dm
itral
tips
itw
as2.
4
* 0.7
mm
Hg.
Thi
sgi
ves
am
eanv
alue
ofth
epr
essu
redi
ffer
ence
from
atriu
mto
apex
of4.
3m
mH
g.T
hust
hepe
akpr
essu
redi
ffer
ence
inth
esi
m-
ulat
ion
isre
ason
ablec
ompa
redt
om
easu
rem
entsw
hen
we
take
into
acco
untth
ede
viat
ions
inth
em
easu
rem
entsa
ndth
efa
ctth
atth
epr
essu
rele
veli
sto
olo
win
the
mod
el,s
eeS
ectio
n4.5
.T
heag
reem
entof
the
velo
city
profi
les
and
velo
city
inde
xesb
etw
eens
imul
a-tio
nan
dM
Rda
tais
reas
onab
lyfai
r.B
ecau
seof
prev
ious
com
paris
onsb
etw
een
120
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
11.2
1.4
1.6
1.8
t[s]
-101234
pressure[mmHg]
Fig
ure
6.20
:Tim
eva
riatio
nof
pres
sure
diff
eren
ce(m
mH
g)be
twee
natr
ium
and
vent
ricle
from
asi
mul
atio
nofa
heal
thyh
uman
.
mod
elan
dM
Rda
ta(C
hapt
er3)
,we
coul
dno
texp
ecti
tto
bebe
tter.
Inth
efo
l-lo
win
gin
vest
igat
ions
we
focu
son
the
rela
tive
chan
gesin
PV,
VI,
TD
andp
ress
ure
diff
eren
cew
hen
we
sim
ulat
ediff
eren
tmec
hani
sms.
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
We
wis
hto
inve
stig
atec
erta
inm
echa
nism
sof
early
dias
tolic
fillin
gus
ing
the
mod
elde
scrib
edin
the
prev
ious
sect
ion.
We
hypo
thes
izet
hata
hear
tdis
ease
whi
chaf
fect
sth
edi
asto
licfu
nctio
nca
nbe
mod
elle
dei
ther
bya
chan
gein
the
rela
xatio
nra
te
Þuß orsi
mpl
yby
letti
nga
regi
onof
the
hear
twal
lbe
inac
tive.
The
form
erco
rres
pond
stoa
chan
gein
Ü seeni
nm
easu
rem
entso
fsev
eral
patie
ntty
pes,
whi
leth
ela
tterc
orre
spon
dsto
am
yoca
rdia
linfa
rctio
n,w
here
part
ofth
ehe
artw
alld
iesd
ueto
abl
ocka
geof
the
coro
nary
arte
riesa
ndsu
bseq
uent
lylose
sits
abili
tyto
cont
ract
and
rela
x.W
ein
clud
eth
rees
cena
riosin
ouri
nves
tigat
ion
1.sl
owre
laxa
tiono
fthe
entir
eve
ntric
le
2.sl
owre
laxa
tiono
fthe
apic
alre
gion
ofth
eve
ntric
le
3.in
activ
e(p
assiv
e)ap
ical
regi
on
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
121
The
first
scen
ario
coul
dha
vebe
enin
vest
igat
edw
ithou
tthe
new
activ
atio
nfu
nctio
nim
plem
ente
din
the
mod
el,b
ecau
seit
corr
espo
ndsto
ach
ange
in
Þuß
for
the
entir
eve
ntric
le.T
his
scen
ario
corr
espo
ndsto
agl
obal
dysf
unct
iono
fth
eve
ntric
le,e
.g.c
ause
dby
agl
obal
isch
emia
.The
seco
ndsc
enar
ioco
rres
pond
stoa
loca
ldys
func
tiono
fth
ehe
artc
ause
dby
anis
chem
icap
ical
regi
ondu
eto
e.g.
CA
Dor
AM
I.T
heth
irdsc
enar
iore
sem
bles
wha
thap
pens
durin
ga
myo
card
ial
infa
rctio
nan
dth
ussh
ould
com
pare
the
best
with
resu
ltsfr
omA
MI
patie
nts.
We
expe
ctto
sees
igni
fican
tcha
nges
insi
mul
atio
nres
ults
inth
eth
isca
se,b
ecau
sean
inac
tive
regi
onis
ave
rydr
astic
chan
geof
the
mod
el.
6.4.
1S
low
rela
xatio
nof
the
vent
ricle
Asl
ower
rela
xatio
nin
the
entir
eve
ntric
leco
rres
pond
stom
odel
ling
adi
seas
edhe
artw
here
the
glob
alfu
nctio
ning
ofth
ehe
artis
redu
ced.
We
achi
eve
asl
ower
rela
xatio
nby
decr
easi
ngth
eva
lue
of
Þuß forth
een
tire
vent
ricle
corr
espo
ndin
gtoan
incr
ease
in
Ü .Nishi
hara
etal
.(20
00)r
epor
t
Ü values
of54
* 12m
sfo
rpa
tient
sw
ithH
CM
,whi
leS
t.G
oare
tal.
(199
1)re
port
Ü values
of59
* 8m
sfo
rpa
tient
sw
ithD
CM
,an
dfin
ally
Oht
eet
al.(
1999
)rep
ortv
alue
sof
47
* 6.3
ms
for
CA
Dpa
tient
s.B
ased
onth
eset
hree
stud
iesw
ead
just
Þ"ß to% $(Ø) co
rres
pond
ing
to
Ü ò 60
ms.
With
Þ"ßò% $(Ø) th
epe
riod
ofth
eac
tivat
ion
func
tion,
and
cons
eque
ntlyt
hele
ngth
ofth
ehe
artc
ycle
,bec
omes
0.94
2s.
Apa
rtfr
omth
ese
adju
stm
ents
,the
para
met
erse
tting
isid
entic
alto
the
refe
renc
esim
ulat
ion.
The
velo
city
profi
lesa
long
the
six
tran
sver
selin
esin
the
vent
ricle
are
show
nin
Fig
ure
6.21
atfiv
etim
es:2
6m
sbe
fore
PV
atth
em
itral
leafl
ettip
s,at
the
time
ofP
Vat
the
mitr
alle
aflet
tips,
and
26,5
2an
d78
ms
afte
rthe
time
ofP
Vat
the
mitr
alle
aflet
tips.
Fro
mth
efig
ure
we
seet
hatt
hesh
apeo
fth
eve
loci
typr
ofile
sha
veno
tcha
nged
com
pare
dto
asi
mul
atio
nofa
heal
thyh
uman
(Fig
ure
6.15
).F
igur
es6.
22–6
.24s
how
the
peak
velo
city
PV,
the
velo
city
inde
xV
I,an
dth
etim
ede
layT
D,r
espe
ctive
ly.I
nea
chof
thes
ethr
eefig
ures
the
corr
espo
ndin
gdat
afr
omM
Rm
easu
rem
entso
fhe
alth
ype
rson
sand
from
asi
mul
atio
nof
ahe
alth
ype
rson
are
incl
uded
.The
data
from
the
MR
mea
sure
men
tsisid
entic
alto
that
ofF
igur
es6.
2–6.
4,w
hile
the
data
from
asi
mul
atio
nofa
heal
thyp
erso
nis
iden
tical
toth
atof
Fig
ures
6.16
–6.1
8.F
rom
both
the
velo
city
profi
lesa
ndth
epl
otof
PV
we
seet
hatt
hepe
akve
loci
-tie
sin
the
pres
ents
imul
atio
nare
low
erth
anin
the
sim
ulat
iono
fahe
alth
yper
son.
The
peak
velo
citie
sst
illha
vea
tend
ency
toin
crea
sefu
rthe
rdow
nstr
eam
ofth
eM
LT
line,
whe
reth
eM
Rda
tash
ows
ade
crea
se.T
here
lativ
ely
high
PV
valu
eat
the
3cm
line
isno
trel
ated
toth
ein
flow
jetb
utis
due
toth
evo
rtex
inth
elo
wer
part
ofth
eve
ntric
le.T
his
can
bese
enby
look
ing
atth
eve
loci
tyfie
ldat
the
time
122
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
+4
+3
+2
+1
ML
T
-2
-26
ms
PV
ÑÒÓ
+26
ms
+52
ms
+78
ms
1.04
7s1.
073
s1.
099
s1.
125s
1.15
1s
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Fig
ure
6.21
:V
eloc
itypr
ofile
sof
abso
lute
velo
city
(m/s
)alo
ngsi
xlin
esat
five
times
from
asi
mul
atio
nwhe
reve
ntric
ular
rela
xatio
nis
slow
.If
the
velo
city
profi
leis
plot
ted
atth
etim
eof
PV
ata
line,
PV
ism
arke
dby
a
Ô .of
PV
atth
eM
LT
line
show
nin
Fig
ure6
.25.
Bec
ause
ofth
isvo
rtex
we
shou
ldno
tpu
ttoo
muc
hem
phas
ison
the
velo
city
inde
xesf
rom
the
3an
d4
cmlin
es,s
ince
they
are
notr
elat
edto
the
inflo
wje
tbut
rath
erto
the
vort
ex.
We
only
see
asm
allc
hang
ein
VI
inth
epr
esen
tsim
ulat
ion
com
pare
dto
asi
mul
atio
nof
ahe
alth
ype
rson
.How
ever
,F
igur
e6.
23sh
ows
that
the
velo
city
inde
xV
Iis
gene
rally
high
erin
the
sim
ulat
ion
than
seen
inM
Rm
easu
rem
ents
ofhe
alth
yper
sons
.We
have
nom
easu
redd
ata
ofV
Iin
patie
ntsw
itha
glob
ally
redu
cedf
unct
ioni
ngof
the
vent
ricle
toco
mpa
rew
ith.
Fig
ure
6.24
ofT
Dsh
ows
that
atlin
es1
and
2cm
dow
nstr
eam
ofth
eM
LT
line
TD
issu
bsta
ntia
llyhi
gher
com
pare
dto
asi
mul
atio
nof
ahe
alth
yper
son.
At
the
3an
d4
cmlin
esT
Dis
noth
ighe
ror
only
mod
erat
elyh
ighe
r.T
Dat
the
3an
d4
cmlin
esis
noto
btai
nedf
orth
ein
flow
jet,
asde
scrib
edab
ove,
butr
athe
r
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
123
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
PV[m/s]
MR
healthy
Healthy
Ventr.
slow
Fig
ure
6.22
:Pea
kvel
ocity
(PV
)att
hesi
xlin
esfr
omM
Rm
easu
rem
entso
fhe
alth
yper
sons
(
Õ ),an
dfr
oma
sim
-ul
atio
nof
ahe
alth
ype
rson
(
& )an
da
sim
ulat
ion
whe
reth
eve
ntric
ular
re-
laxa
tion
issl
ow(
- ).
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
1.2
1.4
1.6
VI
MR
healthy
Healthy
Ventr.
slow
Fig
ure6
.23:
Vel
ocity
inde
x(V
I)at
the
six
lines
from
MR
mea
sure
men
tsof
heal
thyp
erso
ns(
Õ ),an
dfr
oma
sim
-ul
atio
nof
ahe
alth
ype
rson
(
& )an
da
sim
ulat
ion
whe
reth
eve
ntric
ular
re-
laxa
tion
issl
ow(
- ).
-2
01
23
4Distance
from
MLT
[cm]
-500
50
100
TD[ms]
MR
healthy
Healthy
Ventr.
slow
Fig
ure
6.24
:Tim
ede
lay(
TD
)at
the
six
lines
from
MR
mea
sure
men
tsofh
ealth
ype
rson
s(
Õ ),an
dfro
ma
sim
ulat
iono
fahe
alth
yper
son(
& )and
asi
mul
atio
nwhe
reth
eve
ntric
ular
rela
xatio
nis
slow
(
- ).
124
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
from
the
vort
exin
the
vent
ricle
.If
the
vort
exw
asno
tpre
sent
we
wou
ldex
pect
that
TD
wou
ldbe
muc
hhi
gher
atlo
wer
lines
.Stu
gaar
deta
l.(1
994b
)mea
sure
sth
eap
ical
time
dela
yin
dogs
,bot
hin
heal
thy
dogs
and
whe
nbe
ta-b
lock
adeis
indu
ced.
Acc
ordi
ngto
Kim
(200
1)in
duct
iono
fbet
a-bl
ocka
deaf
fect
sthe
vent
ricle
glob
ally
,so
we
can
uset
his
for
com
paris
onw
ithou
rpre
sent
sim
ulat
ion.
Stu
gaar
det
al.(
1994
b)fin
dth
atth
em
eanv
alue
ofth
eap
ical
time
dela
yfo
rhe
alth
ydog
sis
21m
s,an
dit
incr
ease
sto80
ms
whe
nbe
ta-b
lock
adeis
indu
ced.
Thu
sth
esu
bsta
ntia
llyin
crea
sedT
Din
ours
imul
atio
ncom
pare
swel
lwith
mea
sure
men
ts.
Ifw
eco
mpa
reth
eve
loci
tyfie
ldat
the
time
ofpe
akve
loci
tyat
the
mitr
alle
aflet
tips
(Fig
ure
6.25
)with
the
velo
city
field
from
ahe
alth
yhum
an(F
igur
e6.
19)w
eca
nse
etha
tthe
inflo
wje
tis
nota
sst
rong
.It
hasl
ower
velo
citie
sand
does
not
reac
has
deep
lyin
toth
eve
ntric
leas
for
ahe
alth
yhum
an.B
oth
ofth
esefi
ndin
gsar
eid
entic
alto
the
findi
ngs
ofth
eve
loci
typr
ofile
san
din
dexe
sab
ove
and
are
sign
sofp
oord
iast
olic
fillin
g.F
igur
e6.
26sh
ows
the
time
varia
tion
ofth
epr
essu
redi
ffer
ence
betw
eent
heat
rium
and
vent
ricle
durin
gdi
asto
lefr
omth
esi
mul
atio
n.B
yco
mpa
ringt
hefig
ure
with
the
sam
efro
ma
sim
ulat
ion
ofa
heal
thy
pers
on(F
igur
e6.
20)w
ese
eth
atth
epe
akpr
essu
redi
ffer
ence
durin
gea
rlyfil
ling
isre
duce
d.In
ahe
alth
yhum
anit
isap
prox
imat
ely2
mm
Hg,
whi
leit
isap
prox
imat
ely1
.2m
mH
gin
the
pres
ent
sim
ulat
ion,
i.e.a
40%
redu
ctio
n.S
tuga
arde
tal.
(199
4b)fi
nda
peak
pres
sure
diff
eren
cebe
twee
natr
ium
and
vent
ricle
of3.
9m
mH
g(m
ean
valu
e)in
heal
thy
dogs
,whi
leit
falls
to3.
4m
mH
gdu
ring
indu
ctio
nof
beta
-blo
ckad
e,thi
sis
a13
%
Time
1.073
Fig
ure
6.25
:V
eloc
ityfie
ldat
the
time
ofpe
akve
loci
tyat
the
ML
Tlin
efr
oma
sim
ulat
ionw
here
the
vent
ricul
arre
laxa
tioni
ssl
ow.
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
125
11.2
1.4
1.6
1.8
t[s]
-101234
pressure[mmHg]
Fig
ure
6.26
:Tim
eva
riatio
nof
pres
sure
diff
eren
ce(m
mH
g)be
twee
natr
ium
and
vent
ricle
from
asi
mul
atio
nwhe
reth
eve
ntric
ular
rela
xatio
nis
slow
.
redu
ctio
n.Le
mm
onan
dY
ogan
atha
n(20
00a)
have
mad
ean
inve
stig
atio
nof
vent
ricul
ardy
sfun
ctio
nusi
nga
3Dm
odel
ofth
ele
ftat
rium
and
vent
ricle
base
don
the
im-
mer
sedb
ound
arym
etho
d(Le
mm
onan
dY
ogan
atha
n,200
0b).
The
irm
odel
have
been
foun
dto
bein
agre
emen
twith
clin
ical
lyob
serve
ddi
asto
licflo
wco
nditi
ons
(Lem
mon
and
Yog
anat
han,2
000b
),bu
tthe
intr
aven
tric
ular
flow
patte
rnha
snot
been
com
pare
dinde
tailw
ithe.
g.M
Rda
ta.
Lem
mon
and
Yog
anat
han(2
000a
)exa
min
ethe
bloo
dve
loci
tyin
the
vent
ricu-
lari
nflow
trac
t,th
atis
,fro
mth
em
itral
annu
lusa
nd3.
5cm
into
the
vent
ricle
.One
ofth
eirs
imul
atio
ncas
esis
sim
ilart
oou
rpre
sent
sim
ulat
ion:
dela
yedv
entr
icul
arre
laxa
tion.
The
yco
mpa
reth
eirs
imul
atio
nwith
asi
mul
atio
nwhe
reth
ere
laxa
tion
isno
rmal
.The
yfin
dth
atth
epe
akve
loci
tyis
decr
ease
dinth
edi
seas
edca
se:th
epe
akve
loci
tyat
the
mitr
alrin
gin
the
norm
alca
seis
appr
oxim
atel
y0.7
4cm
/san
dit
decr
ease
stoap
prox
imat
ely0
.68
cm/s
inth
edi
seas
edca
se.B
oth
ofth
esev
alue
sar
ehi
gher
than
ours
imila
rres
ults
and
high
erth
anth
eva
lue
from
the
MR
data
,bu
twith
inth
est
anda
rdde
viat
ion
ofth
is.
Lem
mon
and
Yog
anat
han(
2000
a)fin
dth
atth
epe
akve
loci
typr
ogre
sses
slow
erin
toth
eve
ntric
le.
Sum
mar
izin
gour
resu
lts,w
efin
dth
ata
slow
erre
laxa
tiono
fthe
entir
eve
ntri-
cle
resu
ltsin
gene
rally
low
erpe
akve
loci
tiesi
nth
eve
ntric
lean
da
dela
yedfi
lling
.It
can
beex
plai
ned
bya
redu
cedp
ress
ured
iffer
ence
betw
eent
heat
rium
and
the
126
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
vent
ricle
.The
low
erpe
akve
loci
tiesa
ndde
laye
dfilli
ngis
sim
ilart
oth
esi
mul
atio
nre
sults
ofLe
mm
onan
dY
ogan
atha
n(200
0a),w
hous
ea3D
mod
elof
the
left
side
ofth
ehe
art.
The
incr
ease
dtim
ede
lay
and
redu
cedp
ress
ured
iffer
ence
isin
ac-
cord
ance
with
the
findi
ngso
fStu
gaar
deta
l.(1
994b
),w
hom
easu
redth
ech
ange
sin
dogs
durin
gin
duct
ion
ofbe
ta-b
lock
ade,w
hich
affe
ctst
heve
ntric
legl
obal
ly.
6.4.
2S
low
rela
xatio
nof
apic
alre
gion
The
seco
ndsc
enar
iois
asi
mul
atio
nw
here
the
rela
xatio
nof
the
apic
alre
gion
(reg
ion
3in
Fig
ure
6.7)
issl
owed
.Thi
sco
rres
pond
stom
odel
ling
ahe
artw
here
the
apic
alre
gion
isis
chem
ic.S
tein
eeta
l.(1
999)
mea
sure
s
Ü inthe
apic
alre
gion
inhe
alth
ydog
sand
indo
gsw
ithac
utei
sche
mic
failu
re.A
cute
isch
emic
failu
reis
obta
ined
byin
duce
dcor
onar
ymic
roem
boliz
atio
n.S
tein
eeta
l.(1
999)
mea
sure
sa
Ü ã.àä
inth
eap
ical
regi
onin
the
isch
emic
hear
tof4
9
* 6m
s(m
ean
* SD
).T
husw
ech
oose
tole
tthe
rela
xatio
nrat
eco
nsta
nt
Þ ß be% !!(Ø) in
the
apic
alre
gion
,w
hile
itis
still
% +,(Ø) in
the
rest
ofth
eve
ntric
leas
for
ahe
alth
yhum
an.T
hele
ngth
ofa
hear
tcyc
leis
still
0.90
9s.
Fig
ure
6.27
show
sth
eve
loci
typr
ofile
sal
ong
the
six
tran
sver
selin
esat
five
times
:25
ms
befo
reP
Vat
the
mitr
alle
aflet
tips,
atth
etim
eof
PV
atth
em
itral
leafl
ettip
s,an
d25
,50
and
76m
saf
tert
hetim
eof
PV
atth
em
itral
leafl
ettip
s.T
heve
loci
typr
ofile
sar
eno
tver
ydi
ffer
entf
rom
the
ones
inth
esi
mul
atio
nof
ahe
alth
yper
soni
nF
igur
e6.
15.
Fig
ures
6.28
–6.3
0sho
wth
epe
akve
loci
tyP
V,th
eve
loci
tyin
dex
VI,
and
the
time
dela
yT
D,
resp
ectiv
ely.
Inea
chof
thes
eth
ree
figur
esth
eco
rres
pond
ing
data
from
MR
mea
sure
men
tsof
heal
thy
pers
onsa
ndA
MI
patie
ntsa
ndfr
oma
sim
ulat
iono
fahe
alth
yper
sona
rein
clud
ed.T
heda
tafr
omth
eM
Rm
easu
rem
ents
isid
entic
alto
that
ofF
igur
es6.
2–6.
4,w
hile
the
data
from
asi
mul
atio
nofa
heal
thy
pers
onis
iden
tical
toth
atof
Fig
ures
6.16
–6.1
8.F
igur
e6.
28sh
ows
that
PV
isno
tver
ydi
ffer
entin
the
pres
ents
imul
atio
ncom
-pa
red
toa
sim
ulat
ion
ofa
heal
thy
pers
on.O
nly
atth
e3
cmlin
edo
we
see
aco
nsid
erab
lyhig
herP
V.T
his
is,a
ssee
nin
the
prev
ious
sim
ulat
ion,
rela
tedt
oth
evo
rtex
pres
entin
the
vent
ricle
,as
can
bese
enby
look
ing
atth
eve
loci
tyfie
ldat
the
time
ofP
Vat
the
ML
Tlin
ein
Fig
ure
6.31
.The
chan
gein
PV
betw
eena
sim
-ul
atio
nof
ahe
alth
yper
sona
ndth
epr
esen
tsim
ulat
ion
isno
tthe
sam
eas
seen
inM
Rm
easu
rem
entsb
etw
eenh
ealth
yper
sons
and
AM
Ipa
tient
s.In
the
MR
mea
-su
rem
entsP
Vde
crea
sesat
the
3an
d4
cmlin
es,a
ndin
the
sim
ulat
ionw
eac
tual
lyse
ean
incr
ease
(aga
inbe
caus
eoft
hevo
rtex
).T
hesa
mep
atte
rnis
seen
forV
Iin
Fig
ure6
.29.
VI
isth
esa
mef
rom
the
pres
ent
sim
ulat
iona
ndth
esi
mul
atio
nofa
heal
thyp
erso
n,ex
cept
atth
e3
and
4cm
lines
.
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
127
+4
+3
+2
+1
ML
T
-2
-25
ms
PV
ÑÒÓ
+25
ms
+50
ms
+76
ms
0.98
5s1.
010
s1.
035s
1.06
0s1.
086
s
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Fig
ure
6.27
:V
eloc
itypr
ofile
sof
abso
lute
velo
city
(m/s
)alo
ngsi
xlin
esat
five
times
from
asi
mul
atio
nwhe
reth
eap
ical
rela
xatio
nis
slow
.If
the
velo
city
profi
leis
plot
ted
atth
etim
eof
PV
ata
line,
PV
ism
arke
dby
a
Ô .H
ere,
VI
ishi
gher
inth
epr
esen
tsim
ulat
ion.
MR
mea
sure
men
tsofh
ealth
yper
sons
and
AM
Ipa
tient
ssho
wth
atV
Ish
ould
decr
ease
inan
AM
Ipa
tient
satt
he3
and
4lin
esan
dot
herw
iseb
eun
chan
ged.
Fig
ure
6.30
show
sth
atT
Dis
high
erin
the
pres
ents
imul
atio
ntha
nse
enin
the
sim
ulat
ion
ofa
heal
thy
pers
on,e
xcep
tatt
he2
cmlin
ew
here
itis
low
er.
The
MR
data
also
show
sth
atT
Din
crea
sesin
anA
MI
patie
ntco
mpa
redt
oa
heal
thy
pers
on,b
uton
lyat
the
3an
d4
cmlin
es.B
ecau
seof
the
vort
exin
the
sim
ulat
ions
we
shou
ldno
tput
too
muc
hem
phas
ison
the
resu
ltsfr
omth
e3
and
4cm
lines
,so
we
mus
tcon
clud
etha
tTD
does
nots
how
sign
ifica
ntch
ange
sinth
epr
esen
tsi
mul
atio
ncom
pare
dto
the
sim
ulat
iono
fahe
alth
yper
son.
The
plot
ofth
eve
loci
tyfie
ldat
the
time
ofpe
akve
loci
tyat
the
mitr
alle
aflet
tips
(Fig
ure
6.31
)is
very
sim
ilar
toth
eve
loci
tyfie
ldse
enfo
ra
heal
thyh
uman
128
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
PV[m/s]
MR
healthy
MR
AMI
Healthy
Apex
slow
Fig
ure
6.28
:Pea
kvel
ocity
(PV
)att
hesi
xlin
esfr
omM
Rm
easu
rem
entso
fhe
alth
yper
sons
(
Õ )an
dA
MI
patie
nts
(
Ö ),an
dfro
ma
sim
ulat
iono
fahe
alth
ype
rson
(
& )an
da
sim
ulat
ionw
here
the
apic
alre
laxa
tioni
ssl
ow(
- ).
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
1.2
1.4
1.6
VI
MR
healthy
MR
AMI
Healthy
Apex
slow
Fig
ure6
.29:
Vel
ocity
inde
x(V
I)at
the
six
lines
from
MR
mea
sure
men
tsof
heal
thyp
erso
ns(
Õ )an
dA
MI
patie
nts
(
Ö ),an
dfro
ma
sim
ulat
iono
fahe
alth
ype
rson
(
& )an
da
sim
ulat
ionw
here
the
apic
alre
laxa
tioni
ssl
ow(
- ).
-2
01
23
4Distance
from
MLT
[cm]
-500
50
100
TD[ms]
MR
healthy
MR
AMI
Healthy
Apex
slow
Fig
ure
6.30
:Tim
ede
lay(
TD
)at
the
six
lines
from
MR
mea
sure
men
tsofh
ealth
ype
rson
s(
Õ )an
dA
MI
patie
nts(
Ö ),an
dfr
oma
sim
ulat
iono
fahe
alth
yper
son(
& )an
da
sim
ulat
ionw
here
the
apic
alre
laxa
tioni
ssl
ow(
- ).
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
129
Time
1.010
Fig
ure
6.31
:V
eloc
ityfie
ldat
the
time
ofpe
akve
loci
tyat
the
ML
Tlin
efr
oma
sim
ulat
ion
whe
reth
eap
ical
rela
xatio
nis
slow
.
11.2
1.4
1.6
1.8
t[s]
-101234
pressure[mmHg] Fig
ure6
.32:
Tim
eva
riatio
nofp
ress
ure
diff
eren
ce(m
mH
g)be
twee
nat
rium
and
vent
ricle
from
asi
mul
atio
nw
here
the
apic
alre
laxa
tioni
ssl
ow.
(Fig
ure
6.19
),w
hich
agre
esw
ithth
ere
sults
abov
e:th
epr
esen
tsim
ulat
ion
isno
tve
rydi
ffer
entf
rom
the
sim
ulat
iono
fahe
alth
yhum
anpr
esen
tedin
Sec
tion6
.3.3
.T
his
isfu
rthe
rsup
port
edby
the
plot
ofth
epr
essu
redi
ffer
ence
betw
eena
triu
man
dve
ntric
lesh
own
inF
igur
e6.
32.
The
peak
pres
sure
diff
eren
ceis
som
ewha
tlo
wer
than
inth
esi
mul
atio
nof
ahe
alth
yhum
an(F
igur
e6.
20),
itha
sdec
reas
edap
prox
imat
ely1
5%.B
utth
isis
noti
nag
reem
entw
ithw
hatS
tuga
arde
tal.
(199
4b)
foun
dfo
rdo
gsw
ithac
ute
isch
emic
failu
re,w
here
the
peak
pres
sure
diff
eren
cede
crea
seda
ppro
xim
atel
y59%
.In
sum
mar
y,w
efin
dth
atth
esi
mul
atio
nof
asl
owre
laxa
tion
ofth
eap
ical
regi
onis
not
very
diff
eren
tcom
pare
dto
asi
mul
atio
nof
ahe
alth
ype
rson
,an
obse
rvatio
nw
hich
hold
sfo
rbo
thve
loci
tyan
dpr
essu
redi
ffer
ence
data
.T
hese
sim
ulat
ionr
esul
tsar
eno
tin
agre
emen
twith
mea
sure
men
tson
hum
ansa
nddo
gs,
whe
resu
bsta
ntia
lchan
gesa
rese
enin
AM
Ipa
tient
scom
pare
dto
heal
thyp
erso
ns.
6.4.
3A
pica
lreg
ion
inac
tive
The
third
scen
ario
we
inve
stig
atei
sth
eca
seof
anin
activ
e(o
rpas
sive)
apic
alre
-gi
on.T
his
mea
nsth
atth
ere
gion
neith
erco
ntra
ctsn
orre
laxe
s.It
issi
mpl
ypa
ssive
130
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
tissu
efol
low
ing
the
mot
ion
ofth
ebl
ood.
Itco
rres
pond
stoa
myo
card
ialin
farc
-tio
n,w
here
part
ofth
ehe
artw
alld
ies
due
toa
bloc
kage
ofth
eco
rona
ryar
terie
san
dsu
bseq
uent
lyloo
sesi
tsab
ility
toco
ntra
ctan
dre
lax.
Whe
nw
esi
mul
atea
nin
activ
ere
gion
ofth
ehe
artw
alli
n2D
itco
rres
pond
stoa
who
lecr
osss
ectio
nof
the
hear
tbei
ngin
activ
ein
3D.T
hus,
this
type
ofsi
mul
atio
nis
only
reas
onab
lew
hent
hein
activ
ere
gion
islo
cate
datt
heap
ex.C
hoos
inga
nyot
herr
egio
nin
a2D
mod
elw
ould
notb
eve
rym
eani
ngfu
l.W
esi
mul
ate
apa
ssive
regi
onby
letti
ngit
bede
fined
sim
ilarly
toth
eva
lve
tissu
e,i.e
.by
notl
ettin
git
have
any
mus
clep
rope
rty.
We
doth
ison
lyfo
rth
elin
ksal
ong
the
hear
twal
lin
the
apex
regi
on,i
.e.n
otfo
rth
ecr
ossl
inks
inth
ere
gion
.W
ere
duce
the
size
ofth
eap
ical
regi
onbe
caus
ethe
mod
elca
nnot
with
stan
dthe
pres
sure
insy
stol
ewhe
nth
een
tire
apic
alre
gion
isin
activ
e.A
sa
cons
eque
nce
ofth
ehi
ghpr
essu
re,th
ew
alls
leak
and
bloo
dle
aves
the
vent
ricle
thro
ugh
the
vent
ricul
arw
all.
Eve
nw
hen
we
redu
ceth
esi
zeof
the
inac
tive
regi
onw
eha
veto
incr
ease
the
num
bero
fbou
ndar
ypoi
ntst
ow
ithst
andt
hesy
stol
icpr
essu
re.
Dur
ing
dias
tole
the
hear
tbou
ndar
ycon
figur
atio
nfro
mth
esi
mul
atio
nwith
anin
activ
eap
ical
regi
on(le
ftsi
depl
otin
Fig
ure
6.33
)is
notv
ery
diff
eren
tfro
mth
eco
nfigu
ratio
nfor
ahe
alth
yper
son(
Fig
ure
6.19
),th
eve
ntric
lebe
ing
slig
htly
larg
er.B
utw
hen
syst
oles
etsi
nit
iscl
eart
hatt
heap
ical
part
isin
activ
e:it
bulg
esou
tand
hang
slik
ea
ballo
onfil
led
with
wat
erbe
low
the
cont
ract
edup
perp
arto
fth
eve
ntric
le.T
herig
htsi
depl
otin
Fig
ure
6.33
show
sth
eve
loci
tyfie
ldan
dhe
art
Time
1.035
Time
1.666
Fig
ure6
.33:
Vel
ocity
field
andh
eartb
ound
arya
tthe
time
ofpe
akve
loci
tyat
ML
Tlin
e(le
ft)an
ddu
ring
syst
ole(
right
)fr
oma
sim
ulat
ion
whe
reth
eap
ical
regi
onis
inac
tive.
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
131
+4
+3
+2
+1
ML
T
-2
-25
ms
PV
ÑÒÓ
+25
ms
+51
ms
+76
ms
1.01
0s1.
035
s1.
060
s1.
086s
1.11
1s
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Fig
ure
6.34
:V
eloc
itypr
ofile
sof
abso
lute
velo
city
(m/s
)alo
ngsi
xlin
esat
five
times
from
asi
mul
atio
nwhe
reth
eap
ical
regi
onis
inac
tive.
Ifth
eve
loci
typr
ofile
ispl
otte
dat
the
time
ofP
Vat
alin
e,P
Vis
mar
ked
bya
Ô .bo
unda
ryat
æ ò$
$$ s,w
here
the
larg
eve
ntric
leis
clea
rlyse
en.C
ompa
reth
ispl
otw
ithth
eve
loci
tyfie
ldin
the
last
fram
e(a
t
æ ò/ $ s)
inF
igur
e4.
2.T
hetw
opl
ots
are
atth
esa
met
ime
rela
tive
toth
ele
ngth
ofa
hear
tcyc
le.
Fig
ure
6.34
show
sth
eve
loci
typr
ofile
sal
ong
the
six
tran
sver
selin
esin
the
vent
ricle
atfiv
etim
es:
25m
sbe
fore
PV
atth
em
itral
leafl
ettip
s,at
the
time
ofP
Vat
the
mitr
alle
aflet
tips,
and
25,5
1an
d76
ms
afte
rthe
time
ofP
Vat
the
mitr
alle
aflet
tips.
Com
parin
gthe
velo
city
profi
les
with
thos
eobt
aine
dfor
ahe
alth
yhum
an(F
igur
e6.
15)w
ese
esev
eral
chan
ges.
The
profi
les
are
gene
rally
low
er,w
ider
and
mor
esk
ewed
.T
hela
tterc
anal
sobe
seen
from
the
plot
ofth
eve
loci
tyfie
ldat
the
time
ofpe
akve
loci
tyat
the
mitr
alle
aflet
tips
inF
igur
e6.
33.
Ifw
eco
mpa
reth
isto
the
velo
city
field
for
the
heal
thyh
uman
(Fig
ure
6.19
)we
see
that
the
dire
ctio
nof
the
inflo
wje
tis
mor
eto
war
dth
ean
terio
rwal
lin
the
132
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
PV[m/s]
MR
healthy
MR
AMI
Healthy
Apex
inact.
Fig
ure
6.35
:Pea
kvel
ocity
(PV
)att
hesi
xlin
esfr
omM
Rm
easu
rem
entso
fhe
alth
yper
sons
(
Õ )an
dA
MI
patie
nts
(
Ö ),an
dfro
ma
sim
ulat
iono
fahe
alth
ype
rson
(
& )an
da
sim
ulat
ionw
here
the
apic
alre
gion
isin
activ
e(
- ).
-2
01
23
4Distance
from
MLT
[cm]
0.2
0.4
0.6
0.81
1.2
1.4
1.6
VI
MR
healthy
MR
AMI
Healthy
Apex
inact.
Fig
ure6
.36:
Vel
ocity
inde
x(V
I)at
the
six
lines
from
MR
mea
sure
men
tsof
heal
thyp
erso
ns(
Õ )an
dA
MI
patie
nts
(
Ö ),an
dfro
ma
sim
ulat
iono
fahe
alth
ype
rson
(
& )an
da
sim
ulat
ionw
here
the
apic
alre
gion
isin
activ
e(
- ).
-2
01
23
4Distance
from
MLT
[cm]
-500
50
100
TD[ms]
MR
healthy
MR
AMI
Healthy
Apex
inact.
Fig
ure
6.37
:Tim
ede
lay(
TD
)at
the
six
lines
from
MR
mea
sure
men
tsofh
ealth
ype
rson
s(
Õ )an
dA
MI
patie
nts(
Ö ),an
dfr
oma
sim
ulat
iono
fahe
alth
yper
son(
& )an
da
sim
ulat
ionw
here
the
apic
alre
gion
isin
activ
e(
- ).
6.4
Sim
ulat
ions
ofpa
thol
ogic
alhe
artc
ondi
tions
133
pres
ents
imul
atio
n.F
urth
erm
ore,t
hesh
apeo
fthe
profi
lesi
nF
igur
e6.
34ar
emor
etr
iang
ular
than
for
ahe
alth
yper
son.
Fig
ures
6.35
–6.3
7sho
wth
epe
akve
loci
tyP
V,th
eve
loci
tyin
dex
VI,
and
the
time
dela
yT
D,
resp
ectiv
ely.
Inea
chof
thes
eth
ree
figur
esth
eco
rres
pond
ing
data
from
MR
mea
sure
men
tsof
heal
thy
pers
onsa
ndA
MI
patie
ntsa
ndfr
oma
sim
ulat
iono
fahe
alth
yper
sona
rein
clud
ed.T
heda
tafr
omth
eM
Rm
easu
rem
ents
isid
entic
alto
that
ofF
igur
es6.
2–6.
4,w
hile
the
data
from
asi
mul
atio
nofa
heal
thy
pers
onis
iden
tical
toth
atof
Fig
ures
6.16
–6.1
8.F
igur
e6.
35sh
ows
that
dow
nstr
eam
ofth
eM
LT
line
PV
islo
wer
atth
e3
and
4cm
lines
com
pare
dto
the
sim
ulat
ion
ofa
heal
thyp
erso
n.T
his
isin
cont
rast
tow
hatw
esa
win
the
prev
ious
scen
ario(
with
asl
owre
laxa
tiono
fthe
apic
alre
gion
),he
reP
Vw
ashi
gher
atth
osel
ines
.In
the
pres
ents
imul
atio
nP
Vis
low
erde
spite
the
vort
expr
esen
tinth
eve
ntric
le,s
ow
ebe
lieve
that
we
can
trus
tthe
sere
sults
mor
e.T
hech
ange
inth
eva
riatio
nof
PV
with
the
dist
ance
from
the
ML
Tlin
eis
very
sim
ilari
nsi
mul
atio
nsan
dM
Rm
easu
rem
ents
:PV
isge
nera
llylo
wer
ina
dise
ased
hear
t,but
mos
tlyat
the
3an
d4
cmlin
es.
We
find
the
sam
egoo
dagr
eem
entbe
twee
nsim
ulat
ions
andM
Rm
easu
rem
ents
whe
nw
elo
okat
VI
inF
igur
e6.
36.V
Iis
low
erat
the
4cm
line
ina
dise
ased
hear
t(b
oth
from
sim
ulat
ions
andM
Rda
ta)th
anin
ahe
alth
yhea
rt,o
ther
wis
eitd
oesn
otch
ange
muc
h.It
shou
ldbe
note
d,ho
wev
er,t
hati
nM
Rm
easu
rem
entsV
Iis
also
low
erat
the
3cm
line
inan
AM
Ipa
tient
than
ina
heal
thyp
erso
nF
igur
e6.
37sh
ows
that
TD
ishi
gher
inth
epr
esen
tsim
ulat
iont
han
inth
esi
m-
ulat
ion
ofa
heal
thyp
erso
n,es
peci
ally
for
the
-2an
d4
cmlin
es.T
his
isin
reas
on-
able
agre
emen
twith
the
MR
mea
sure
men
ts.Her
e,T
Dis
high
erat
the
3an
d4
cmlin
esfo
rA
MI
patie
ntsc
ompa
redt
ohe
alth
yper
sons
,oth
erw
isei
tis
unch
ange
d.T
Dfr
omth
epr
esen
tsim
ulat
ion
isev
eryw
here
with
inth
est
anda
rdde
viat
ion
ofth
eM
Rre
sults
.Am
uch
high
erT
Dat
the
line
clos
estto
the
apex
isin
good
agre
e-m
entw
ithth
efin
ding
sof
Stu
gaar
deta
l.(1
994b
).T
hey
mea
sure
the
time
dela
yin
heal
thyd
ogsa
nddo
gsw
ithac
utei
sche
mic
failu
rean
dfin
dth
atth
etim
ede
lay
incr
ease
sapp
roxi
mat
elyb
ya
fact
orof
5(f
rom
20m
sto
101
ms)
.T
hesi
mul
atio
nre
sults
can
part
lybe
expl
aine
dby
the
pres
sure
diff
eren
cebe
-tw
een
atriu
man
dve
ntric
le,w
hich
ispl
otte
din
Fig
ure
6.38
.The
figur
esh
ows
asm
allp
ositi
vepr
essu
redi
ffer
ence
first
,and
then
,del
ayed
com
pare
dto
ahe
alth
yhu
man
(Fig
ure
6.20
),a
larg
erpo
sitiv
epr
essu
repe
akoc
curs
.The
peak
pres
sure
diff
eren
ceis
larg
erth
anfo
rth
ehe
alth
yhum
an.T
hela
tero
ccur
renc
eoft
hepe
akpr
essu
redi
ffer
ence
expl
ains
why
TD
incr
ease
s,but
sinc
eth
epe
akis
larg
erth
anfo
ra
heal
thyp
erso
nit
does
note
xpla
inw
hyP
Van
dV
Iar
elo
wer
atth
e4
cmlin
e.T
hela
rger
peak
pres
sured
iffer
ence
inth
epr
esen
tsim
ulat
ionc
ompa
redto
the
sim
-ul
atio
nof
ahe
alth
yper
soni
sno
tin
agre
emen
twith
the
findi
ngso
fStu
gaar
deta
l.
134
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
11.2
1.4
1.6
1.8
t[s]
-101234
pressure[mmHg]
Fig
ure
6.38
:Tim
eva
riatio
nof
pres
sure
diff
eren
ce(m
mH
g)be
twee
natr
ium
and
vent
ricle
from
asi
mul
atio
nwhe
reth
eap
ical
regi
onis
inac
tive.
(199
4b).
The
yfin
dth
atth
epe
akpr
essu
rede
crea
sesw
ithap
prox
imat
ely5
8%in
dogs
durin
gac
utei
sche
mic
failu
reco
mpa
redt
ohe
alth
ydog
s.T
hesi
mul
atio
nof
anin
activ
eap
ical
regi
onco
mpa
resr
easo
nabl
ywel
lw
ithM
Rm
easu
rem
entso
fAM
Ipa
tient
s.W
ese
ethe
sam
echa
nges
inth
eve
loci
typr
o-fil
esan
din
dexe
sfou
ndby
Hou
lind
etal
.(20
01)i
nM
Rm
easu
rem
entso
fhea
lthy
pers
onsa
ndA
MI
patie
nts.
The
shap
esof
the
velo
city
profi
lesa
rem
ore
tria
ngul
arin
adi
seas
edhe
art,
and
PV
and
VI
are
low
erne
arth
eap
exw
hile
TD
ishi
gher
near
the
apex
.T
heva
riatio
nof
the
pres
sure
diff
eren
cebe
twee
nthe
atriu
man
dth
eve
ntric
leex
plai
nsth
ehi
gher
TD
valu
ebu
tnot
the
low
erva
lues
ofP
Van
dV
I.F
urth
erm
ore,t
hepe
akpr
essu
rediff
eren
ceis
noti
nag
reem
entw
ithm
easu
rem
ents
.
6.5
Con
clus
ions
We
have
impl
emen
teda
grad
uala
ctiv
atio
nof
the
hear
tmus
clei
nth
em
odel
and
com
pare
dasi
mul
atio
nw
ithth
egr
adua
lact
ivat
ion
with
the
refe
renc
esim
ulat
ion
from
Cha
pter
4.T
heco
mpa
rison
show
edth
atth
em
odel
isve
ryse
nsitiv
eto
the
ac-
tivat
ion
time
ofth
epa
pilla
rym
uscl
e,be
caus
ethe
latte
rinfl
uenc
esth
eco
ntra
ctio
npa
ttern
ofth
eve
ntric
ular
wal
land
the
mov
emen
toft
hem
itral
valv
e.T
here
sults
from
asi
mul
atio
nw
here
we
adju
sted
the
activ
atio
ntim
eof
the
papi
llary
mus
cle
6.5
Con
clus
ions
135
and
the
prop
agat
ionv
eloc
ityin
the
vent
ricle
show
edth
atth
egr
adua
lact
ivat
ion
does
noti
ntro
duce
maj
ordi
ffer
ence
sinth
eflo
wpa
ttern
.The
bigg
estd
iffer
ence
isse
enat
the
aort
icou
tflow
asa
dela
yedo
peni
ng/c
losi
ngof
the
aort
icsi
nkan
da
low
erpe
akou
tflow
rate
.We
choo
seno
tto
use
the
grad
uala
ctiv
atio
nbe
caus
eitin
trod
uces
only
smal
lcha
nges
,and
beca
useo
fth
ese
nsitiv
ityto
the
activ
atio
nof
the
papi
llary
mus
cle.
We
also
impl
emen
teda
regi
onal
activ
atio
n,w
here
itis
poss
ible
tode
fine
the
para
met
ersd
eter
min
ingt
heac
tivat
ion
func
tion
inde
pend
ently
for
5re
gion
sin
the
vent
ricul
arw
all.
Thi
sw
asou
rbas
isfo
rco
nduc
tingi
nves
tigat
ions
ofth
em
echa
-ni
smso
fthe
early
dias
tolic
fillin
gph
ase.
We
sim
ulat
eda
heal
thyh
uman
and
com
pare
dthe
resu
ltsw
ithM
Rm
easu
re-
men
ts.T
heve
loci
typr
ofile
alon
gsi
xtr
ansv
erse
lines
inth
eve
ntric
lean
dth
eve
-lo
city
inde
xesd
eriv
edfr
omth
epr
ofile
sco
mpa
rere
ason
ablyw
ellw
ithM
Rm
ea-
sure
men
ts.T
hepr
essu
redi
ffer
ence
betw
eena
triu
man
dve
ntric
le,w
hich
driv
esth
ebl
ood
into
the
vent
ricle
,als
oco
mpa
redw
ellw
ithm
easu
rem
ents
.W
eha
vein
vest
igat
edth
ree
scen
ario
swhi
chre
flect
the
mec
hani
smso
fea
rlydi
asto
licfil
ling:
slow
rela
xatio
nof
the
vent
ricle
,slo
wre
laxa
tion
ofth
eap
ical
regi
on,a
ndan
inac
tive
apic
alre
gion
.We
hypo
thes
ized
that
dias
tolic
dysf
unct
ion
coul
dbe
mod
elle
dby
alo
wer
rela
xatio
nrat
e
Þ"ß (corr
espo
ndin
gtoa
high
er
Ü inm
easu
rem
ents
)orsi
mpl
yby
letti
nga
regi
onof
the
hear
twal
lbe
inac
tive.
Inth
efir
stsc
enar
iowe
saw
low
erpe
akve
loci
tiesa
nda
dela
yedfi
lling
,an
dthi
sco
uld
beex
plai
ned
bya
redu
cedp
ress
ured
iffer
ence
betw
eent
heat
rium
and
the
vent
ricle
.T
hese
resu
ltsw
ere
inag
reem
entw
ithth
em
easu
rem
entso
fS
tuga
ard
etal
.(19
94b)
,who
inve
stig
ated
dogs
durin
gin
duct
ion
ofbe
ta-b
lock
ade,w
hich
affe
cts
the
vent
ricle
glob
ally
.T
hus
itis
poss
ible
tosi
mul
ateg
loba
lisc
hem
iain
the
vent
ricle
byde
crea
sing
Þuß .T
hese
cond
scen
ario
reve
aled
that
the
sim
ulat
ion
resu
ltsdo
notc
hang
ever
ym
uch
whe
non
ere
gion
rela
xes
slow
erth
anth
eot
herr
egio
ns.
The
sesi
mul
atio
nre
sults
are
noti
nag
reem
entw
ithm
easu
rem
entso
nbo
thhu
man
and
dogs
,whe
resu
bsta
ntia
lchan
gesa
rese
en.T
husd
ecre
asin
g
Þuß regio
nally
does
nots
imul
atea
loca
lisc
hem
icsi
tuat
ion.
The
resu
ltsof
the
third
scen
ario
whe
reth
eap
ical
regi
onis
inac
tive
com
pare
dw
ellw
ithM
Rm
easu
rem
entso
fAM
Ipa
tient
s.B
oth
velo
city
profi
lesa
ndve
loci
tyin
dexe
ssho
wed
the
sam
echa
ngeb
etw
eena
heal
thya
nda
dise
ased
hear
t(AM
I)as
seen
inM
Rda
ta.H
owev
er,t
hese
chan
gesc
ould
only
part
lybe
expl
aine
dby
the
pres
sured
iffer
ence
betw
eena
triu
man
dven
tric
lese
enin
the
sim
ulat
ion.
The
peak
pres
sure
diff
eren
cew
as,fu
rthe
rmor
e,no
tin
agre
emen
twith
mea
sure
men
ts.Thu
sle
tting
the
apic
alre
gion
bein
activ
ese
emst
obe
are
ason
ablew
ayof
mod
ellin
gA
MI.
136
Sim
ulat
ing
path
olog
ical
hear
tcon
ditio
ns
We
conc
lude
that
the
mod
elca
nbe
used
tost
udyc
erta
inm
echa
nism
sinvo
lved
inth
eea
rlydi
asto
licfil
ling
phas
e.T
hem
odel
canc
aptu
rem
echa
nism
sthat
chan
geth
ere
laxa
tionp
atte
rndr
astic
ally,
buts
mal
lerc
hang
esin
the
rela
xatio
npat
tern
dono
tcha
nget
hem
odel
resu
lts.W
eha
vesh
own
that
itis
poss
ible
tom
odel
agl
obal
dias
tolic
dysf
unct
ionb
ych
angi
ngth
ere
laxa
tionr
ate
Þ ß .Buti
nth
eca
seof
anis
chem
icap
ical
regi
on(a
loca
ldys
func
tioni
ngre
gion
),w
eco
uld
notm
odel
itby
chan
ging
Þ ß ,butw
ew
ere
able
tosi
mul
atea
nin
farc
tion
byle
tting
the
regi
onbe
inac
tive,
even
thou
ghth
edr
ivin
gpr
essu
redi
ffer
ence
was
noti
nag
reem
entw
ithm
easu
rem
ents
.
Cha
pter
7
The
3Dhe
artm
odel
Orig
inal
lyit
was
hope
dtha
tthe
2Dm
odel
coul
dbe
used
asa
prel
imin
aryt
estin
gto
olfo
rex
perim
ents
even
tual
lyto
beca
rrie
dout
inth
e3D
hear
tmod
el.H
owev
er,
ittu
rned
outt
hatt
he3D
mod
elw
asco
mpu
tatio
nally
too
dem
andi
ngan
ddi
dno
tm
atch
MR
mea
sure
men
tswel
leno
ught
obe
used
for
expe
rimen
tsan
din
vest
iga-
tions
ofth
ety
pepr
esen
tedin
Cha
pter
6fo
rth
e2D
mod
el.
Thi
sch
apte
rdes
crib
esbr
iefly
the
3Dhe
artm
odel
and
pres
entsa
com
paris
onof
the
outp
utfr
oma
sim
ulat
ionw
ithM
Rda
ta.T
heco
mpu
tatio
nalde
man
doft
he3D
mod
elis
also
pres
ente
d.
7.1
Des
crip
tion
ofth
em
odel
The
prin
cipl
esof
the
3Dm
odel
are
sim
ilart
oth
oseo
fthe
2Dm
odel
.The
hear
tisim
mer
sedi
na
cube
fille
dw
itha
fluid
ofth
esa
med
ensi
tyan
dvi
scos
ityas
bloo
d.T
hehe
artb
ound
ary(
mus
cula
rwal
land
the
valv
ele
aflet
s)ar
etr
eate
dasm
assl
ess
one-
dim
ensi
onalb
ound
arie
stota
llyim
mer
sedi
nflu
id(b
lood
)with
the
pote
ntia
lof
exer
ting
forc
eson
the
fluid
.In-
and
outfl
owto
the
hear
tism
odel
ledb
yso
urce
san
dsi
nks
and
the
mod
elin
clud
esan
exte
riors
ink/
sour
ceto
mak
ero
omfo
rth
eva
riatio
nin
tota
lhea
rtvo
lum
edu
ring
ahe
artb
eat.
The
maj
ordi
ffer
ence
betw
eent
he2D
and
3Dm
odel
sis
that
the
3Dm
odel
isof
the
full
hear
t(al
lfou
rhea
rtch
ambe
rsand
the
near
bygr
eatv
esse
ls),w
hile
the
2Dm
odel
isof
the
left
side
only
.F
urth
erm
ore,
som
ecom
puta
tiona
ldiff
eren
ces
exis
tand
thes
eare
desc
ribed
inth
efo
llow
ing.
138
The
3Dhe
artm
odel
7.1.
1E
quat
ions
ofm
otio
n
The
gove
rnin
geq
uatio
nsof
mot
ion
are
01 232 æ 3 4 3
5ò ó476 84î 39
(7.1
)
4 3 ò:ì<; ûæí
(7.2
)
9ì<; ûæí ò
=?>ìA@ ûB ûûæí C
ì ; óDì @ ûB û ûæíí ñ @ñ Bñ
(7.3
)
2D
2 æìA@ ûB ûûæíò3ìDìA@ ûB û ûæí ûæí
(7.4
a)
ò= 3ì ; ûæí Cì<; óDì @ ûB û ûæíí ñ ;
(7.4
b)
E òF1GHGIGIG2
D2
GHGIGIGJ@ ûB û ûæ
5
(7.5
)
K ò2D% 2
2D% 2
(7.6
)
>ò2 2 ì E Kí
(7.7
)
The
3DE
uler
ian
varia
bles
are
the
velo
city
3ì ; ûæí ,
the
pres
sure
6ì<; ûæí ,
the
forc
e
9ì<; ûæí ñ ; a
ndth
ede
nsity
dist
ribut
ion
ofth
eso
urce
sand
sink
s
:ì<; ûæí ,
whe
re
; den
otes
posi
tion
insp
ace,
ñ ; isth
evo
lum
eel
emen
t,and
æ istim
e.T
heLa
gran
gian
varia
bles
are
the
fiber
poin
tpos
ition
DìA@ ûB û ûæí ,
the
fiber
tens
ion
EìA@ ûB û ûæíñ @ñ B ,
the
unit
tang
entto
the
fiber
s
Kì @ ûB û ûæí
and
the
resu
ltant
fiber
forc
e>ìA@ ûB ûûæí
ñ @ñ Bñ .
9 and>
are
corr
espo
ndin
gforc
ede
nsiti
es.H
ere@ ,
B
and
are
curv
iline
arco
ordi
nate
scho
seni
nsu
cha
way
that
ì @ ûBí
labe
lsa
part
ic-
ular
fiber
and
ì @ ûB ûí la
bels
apa
rtic
ular
mat
eria
lpoi
nt.T
heeq
uatio
ns,as
writ
ten
here
,des
crib
eath
ree-
dim
ensi
onals
yste
mof
fiber
sasi
sap
prop
riate
for
the
thic
khe
artw
all.
Ava
lve
leafl
etis
afib
er-r
einf
orce
dsur
face
and
can
bede
scrib
edby
drop
ping
one
ofth
ein
dice
s@ or
B .E
quat
ion(
7.1)
isth
eth
ree-
dim
ensi
onalN
avie
r-S
toke
sequ
atio
nand
equa
tion
(7.2
)is
the
cont
inui
tyeq
uatio
n,th
e3D
vers
ions
ofeq
uatio
ns(3
.1)a
nd(3
.2),
re-
spec
tivel
yfo
rth
e2D
case
.Pes
kin
and
McQ
ueen
have
not,
toou
rkn
owle
dge,
publ
ishe
dade
scrip
tiono
fthe
dens
itydi
strib
utio
n
: in3D
,but
sinc
esou
rces
and
sink
sare
hand
ledi
dent
ical
lyin
the
2Dan
d3D
mod
els,
the
dens
itydi
strib
utio
nin
the
3Dm
odel
will
beof
the
sam
efor
mas
inth
e2D
mod
el(s
eeeq
uatio
n(3.
8)).
The
only
diff
eren
ceis
that
the
sum
mat
ioni
sov
erth
eto
taln
umbe
rofs
ourc
esan
dsi
nksi
nth
e3D
mod
el.
7.1
Des
crip
tion
ofth
em
odel
139
Rec
ently
McQ
ueen
and
Pes
kin
(200
0)in
trod
uced
the
new
form
ofth
em
o-m
entu
meq
uatio
ndis
cuss
edin
Sec
tion4
.6,w
hich
take
sint
oac
coun
tthe
diff
eren
tfo
rmof
the
cont
inui
tyeq
uatio
nand
cons
eque
ntlyt
hedi
ffer
entf
orm
ofth
em
o-m
entu
meq
uatio
n,w
hen
dist
ribut
edso
urce
sand
sink
sare
incl
uded
.How
ever
,thi
sit
isno
tinc
lude
din
the
vers
iono
fthe
3Dco
dew
ew
ork
with
.E
quat
ions
(7.3
)and
(7.4
b)ar
eth
ein
tera
ctio
nequ
atio
nsth
atco
nnec
tEul
eria
nan
dLa
gran
gian
quan
titie
s,an
alog
ousto
equa
tions
(3.3
)and
(3.4
)in
the
2Dca
se.
Her
eCì ; óDìA@ ûB û ûæíí is
the
thre
e-di
men
sion
alDira
cde
ltafu
nctio
n.E
quat
ions
(7.6
)–(7
.7)a
reth
efib
ereq
uatio
nsth
atar
eus
edto
dete
rmin
ethe
Lagr
angi
anfib
erfo
rce
dens
ity
>
from
the
fiber
confi
gura
tion
D attim
e
æ .In
equa
tion(
7.5)
, 2D% 2 is
the
fiber
stra
inan
d
F isthe
stre
ss-s
trai
nrela
tion
ofth
efib
ers.
Not
eth
eex
plic
ittim
ede
pend
encei
n
F ,whi
chal
low
sth
ehe
artto
cont
ract
and
rela
x.
Bou
ndar
yfo
rces
Inth
e3D
case
the
hear
tmus
clei
sno
trep
rese
nted
byth
e3
elem
entm
odel
de-
scrib
edin
Sec
tion3
.3.5
for
the
2Dm
odel
,nor
isth
epa
ssive
tissu
emod
elle
dby
linea
rspr
ings
.Inth
e3D
mod
elth
ete
nsio
n
E ina
shor
tfibe
rseg
men
tisgi
ven
bya
non-
linea
rrel
atio
nsh
ip
E òLNM<O MAPQ å1 R óR å R å5î RSRå
RTRå
whe
re
R isth
ele
ngth
ofth
efib
erse
gmen
t,Rå isth
ere
stin
gle
ngth
ofth
ese
g-m
enta
nd
Q å isthe
stiff
ness
.In
the
case
ofac
tive,
cont
ract
ilehe
artm
uscl
eRå
and
Q å aretim
ede
pend
ent,w
hile
they
are
cons
tant
for
pass
ivetis
sue.
Inth
eor
ig-
inal
2Dm
odel
the
entir
eat
rium
cont
ract
ssim
ulta
neou
sly,a
nd,a
ftera
dela
y,th
een
tire
vent
ricle
cont
ract
ssim
ulta
neou
sly,th
isis
also
the
case
inth
e3D
mod
el.
The
prop
agat
iono
fthe
activ
atio
nw
ave
intr
oduc
edin
the
2Dm
odel
inC
hapt
er6
coul
dea
sily
bein
trod
uced
in3D
byde
finin
gth
etim
ede
pend
ence
of
R å and
Q å
diff
eren
tlyfo
rdi
ffer
entp
arts
ofth
ehe
art.
Sou
rces
and
sink
s
The
3Dm
odel
incl
udes
allt
hene
arby
grea
tves
selso
fthe
hear
t:th
eao
rta,
the
pul-
mon
arya
rter
y,th
efo
urpu
lmon
aryv
eins
and
the
supe
riora
ndin
ferio
rven
acav
a.A
llth
eve
ssel
shav
ebl
ind
ends
,but
are
equi
pped
with
eith
era
sour
ce(p
ulm
onar
y
140
The
3Dhe
artm
odel
vein
san
dve
naca
va)
ora
sink
(aor
taan
dpu
lmon
arya
rter
y).
Fur
ther
mor
e,an
exte
riors
ourc
e/si
nkal
low
sfo
rvo
lum
ech
ange
sast
hehe
artfi
llsan
dej
ects
.A
sin
the
orig
inal
2Dm
odel
the
rela
tion
betw
eenfl
owan
dpr
essu
refo
ral
lso
urce
sand
sink
sis
chos
ento
bea
linea
rres
ista
ncem
odel
pum
ping
agai
nsta
cons
tant
pres
sure
rese
rvoir,
seeS
ectio
n3.2
.1fo
ra
desc
riptio
noft
helin
earr
esis
-ta
ncem
odel
ofth
eat
rials
ourc
ein
the
2Dm
odel
.
7.1.
2C
ompu
tatio
nalm
etho
d
The
num
eric
alm
etho
duse
din
the
3Dm
odel
isdi
ffer
entf
rom
the
one
inth
e2D
mod
el.
Itis
base
don
the
follo
win
gdi
scre
tizat
iono
fth
eeq
uatio
nsof
mot
ion.
Tim
eis
divi
ded
into
inte
rval
sof
leng
th
æ .T
heflu
ideq
uatio
nsar
edi
scre
tized
ona
regu
larc
ubic
latti
cein
the
phys
ical
spac
eof
the
varia
ble;
òìAU ) ûU î ûU'í ,
and
the
fiber
equa
tions
are
disc
retiz
edon
are
ctan
gula
rlatti
cein
the
spac
eoft
heLa
gran
gian
para
met
ers@ ,
B , .T
hem
eshw
idth
sof
thes
elat
tices
are
V òU )ò
U îò U' an
d
@ ,
B ,
,res
pect
ively
.O
nth
eflu
idla
ttice
the
forw
ard,
back
ward
and
cent
rald
ivid
eddi
ffer
ence
oper
-at
orsa
rede
note
d
Wï ,
W Ø and
Wå ,res
pect
ively
.Asu
bscr
ipt
Ý ò û" û+ in
dica
tes
the
dire
ctio
nin
whi
chth
edi
ffer
ence
isap
plie
d.
Xå isus
edfo
rth
eve
ctor
diff
er-
ence
oper
ator
ì Xå ) ûXå î ûXå 'í ,w
hich
corr
espo
ndsto
4 .O
nth
eLa
gran
gian
fiber
latti
ceth
ece
ntra
ldiff
eren
ceop
erat
orin
the
fiber
di-
rect
ion
isde
note
d
W ç andde
fined
by
ì W çYíìAíòì YìA % "í óYì ó%"íí%
The
disc
retiz
edeq
uatio
nsar
e
0Z 3êï) ó3ê æ'\[ ] ^)_ê ]W` ]3êa ò ó
Xå 6êï) 8
'\[ ] ^)Wï ]WØ ]3êï) 9ê
(7.8
)
Xå 3êï)ò:ê
(7.9
)
9ê ì ;í ò[ b éß éç>ê ìA@ûB û
í Cdcì ;óDê ì @ ûB ûíí@ B
(7.1
0)
Dêï)ìA@ ûB ûí òDê ì @ ûB ûíæ[ e3êï)ì<;í Cdcì ; óDê ì @ ûB ûííV' (7
.11)
7.1
Des
crip
tion
ofth
em
odel
141
Eê òFì W çDê J@ûB û
í
(7.1
2)
Kê òW çDê W ç
Dê
(7.1
3)
>êòW çì EêKêí
(7.1
4)
Inth
esee
quat
ions
_ê ]W` ] re
fers
toup
win
ddi
ffer
enci
ng:t
hefo
rwar
ddi
ffer
-en
ceis
used
ifth
eco
nvec
tionv
eloc
ity
_ê isne
gativ
ean
dth
eba
ckwa
rddi
ffer
ence
ifth
eco
nvec
tionv
eloc
ityis
posi
tive.
The
nota
tion
f b éß éçre
fers
toa
sum
over
the
fiber
latti
ce,w
hile
f e ref
erst
oa
sum
over
the
fluid
latti
ce.F
inal
ly,t
heno
tatio
n
C c
refe
rsto
anap
prox
imat
iono
fthe
thre
e-di
men
sion
alDira
cde
ltafu
nctio
n.T
he
C c
func
tion
used
inth
e3D
mod
elis
C cì<;íò V'C c
g U ) Vh C cg U î Vh C cg U ' Vh
whe
re
Cdcì BíòLNMAMAMAMAO MAMAMAMAP+ ó
" Bi jB ój Bî
,
û B k
! ó" B ói ó/" B ój Bî
,
û k B k"
û
" k B
The
com
puta
tiona
lmet
hod
proc
eeds
asfo
llow
s.G
iven
the
fiber
confi
gura
-tio
nDê ìA@ ûB ûí an
dth
eflu
idve
loci
ty
3êì ;í ev
alua
teth
efib
erfo
rces
>ê ìA@ûB û
í
thro
ught
hesu
cces
sive
useo
fequ
atio
ns(7
.12)
,(7.
13)a
nd(7
.14)
.The
nap
ply
the
fiber
forc
esto
the
com
puta
tiona
llatti
ceof
the
fluid
usin
geq
uatio
n(7.
10).
The
nup
date
the
fluid
velo
city
byso
lvin
gth
esy
stem
ofeq
uatio
ns(7
.8)a
nd(7
.9)f
or
ì 3êï) û6êï) í .
Thi
sis
done
byFo
urie
rtra
nsfo
rmin
gthe
equa
tions
,sol
ving
for
the
Four
iert
rans
form
sof
3 and6 ,
and
then
appl
ying
the
inve
rseF
ourie
rtra
nsfo
r-m
atio
n.T
hede
tails
ofth
isis
expl
aine
din
Pes
kina
ndM
cQue
en(1
996)
.Fin
ally
,in
terp
olat
ethe
new
velo
city
field
and
mov
eth
efib
ers
usin
geq
uatio
n(7.
11),
and
the
time
step
isco
mpl
ete.
Thi
sne
wco
mpu
tatio
nalm
etho
dre
mov
esth
ene
edto
scal
eal
lle
ngth
sand
time
bya
fact
or
l asdo
nein
the
2Dm
odel
.Thu
s,in
the
3Dm
odel
the
visc
os-
ity(a
ndco
nseq
uent
lythe
Rey
nold
snu
mbe
r)is
kept
atits
phys
iolo
gica
llyco
r-re
ctva
lue.
Pes
kina
ndM
cQue
enha
veim
prov
edth
eco
mpu
tatio
nalm
etho
deve
nm
ore
rece
ntly
byre
duci
ngth
enu
mer
ical
visc
osity
thro
ugh
the
intr
oduc
tion
ofa
(for
mal
ly)
seco
ndor
dera
ccur
atem
etho
d(M
cQue
enan
dP
eski
n,20
00;L
aian
dP
eski
n,20
00).
142
The
3Dhe
artm
odel
7.1.
3In
itial
geom
etry
The
initi
alge
omet
ryof
the
hear
tinth
e3D
mod
elis
muc
hm
ore
real
istic
than
inth
e2D
mod
el.I
tis
base
don
the
wor
kby
Tho
mas
(195
7),w
hoga
vea
deta
iled
desc
riptio
nof
the
fiber
layo
utof
mus
cle
fiber
sin
the
vent
ricle
sof
mam
mal
ian
hear
tsan
don
the
wor
kby
Str
eete
reta
l.(1
969;
1978
)who
mad
emea
sure
men
tsof
the
dist
ribut
ion
offib
eran
gles
inth
ele
ftve
ntric
ular
wal
l.P
eski
nand
McQ
ueen
cons
truc
tsthe
initi
alla
yout
offib
ers(
asbe
stas
poss
ible
)tom
imic
the
fiber
layo
utde
scrib
edin
Tho
mas
(195
7)an
dS
tree
tere
tal.
(196
9;19
78).
The
deta
ilsof
the
initi
alhe
artg
eom
etry
are
desc
ribed
inP
eski
nand
McQ
ueen
(199
6).T
hein
itial
layo
utof
the
hear
tcon
sist
ofm
ore
than
4000
fiber
san
dap
prox
imat
ely6
0000
0bo
unda
rypo
ints
.F
igur
e7.
1sh
ows
the
initi
alge
omet
ryof
the
hear
t.F
rom
the
figur
eit
isse
en
vein
sP
ulm
onar
y
Aor
ta
Left
atriu
m
Left
vent
ricle
Rig
ht
Pul
mon
ary
arte
ry
vent
ricle
Fig
ure
7.1:
Initi
alge
omet
ryof
the
3Dhe
art,t
herig
htat
rium
and
the
infe
riora
ndsu
perio
rven
acav
aca
nnot
bese
en.
7.2
Res
ults
143
Mitr
alva
lve
Rig
htLe
ftve
ntric
le
Left
atriu
m
Aor
ta
Aor
ticva
lve
Pul
mon
ary
vein
s
vent
ricle
Fig
ure
7.2:
Cut
-aw
ayvi
ewof
the
initi
alfib
erco
nfigu
ratio
n.
that
the
initi
alge
omet
ryis
still
sim
plifi
edco
mpa
redt
oa
real
istic
shap
eof
the
hear
t,but
the
geom
etry
chan
gesq
uick
lyto
am
ore
real
istic
shap
ewhe
nth
em
odel
isru
n.F
rom
Fig
ure
7.1
itis
diffi
cult
tose
ethe
inte
rioro
fth
ehe
art,
buti
nF
ig-
ure
7.2
acu
t-aw
ayvi
ewof
the
initi
alge
omet
ryis
show
n.H
ere,
the
mitr
alan
dao
rtic
valv
esca
nbe
seen
,asw
ella
sthe
initi
alco
nesh
apeo
fthe
left
vent
ricle
.T
hehe
artin
the
3Dm
odel
isal
sote
ther
edto
fixed
poin
ts,in
aw
aysi
mila
rto
that
desc
ribed
for
the
2Dm
odel
inS
ectio
n3.3
.7.I
nth
e3D
mod
elth
efix
poin
tsar
elo
cate
dal
ong
the
initi
alpo
sitio
nsof
the
outfl
owve
ssel
sof
the
hear
t.T
hete
ther
ings
inth
e2D
and
3Dm
odel
swer
ein
trod
uced
inde
pend
ently.
7.2
Res
ults
Asi
mul
atio
nw
ithth
e3D
mod
elst
arts
indi
asto
le,ju
stas
the
2Dm
odel
,but
inth
e3D
mod
elno
neof
the
fiber
sar
eun
dert
ensi
onat
initi
altim
e(t
heir
rest
ing
144
The
3Dhe
artm
odel
leng
thsa
reeq
ualt
oth
eir
initi
alle
ngth
s).
Thu
sth
ein
itial
setu
pis
even
mor
eun
real
istic
than
inth
e2D
mod
el.A
sa
cons
eque
nceof
this
,the
mod
elst
arts
with
alo
ngfil
ling
phas
e(it
last
sfor
3/4
ofa
hear
tcyc
le),
then
atria
lsys
tole
sets
inan
dth
esi
mul
atio
nco
mpu
teso
nehe
artb
eat.
The
tota
lsim
ulat
ion
time
isth
us1.
75he
artb
eats
.The
long
fillin
gin
the
begi
nnin
gof
asi
mul
atio
nis
driv
ensi
mpl
yby
the
flow
ofbl
ood
ente
ringf
rom
the
sour
cesin
the
atria
,bec
ause
neith
erst
iffne
sses
norr
estin
glen
gths
(whi
chde
term
iner
elax
atio
nand
cont
ract
ion)
ofth
eco
ntra
ctile
fiber
scha
ngeu
ntil
the
onse
tofa
tria
lsys
tole
.W
epe
rfor
med
one
sim
ulat
ionw
ithth
e3D
mod
elw
here
the
para
met
ersd
eter
-m
inin
gth
etim
eco
urse
inth
em
odel
(ons
etan
ddu
ratio
ntim
esof
vent
ricul
aran
dat
rials
ysto
le)a
reid
entic
alto
the
para
met
ersin
ourr
efer
ence
sim
ulat
ionw
ithth
e2D
mod
el(s
eeS
ectio
n3.3
.6).
We
chos
ethe
sepa
ram
eter
sto(h
opef
ully
)be
able
tom
atch
the
sim
ulat
ionr
esul
tsw
ithth
eM
Rda
tase
twe
com
pare
dthe
2Dm
odel
with
(see
Cha
pter
5).
We
did
notc
hang
eany
othe
rpar
amet
erva
lues
inth
em
odel
,so
the
rem
aini
ngva
lues
are
thus
aspu
blis
hedb
yP
eski
nand
McQ
ueen
(see
e.g.
Pes
kina
ndM
c-Q
ueen
,199
6;P
eski
nand
McQ
ueen
,199
4).T
his
mea
nsth
atth
eco
mpu
tatio
nsar
epe
rfor
med
ona
128
m 128
m 128
grid
with
3276
8tim
est
epsp
erhe
artb
eat,g
ivin
ga
tota
lof5
7344
time
step
scor
resp
ondi
ngto
1.75
hear
tbea
ts.
Sim
ulat
ions
with
the
3Dm
odel
are
com
puta
tiona
llyve
ryde
man
ding
.The
sim
ulat
ionp
rese
nted
here
take
sapp
roxi
mat
ely5
80cp
uho
urso
na
Cra
yC
90an
dus
es51
2M
Bof
mem
ory.
The
impl
emen
tatio
nof
the
code
one.
g.th
eC
ray
isde
scrib
edin
McQ
ueen
and
Pes
kin(
1997
).F
rom
this
pape
ritis
also
seen
that
the
com
puta
tiona
llyde
man
ding
part
sare
the
inte
grat
ion
ofth
eN
avie
r-S
toke
seq
ua-
tions
and
the
inte
ract
ions
betw
eenfl
uid
and
fiber
latti
ces.
Fur
ther
mor
e,du
ring
asi
mul
atio
nm
any
outp
utfil
esar
epr
oduc
edan
dsu
bseq
uent
lythe
post
proc
essi
ngis
also
com
puta
tiona
llyve
ryde
man
ding
.In
Fig
ure
7.3
the
flow
thro
ugh
the
mitr
alan
dao
rtic
valv
esdu
ring
the
sim
u-la
tion
issh
own.
The
flow
thro
ugh
the
aort
icva
lve
look
sas
expe
cted
:zer
oflo
wun
tilth
eon
seto
fsy
stol
e,th
ena
larg
efo
rwar
dpe
ak,f
ollo
wed
bya
smal
lbac
k-w
ard
peak
atth
etim
eof
clos
ureo
fthe
valv
e,an
dfin
ally
zero
flow
.For
the
mitr
alva
lve
flow
ther
eis
ape
akat
the
time
ofat
rials
ysto
le,a
sex
pect
ed,b
utth
isis
follo
wed
bya
long
perio
dof
time
with
back
flow
(neg
ativ
eflo
w),
until
the
onse
tof
the
next
dias
tolic
phas
e,w
hen
afo
rwar
dpe
ak(t
heE
peak
)is
seen
.The
long
perio
dof
back
flow
isno
texp
ecte
dbut
mig
htbe
caus
edby
the
mitr
alva
lves
bein
gcl
osed
and
bulg
ing
into
the
left
atriu
m,i.
e.th
eba
ckwa
rdflo
wm
aybe
bloo
dflo
w-
ing
tow
ard
the
mitr
alle
aflet
sfro
mth
eve
ntric
le.W
ew
ould
have
expe
cted
ala
rger
back
flow
atth
etim
eof
the
clos
ureo
fth
em
itral
valv
e,as
seen
inth
e2D
mod
el.
Thi
sha
sbee
nsee
nin
othe
rsim
ulat
ions
with
the
3Dm
odel
.The
last
peak
inth
e
7.2
Res
ults
145
0.25
0.5
0.75
11.25
1.5
t[s]
0
100
200
300
400
flow[cm^3/s]
Fig
ure7
.3:F
low
time
profi
leof
the
flow
(cm
' /s)t
hrou
ghth
em
itral
valv
e(d
ashe
d)an
dth
roug
hthe
aort
icva
lve
(sol
id).
Ver
tical
dash
edlin
ein
dica
test
hetim
eof
the
long
fillin
gph
ase(
æ ò /"
s).
mitr
alflo
wpr
ofile
corr
espo
ndsto
the
Epe
ak,a
ndth
issh
ould
have
been
larg
erth
anth
efir
stpe
akin
the
flow
profi
le.
Fig
ure
7.4
show
sth
etim
eve
loci
typr
ofile
for
the
abso
lute
cent
reve
loci
tyat
the
aort
icva
lve.
Com
pare
dto
the
velo
city
profi
lefr
omM
Rda
ta(F
igur
e5.
5)w
ese
eth
atth
epe
akve
loci
tyis
muc
hla
rger
inth
e3D
mod
el,4
60cm
/sin
the
3Dm
odel
and
90cm
/sin
MR
data
.T
hese
cond
peak
inth
esi
mul
atio
npr
ofile
isca
used
byba
ckflo
wat
the
time
ofcl
osur
eoft
heva
lve.
Thu
sthe
time
inte
rval
the
valv
eis
open
isap
prox
imat
ely2
50m
s,w
hich
isle
ssth
anse
enin
the
MR
data
(app
roxi
mat
ely3
65m
s).
Fig
ure
7.5
show
sth
etim
eve
loci
typr
ofile
for
the
abso
lute
cent
reve
loci
tyat
the
mitr
alva
lve.
Thi
sis
mor
edi
fficu
ltto
judg
eth
anth
eao
rtic
velo
city
profi
lebe
caus
eitis
notc
lear
whi
chpe
aksa
rere
late
dto
forw
ard
flow
and
whi
char
ere
late
dto
back
ward
flow
.B
utth
efir
stpe
akaf
tert
heve
rtic
allin
em
ustb
eca
used
byat
rials
ysto
le,a
ndw
eex
pect
the
follo
win
gpe
akto
bere
late
dto
back
flow
byth
etim
eof
clos
ureo
fthe
mitr
alva
lve.
By
com
parin
gthe
time
cour
sein
Fig
ure7
.3w
ithth
isfig
ure,
we
belie
veth
atth
ela
stpe
akse
enm
ustb
eth
eE
peak
,whi
chhe
reis
appr
oxim
atel
y25
cm/s
.In
the
MR
data
the
Epe
akis
appr
oxim
atel
y65
cm/s
(Fig
ure
5.4)
.Eve
nif
the
high
estp
eaks
een(
appr
oxim
atel
y43
cm/s
)was
caus
edby
max
imum
forw
ard
flow
,thi
sis
less
than
the
65cm
/sse
enin
the
MR
data
.The
low
peak
velo
city
coul
dbe
caus
edby
ask
ewed
inflo
w,
but
othe
rdat
afr
omth
e
146
The
3Dhe
artm
odel
0.25
0.5
0.75
11.25
1.5
t[s]
100
200
300
400
velocity[cm/s]
Fig
ure
7.4:
Vel
ocity
time
profi
lefo
rth
eab
solu
tece
ntre
velo
city
(cm
/s)a
tthe
aort
icva
lve.
Ver
tical
dash
edlin
ein
dica
test
hetim
eof
the
long
fillin
gph
ase(
æ ò
/" s)
.
0.25
0.5
0.75
11.25
1.5
t[s]
10
20
30
40
velocity[cm/s]
Fig
ure
7.5:
Vel
ocity
time
profi
lefo
rth
eab
solu
tece
ntre
velo
city
(cm
/s)a
tthe
mitr
alva
lve.
Ver
tical
dash
edlin
ein
dica
test
hetim
eof
the
long
fillin
gph
ase(
æ ò
/" s)
.
7.2
Res
ults
147
sim
ulat
iond
oesn
otsu
ppor
tthis
.T
hepr
essu
rein
the
mod
elis
notr
ealis
tic.E
spec
ially
the
pres
sure
inth
eve
n-tr
icle
sis
fart
oohi
gh,w
here
am
axim
umpr
essu
reab
ove
1200
mm
Hg
isre
ache
d.T
his
isa
gene
ralp
robl
emw
ithth
e3D
mod
elth
atha
snot
yetb
eens
olve
d.E
ven
thou
ghth
ehe
arti
ste
ther
ed,it
isdi
fficu
ltto
exam
ine
the
flow
patte
rnin
side
the
hear
tbec
ause
ofth
em
ovem
ento
fth
ehe
art.
Thu
sit
isdi
fficu
ltto
see
the
vort
expa
ttern
inth
ele
ftve
ntric
le.
But
inF
igur
e14
.12
inP
eski
nan
dM
cQue
en(1
996)
the
flow
durin
gea
rlydi
asto
licfil
ling
isde
pict
edby
stre
aklin
es,
i.e.t
raje
ctor
ieso
fpa
rtic
lesi
mm
erse
din
the
flow
.T
hefig
ure
isre
prod
uced
inF
igur
e7.
6.H
ere
we
clea
rlyse
eavo
rtex
ring
form
ing
belo
wth
em
itral
ring.
Thi
sis
also
seen
inF
igur
e1in
McQ
ueen
andP
eski
n(20
00),
whi
chw
eha
vere
prod
uced
Fig
ure
7.6:
Cut
-aw
ayvi
ewof
the
hear
tdur
ing
early
vent
ricul
ardi
-as
tole
.Not
eth
erin
gvo
rtex
visi
ble
inth
ele
ftve
ntric
lebe
low
the
mi-
tral
valv
e.F
rom
Pes
kin
and
Mc-
Que
en(1
996)
.
Fig
ure7
.7:S
ectio
nthr
ough
the
left
vent
ri-cl
esh
owin
gst
ream
lines
ofve
ntric
ular
fill-
ing.
Apa
irof
vort
ices
(cro
ss-s
ectio
nof
avo
rtex
ring)
hasb
eens
hedf
rom
the
mitr
alle
aflet
san
dha
sm
igra
ted
dow
nwar
dsto
-w
ards
the
apex
ofth
ele
ftve
ntric
le.F
rom
McQ
ueen
and
Pes
kin(
2000
).
148
The
3Dhe
artm
odel
inF
igur
e7.
7.In
this
figur
eth
eflo
wis
show
nas
stre
amlin
es,w
hich
are
com
pute
dby
choo
sing
atim
eof
inte
rest
,fixi
ngth
eve
loci
tyfie
ldat
that
give
ntim
e,an
dco
mpu
tinga
ndpl
ottin
gw
hatt
hetr
ajec
torie
sofp
artic
lesw
ould
beif
the
part
icle
sw
ere
tom
ove
inth
at(t
ime-
inde
pend
ent)v
eloc
ityfie
ld.B
oth
ofth
efig
ures
clea
rlysh
owth
efo
rmat
ion
ofa
vort
exrin
gbe
low
the
mitr
alva
lve,
and
whe
ncu
tting
thro
ught
herin
gw
ese
eapa
irof
vort
ices
,sim
ilart
ow
hatw
ese
ein
the
2Dm
odel
,se
eFig
ure
5.1.
7.3
Dis
cuss
ion
The
resu
ltspr
esen
tedin
this
chap
ters
how
that
the
3Dm
odel
hast
obe
adju
sted
and
impr
oved
befo
rea
reas
onab
lecom
paris
onw
ithM
Rda
taca
nbe
obta
ined
.Thi
sis
nota
surp
rise,
sinc
eit
took
alo
ngtim
eto
find
ago
odpa
ram
eter
setti
ngin
the
2Dm
odel
,and
the
3Dm
odel
isev
enm
ore
com
plex
.B
utit
was
hope
dtha
tbec
ause
the
3Dm
odel
was
base
don
are
alis
tic,h
uman
hear
tthe
resu
ltsw
ould
have
been
reas
onab
lewith
outt
oom
any
para
met
erch
ange
s.Pa
ram
eter
sdet
erm
inin
gthe
time
cour
se,th
eco
ntra
ctio
nand
rela
xatio
noft
hefib
ers,
the
pre-
and
afte
rload
and
poss
ibly
the
geom
etry
ofth
eva
lves
have
tobe
adju
sted
toim
prov
eth
eco
mpa
rison
betw
eenm
odel
and
MR
data
.But
sim
ula-
tions
with
the
3Dm
odel
are
com
puta
tiona
llyve
ryde
man
ding
,als
oin
the
post
proc
essi
ngst
ep.T
hus
itis
notr
ealis
ticto
achi
eve
abe
tterp
aram
eter
setti
ngfo
rth
em
odel
with
ina
reas
onab
letim
epe
riod.
Eve
nif
the
com
paris
onw
ithon
esp
ecifi
cM
Rda
tase
tis
notg
ood,
we
can
still
obta
inva
luab
lein
sigh
tsin
toth
eflo
wpa
ttern
,par
ticul
arly
inth
ele
ftve
ntric
le,
from
the
3Dm
odel
.Bec
ause
the
mod
elis
3Dan
dthe
geom
etry
ofth
ele
ftve
ntric
leis
very
real
istic
we
can
uset
hem
odel
toex
amin
ethe
vort
expa
ttern
durin
gfil
ling.
Fro
mth
ere
sults
repr
oduc
edhe
re,w
ese
etha
tavo
rtex
ring
form
sdu
ring
dias
tole
and
acu
t-aw
ayvi
ewof
this
show
sa
vort
expa
ir,ex
actly
asse
enin
the
2Dm
odel
.T
his
supp
orts
the
idea
ofth
evo
rtex
patte
rnbe
ing
a3D
phen
omen
onas
indi
cate
dby
the
MR
data
inC
hapt
er5.
Cha
pter
8
Con
clus
ions
and
sugg
estio
ns
The
purp
oseo
fth
isP
h.D
.pro
ject
hasb
eent
oac
hiev
ea
bette
rund
erst
andi
ngof
the
hem
odyn
amic
sof
the
hear
tusi
nga
com
puta
tiona
lmod
elan
dM
Rda
ta.W
eha
vepr
esen
teda
2Dco
mpu
tatio
nalm
odel
ofth
ebl
ood
flow
inth
ele
ftsi
deof
the
hum
anhe
art,
whi
chw
asco
nstr
ucte
dby
intr
oduc
ing
mod
ifica
tions
and
impr
ove-
men
tsto
Pes
kin
and
McQ
ueen
’s2D
hear
tmod
eldi
men
sion
edto
ado
g’s
hear
t.T
hese
mod
ifica
tions
and
impr
ovem
ents
wer
eba
sedo
nph
ysio
logi
calk
now
ledg
ean
dco
mpa
rison
swith
MR
data
onhu
man
s.
8.1
Sum
mar
yof
conc
lusi
ons
We
com
pare
dout
putf
rom
our2
Dm
odel
with
anM
Rve
loci
tyda
tase
t.In
par-
ticul
ar,
we
exam
ined
the
intr
aven
tric
ular
velo
city
field
san
dth
eve
loci
tytim
escu
rves
over
the
mitr
alrin
gan
dac
ross
the
aort
icou
tflow
trac
t.W
efo
und
reas
on-
able
agre
emen
tbetw
eent
hege
nera
lflow
patte
rnfr
omth
esi
mul
atio
nand
the
MR
data
:aw
ide
inflo
wje
tdur
ing
fillin
g,th
efo
rmat
ion
ofan
ante
riorv
orte
x,an
da
wid
eou
tflow
jetd
urin
gej
ectio
n.H
owev
er,t
here
wer
ediff
eren
cesin
the
deta
ilsof
the
two
flow
patte
rns,p
artic
ular
lyin
the
vort
exfo
rmat
ions
.Inth
esi
mul
atio
nsth
evo
rtic
esw
ere
mor
edi
stin
ctan
dth
eve
loci
tyin
the
vort
ices
was
high
er.
Fur
ther
-m
ore,
we
saw
thre
evo
rtic
esin
the
sim
ulat
ion
buto
nly
two
inth
eM
Rda
ta.T
heth
irdvo
rtex
inth
esi
mul
atio
nis
prob
ably
unph
ysio
logi
cal,a
ndit
wou
ldim
prov
eth
em
odel
cons
ider
ablyi
fit
coul
dbe
rem
oved
.T
hetim
eco
urse
ofdi
asto
lean
dsy
stol
ein
the
sim
ulat
ion
was
fairl
ygo
od:
the
timin
gof
atria
lsys
tole
and
the
onse
tof
vent
ricul
arsy
stol
ewer
ese
trea
son-
ably
wel
l,bu
tthe
dura
tion
ofsy
stol
ecou
ldbe
furt
heri
mpr
oved
byex
tend
ing
it.
150
Con
clus
ions
and
sugg
estio
ns
The
velo
city
time
profi
les
atth
em
itral
ring
and
aort
ictr
actf
rom
the
sim
ulat
ion
mat
ched
reas
onab
lywel
lwith
MR
data
.The
timin
gof
the
profi
lesw
asve
rygo
od,
whi
leth
esh
apes
and
max
imum
valu
esdi
ffer
edso
mew
hat.
The
conn
ectio
nbet
wee
nthe
sim
ulat
ionr
esul
tsan
dthe
para
met
ersin
the
mod
elis
very
com
plex
,and
we
belie
veth
atim
prov
emen
tscan
beac
hiev
edby
furt
herp
a-ra
met
erad
just
men
ts.H
owev
er,t
here
ason
ablec
ompa
rison
with
MR
data
just
ifies
usin
gth
em
odel
for
inve
stig
atio
nsof
the
hem
odyn
amic
soft
hehe
art.
We
impl
emen
teda
grad
uala
ctiv
atio
nof
the
hear
tmus
cle,
i.e.p
ropa
gatio
nof
the
activ
atio
nac
ross
the
hear
twal
lwith
agi
ven
velo
city
,whi
chre
sem
bles
the
activ
atio
nof
the
hear
tmus
clei
nth
eno
rmal
hear
t.We
foun
dth
atth
isdi
dno
taffe
ctth
esi
mul
atio
nre
sults
muc
h,an
dth
ere
sults
wer
eve
ryse
nsitiv
eto
the
activ
atio
nof
the
papi
llary
mus
cle.
We
perf
orm
edan
inve
stig
atio
nof
cert
ain
mec
hani
smsa
ffect
ing
the
dias
tolic
func
tioni
ngof
the
hear
tby
exam
inin
gth
eflo
wpa
ttern
durin
gea
rlyfil
ling.
We
foun
dre
ason
ablea
gree
men
tbetw
eens
imul
atio
nres
ults
andM
Rda
tafo
rahe
alth
yhu
man
.Inpa
rtic
ular
,the
pres
sure
diff
eren
ceth
atdr
ives
the
fillin
gof
the
vent
ricle
was
inag
reem
entw
ithcl
inic
alda
ta.
We
wer
eno
tabl
eto
sim
ulat
ethe
chan
gess
eenc
linic
ally
inth
eflo
wpa
ttern
ina
loca
lisc
hem
icco
nditi
onby
letti
ngth
eap
ical
regi
onre
lax
mor
esl
owly
than
the
rest
ofth
eve
ntric
le.H
owev
er,w
esu
cces
sful
lysim
ulat
eda
glob
alis
chem
icve
ntric
leby
letti
ngth
een
tirev
entr
icle
rela
xmor
eslo
wly
than
inth
ehe
alth
yhea
rt.
We
foun
dlo
wer
peak
velo
citie
sand
dela
yedfi
lling
inth
issi
tuat
ionc
ompa
redw
itha
sim
ulat
iono
fahe
alth
yhum
an.T
hese
chan
gesa
resi
mila
rto
mea
sure
ddat
aon
heal
thy
dogs
and
dogs
ina
glob
alis
chem
icco
nditi
on.
Fur
ther
mor
e,w
ew
ere
able
tosi
mul
atea
cute
myo
card
ialin
farc
tion(
AM
I)by
letti
ngth
eap
ical
regi
onbe
inac
tive.
Her
eth
eve
loci
tyda
tafr
omth
esi
mul
atio
nm
atch
edth
oseo
btai
nedb
yM
Rm
easu
rem
entsin
AM
Ipa
tient
s.H
owev
er,t
hedr
ivin
gpr
essu
rediff
eren
cew
asno
tin
good
agre
emen
twith
clin
ical
data
.W
eha
vesh
own
that
oura
nato
mic
allys
impl
e2D
mod
elis
able
toca
ptur
esom
eof
the
chan
gess
eend
urin
gpa
thol
ogic
alhe
artc
ondi
tions
.The
mod
elca
nbe
used
tost
udy
mec
hani
smsth
atin
trod
ucem
ajor
diff
eren
cesin
the
flow
patte
rn,w
hile
sim
ulat
ions
ofle
ssse
vere
cond
ition
sdo
nota
ltert
heflo
wpa
ttern
com
pare
dwith
the
flow
patte
rnfr
oma
sim
ulat
iono
fahe
alth
yper
son.
We
have
pres
ente
dPes
kina
ndM
cQue
en’s
3Dm
odel
ofth
een
tire
hear
tand
brie
flyco
mpa
redi
tsre
sults
with
anM
Rve
loci
tyda
tase
t.T
his
show
edth
atth
em
odel
hast
obe
furt
heri
mpr
oved
befo
rea
reas
onab
leagr
eem
entca
nbe
obta
ined
.In
part
icul
ar,t
heflo
wan
dve
loci
tyov
erth
em
itral
ring
isno
tin
agre
emen
t.Fur
-th
erm
ore,
sinc
eth
e3D
mod
elis
com
puta
tiona
llyve
ryde
man
ding
itis
notv
ery
usef
ulas
ato
olfo
rhe
mod
ynam
icinv
estig
atio
nsof
the
hear
t.H
owev
er,f
rom
the
8.2
Sug
gest
ions
for
futu
rew
ork
151
sim
ulat
ion
resu
ltsw
est
illga
inin
sigh
tint
oth
e3D
natu
reof
the
flow
patte
rnin
-si
deth
ehe
art.
The
resu
ltssh
owa
clea
rvor
tex
ring
form
ing
belo
wth
em
itral
ring
durin
gfil
ling.
8.2
Sug
gest
ions
for
futu
rew
ork
Sug
gest
ions
for
futu
rew
ork
can
bedi
vide
din
two
part
s:fu
ture
wor
kw
ithth
em
odel
pres
ente
dinth
isth
esis
and
futu
rew
ork
with
hear
tsim
ulat
ions
inge
nera
l.W
ebe
lieve
that
the
2Dm
odel
pres
ente
dinth
isth
esis
cans
tillb
eof
impo
rtan
cein
unde
rsta
ndin
gthe
hem
odyn
amic
soft
hehe
artif
itslim
itatio
nsar
ekep
tinm
ind.
The
follo
win
gsc
enar
iosc
ould
besi
mul
ated
and
inve
stig
ated
inth
e2D
mod
el:
Ô Diast
olic
func
tioni
ng.T
hein
vest
igat
ions
pres
ente
dinth
isth
esis
coul
dbe
exte
nded
e.g.
bysi
mul
atin
ginc
reas
edst
iffne
ssin
part
ofth
eve
ntric
ular
wal
ldu
eto
anin
farc
tion.
Ô Hyper
trop
hicc
ardi
omyo
path
y(HC
M).
Thi
sco
nditi
ons,
whi
chis
char
acte
r-iz
edby
incr
ease
dgro
wth
ofth
ele
ftve
ntric
ular
wal
l,ca
nbe
sim
ulat
edin
the
mod
elby
rais
ing
the
stiff
ness
es
Q .à ,
Q çà and
Q å inthe
mus
cula
rmod
el(s
eeS
ectio
n3.3
.5).
Ô Mitral
valv
est
enos
is.T
his
narr
owin
gof
the
mitr
alva
lve
can
besi
mu-
late
dby
intr
oduc
ingo
neor
seve
ralc
ross
links
betw
eent
hetw
om
itral
valv
ele
aflet
s.Ô Mitr
alva
lve
regu
rgita
tion.
Thi
sis
back
flow
ofbl
ood
from
the
vent
ricle
toth
eat
rium
due
tom
itral
valv
ein
suffi
cien
cyan
dca
nbe
sim
ulat
edby
intr
o-du
cing
bend
ingr
esis
tanc
einth
em
itral
leafl
ets,
whi
chw
ould
caus
ethe
mno
tto
clos
e.B
endi
ngre
sist
ance
can
bein
trod
uced
byth
epr
oced
ured
escr
ibed
inP
eski
nand
McQ
ueen
(198
0)on
mod
ellin
gpr
osth
etic
hear
tval
ves.
The
resu
ltspr
esen
tedin
this
thes
issu
ppor
tthe
idea
ofco
mpu
tatio
nalm
odel
sas
ato
olfo
run
ders
tand
ingth
ehe
mod
ynam
icso
fth
ehe
art.
How
ever
,si
nce
the
intr
aven
tric
ular
flow
patte
rnis
thre
edi
men
sion
al,fu
ture
wor
ksh
ould
even
tual
lybe
perf
orm
edw
itha
3Dm
odel
.W
ebe
lieve
that
this
wou
ldno
tha
veto
bea
com
puta
tiona
llyde
man
ding
mod
elof
the
entir
ehe
art,
like
the
mod
elof
Pes
kin
and
McQ
ueen
.Ins
tead
,asi
mpl
er3D
mod
elof
the
left
side
ofth
ehe
artw
ould
been
ough
tom
eett
hede
man
dsof
ahe
mod
ynam
ictoo
l.T
hew
ork
pres
ente
dhe
reha
ssho
wn
that
an(a
nato
mic
ally
)sim
ple2
Dm
odel
can,
with
inlim
its,p
redi
ct
152
Con
clus
ions
and
sugg
estio
ns
phys
iolo
gica
llyco
rrec
tflow
patte
rns.
Thu
sasi
mpl
e3D
mod
elsh
ould
also
beab
leto
doth
isan
dev
enbe
tter.
Suc
ha3D
mod
elsh
ould
beev
alua
teda
ndva
lidat
edin
deta
ilby
com
paris
onw
ithM
Rve
loci
tyda
ta.O
ncet
his
isdo
ne,th
em
odel
coul
dbe
used
for
inve
stig
atio
nssi
mila
rto
thos
epr
esen
tedin
this
thes
isan
dto
thos
esu
gges
teda
bove
for
futu
rew
ork
with
the
2Dm
odel
.
App
endi
xA
Mag
netic
reso
nanc
eimag
ing
Thi
sap
pend
ixpr
ovid
esan
intr
oduc
tion
tom
agne
ticre
sona
nceim
agin
g(M
RI)
.It
prov
ides
the
read
erw
itha
know
ledg
eon
how
the
MR
Ida
taus
edin
the
thes
isis
reco
rded
.Itde
scrib
esth
eba
sics
ofM
RI:
spin
ofnu
clei
,exc
itatio
n,re
laxa
tion
and
puls
eseq
uenc
esuse
din
MR
scan
ning
.Ital
sode
scrib
esth
eim
agin
gpr
oces
s:ho
wM
RI
iscr
eate
d,ch
oice
ofsc
anar
ea,s
patia
lenc
odin
gand
imag
ing
sequ
ence
sin
MR
I.F
inal
ly,v
eloc
itym
easu
rem
entsw
ithM
RI
usin
gve
loci
tyen
codi
ngis
de-
scrib
ed.T
hem
ain
sour
cefo
rth
isap
pend
ixis
Phi
lips
(n.d
.).
A.1
Bas
icso
fMR
I
The
prin
cipl
eof
MR
scan
ning
isba
sedo
nth
em
agne
ticpr
oper
tieso
fnuc
leiw
ithan
unev
enat
omic
mas
sor
atom
icnu
mbe
r,e.g
.
) n .S
uch
nucl
eipo
sses
sang
ular
mom
entu
man
dhav
ea
char
acte
ristic
spin
quan
tum
num
berg
reat
erth
anze
ro.T
hesp
inin
duce
sam
agne
ticfie
ldw
ithan
axis
coin
cide
ntw
ithth
eax
isof
spin
,and
with
am
agni
tude
and
dire
ctio
nre
pres
ente
dby
the
mag
netic
dipo
lem
omen
t,or
dipo
leve
ctor
,8 ,se
eFig
ure
A.1
.N
orm
ally
,the
dipo
leve
ctor
ofa
nucl
eusi
sor
dere
dran
dom
ly.B
ut,w
hen
anex
tern
alm
agne
ticfie
ldis
appl
iedt
hedi
pole
vect
oris
alig
nedi
npa
ralle
lora
ntip
ar-
alle
ldire
ctio
nw
ithth
eap
plie
dfiel
d.In
the
pres
ence
ofan
exte
rnal
mag
netic
field
the
dipo
leve
ctor
ofth
enu
cleu
sexp
erie
nces
ato
rque
,whi
chca
uses
itto
rota
te(o
rpr
eces
s)ar
ound
the
axis
ofth
eex
tern
alfie
ldw
itha
cert
ain
freq
uenc
y.T
his
phen
omen
onis
calle
dLa
rmor
prec
essi
on,se
eFig
ure
A.2
.T
hera
teof
prec
essi
onde
pend
supo
nth
eph
ysic
alch
arac
teris
ticso
fth
enu
-cl
eusa
ndth
est
reng
th,
o å ,ofth
em
agne
ticfie
ld.T
his
isex
pres
sedin
the
Larm
or
154
Mag
netic
reso
nanc
eimag
ing
µ
Fig
ureA
.1:A
spin
ning
nucl
eusw
ithits
dipo
leve
ctor
8 whe
nno
exte
rnal
mag
-ne
ticfie
ldis
appl
ied.
The
spin
axis
co-
inci
desw
ithth
edi
pole
vect
or.
B0
B0
Fig
ure
A.2
:P
rece
ssio
nof
the
dipo
leve
ctor
ofa
spin
ning
nucl
eusa
bout
anex
tern
alm
agne
ticfie
ld
o å .eq
uatio
n
p òlo å û
whe
re
p isth
eLa
rmor
freq
uenc
y,
l isth
egy
rom
agne
ticra
tiosp
ecifi
cto
the
nu-
cleu
sand
o å isthe
stre
ngth
ofth
eex
tern
alm
agne
ticfie
ld.
Sin
ce
l issp
ecifi
cto
each
nucl
eust
heLa
rmor
freq
uenc
yis
ther
ebya
lso
spec
ifict
oea
chnu
cleu
s.e.
g.fo
rth
ehy
drog
ennu
cleu
s,
) n ,th
eLa
rmor
freq
uenc
yis
42.5
759a
tan
exte
rnal
mag
netic
field
of1
T(S
chla
ikje
rand
Kol
enda
,199
7).
Larm
orpr
eces
sion
isa
reso
nanc
ephe
nom
enon
.Ifa
syst
emha
sana
tura
lreso
nanc
e,ene
rgy
can
bem
ost
effic
ient
lytr
ansf
erre
dto
the
syst
emat
this
freq
uenc
y.In
MR
imag
ing
itis
mai
nly
the
) n
nucl
eus(
the
prot
on)t
hati
sco
nsid
ered
.W
eth
eref
orel
imit
ours
elve
sto
this
case
inth
efo
llow
ing.
Con
side
ringa
colle
ctio
nof
prot
onsi
na
volu
me
inth
eab
senc
eofa
nex
tern
alm
agne
ticfie
ld.
The
indi
vidu
aldi
pole
vect
orsa
reor
dere
dran
dom
lyan
dth
usth
ene
tmag
netic
mom
ent,
q å ,iszer
o.In
the
follo
win
gw
ere
fert
oth
ene
tmag
netic
mom
enta
sth
em
agne
tizat
ionv
ecto
r.W
hen
anex
tern
alm
agne
ticfie
ld
o å isap-
plie
dth
edi
pole
vect
orsw
illte
ndto
alig
npa
ralle
lor
antip
aral
lelto
the
appl
ied
field
.T
here
isa
slig
htex
cess
favo
urin
gth
epa
ralle
ldire
ctio
n(t
helo
wen
ergy
stat
e),a
ndth
usth
em
agne
tizat
ionv
ecto
r
q å isnon
-zer
oand
isor
ient
edin
the
dire
ctio
nof
the
appl
ied
mag
netic
field
.T
here
isno
tran
sver
seco
mpo
nent
sinc
eth
ein
divi
dual
prec
essi
ngpr
oton
sare
rand
omly
orie
nted
with
resp
ectto
the
phas
eof
thei
rpre
cess
iona
lmot
ion.
A.1
Bas
icso
fMR
I15
5
We
now
intr
oduc
eaco
ordi
nate
syst
emw
here
the
r ó axis
ispa
ralle
ltoth
edi
-re
ctio
nof
the
exte
rnal
mag
netic
field
o å .Weca
nth
enth
ink
ofth
em
agne
tizat
ion
vect
oras
havi
ngtw
oco
mpo
nent
s:
q äs in
the
tran
sver
sepl
ane
and
qut inth
elo
ngitu
dina
lpla
ne.I
nth
eeq
uilib
rium
stat
ewe
have
q tòq å
and
q äsò .
Inth
isst
atet
hem
agne
tizat
ionv
ecto
ris
stat
ican
ddo
esno
tind
ucea
curr
ent,i
.e.w
eca
nnot
obta
inan
ysi
gnal
from
the
volu
me
unde
robs
erva
tion.
Inor
dert
oob
tain
asi
gnal
,the
spin
shav
eto
bepe
rtur
beda
ndle
ave
the
equi
libriu
mst
ate.
A.1
.1E
xcita
tion
The
equi
libriu
mst
atec
anbe
left
byex
posi
ngth
esp
insy
stem
toa
radi
ofr
eque
ncy
(RF
)pu
lse
inth
efo
rmof
atim
eva
ryin
gm
agne
ticfie
ld
o ) .
o ) ism
ade
tobe
perp
endi
cula
rtoth
est
atic
field
o å and
varie
swith
the
Larm
orfr
eque
ncy.
The
o ) fiel
d,w
hich
istu
rned
onon
lyfo
ra
shor
tper
iod
oftim
e,is
prod
uced
bya
coil
with
itsax
isin
the
Uv ópl
ane.
The
stat
icfie
ldis
typi
cally
ofth
eor
dero
f0.
1-1.
5T,
whi
leth
eex
cita
tion
field
isof
the
orde
rof1
mT.
The
effe
ctof
appl
ying
the
o ) fiel
dis
that
the
mag
netiz
atio
nvec
tora
lso
pre-
cess
esab
outth
isse
cond
field
and
rota
tesa
way
from
the
vert
ical
dire
ctio
nto
war
dth
e
Uv ópl
ane.
The
o å field
still
affe
cts
the
mag
netiz
atio
nvec
tora
ndm
akes
itpr
eces
sabo
utth
e
r ó axis
,so
the
tota
leffe
ctis
that
,whi
leth
eR
Fpu
lse
istu
rned
on,t
hem
agne
tizat
ionv
ecto
rper
form
sasp
iralli
ngm
otio
non
asp
here
assk
etch
edin
Fig
ure
A.3
.W
hen
obse
rvin
gthe
situ
atio
nin
aco
ordi
nate
syst
em
ì Uw ûvw ûrí th
atro
tate
sab
outth
e
r ó axis
with
the
Larm
orfr
eque
ncy,
the
effe
ctof
appl
ying
o ) isse
enas
sim
ply
flipp
ing
the
mag
netiz
atio
nvec
tort
owar
dth
eUw vw ó pla
ne.T
hele
ngth
ofth
em
agne
tizat
ionv
ecto
rispr
actic
ally
cons
tant
durin
gth
esh
ortp
erio
dof
time
the
RF
0x
0 y
0
z
0
z
z
y
x
Fig
ure
A.3
:S
pira
lling
mot
ion
ofth
em
agne
tizat
ionv
ecto
rdur
ing
anR
Fpu
lse.
156
Mag
netic
reso
nanc
eimag
ing
puls
eis
on.T
hefin
alfli
pan
gle
x can
beca
lcul
ated
by
x òlo )æ ,w
here
l isth
egy
rom
agne
ticra
tioan
d
æ isth
edu
ratio
nof
the
RF
puls
e.T
hus
the
final
posi
tion
ofth
em
agne
tizat
ionv
ecto
rin
spac
eafte
rthe
RF
puls
eis
turn
edof
fde
pend
son
the
mag
nitu
deof
the
o ) field
and
the
dura
tiono
fthe
RF
puls
e.A
ftere
xcita
tion
the
mag
netiz
atio
nvec
tor’s
prec
essi
onin
duce
sacu
rren
tcal
led
the
MR
sign
al.T
heco
ilpr
oduc
ingt
he
o ) fiel
dca
nal
sobe
used
tosa
mpl
ethi
ssi
gnal
.Whe
nsa
mpl
inga
nM
Rsi
gnal
only
the
com
pone
ntin
the
Uv ó
plan
eca
nbe
dete
cted
.Thu
sth
em
axim
umM
Rsi
gnal
isob
tain
edat
afli
pan
gle
of
yz .In
MR
Ity
pica
llytw
oex
cita
tions
are
inte
rest
ing:
the
yz (
% " )an
dth
e
,z (
)pu
lses
.The
nam
esre
fert
oth
efli
pan
gleo
fthe
mag
netiz
atio
nwhe
nth
eR
Fpu
lse
istu
rned
off.
Thu
sthe
180
z puls
eis
anin
vers
iono
fthe
mag
netiz
atio
nvec
tor.
A.1
.2R
elax
atio
n
Afte
rex
cita
tion
the
nucl
eire
turn
toth
eeq
uilib
rium
stat
e.T
his
proc
essis
calle
dre
laxa
tiona
ndbe
gins
atth
ete
rmin
atio
nof
the
RF
puls
e.D
urin
gre
laxa
tionb
oth
the
long
itudi
nal(
qt )and
the
tran
sver
se(
q äs )co
mpo
nent
sretu
rnto
thei
requ
i-lib
rium
valu
es.T
helo
ngitu
dina
land
tran
sver
sere
laxa
tions
are
mut
ually
inde
pen-
dent
.In
Fig
ure
A.4
the
time
varia
tion
ofth
ele
ngth
ofth
elo
ngitu
dina
land
tran
s-ve
rsec
ompo
nent
sof
the
mag
netiz
atio
nvec
tord
urin
gre
laxa
tion
issh
own.
The
mag
netiz
atio
nvec
torr
etur
nsto
itseq
uilib
rium
valu
ein
anex
pone
ntia
lfash
iona
sin
dica
tedi
nF
igur
eA
.4.
The
long
itudi
nalr
elax
atio
n(of
ten
refe
rred
toas
spin
-latti
cere
laxa
tion)
isre
-la
tedt
oho
wfa
stth
enu
clei
emit
ener
gyth
roug
hint
erac
tionb
etw
eent
hesp
insa
ndth
ela
ttice
.In
MR
Ith
ela
ttice
isth
em
agne
tican
dth
erm
alen
viro
nmen
toft
henu
-cl
ei.T
hetim
eva
riatio
noft
helo
ngitu
dina
lmag
netiz
atio
nvec
tord
urin
gre
laxa
tion
isgi
ven
by
q tòq å| ó×ØÙÚ 7~ ch
arac
teriz
edby
the
time
cons
tant
E ) ,see
the
top
pane
lofF
igur
eA.4
.In
biol
ogic
altis
sue
E ) rang
esfr
omab
out5
0m
illis
econ
dsto
afe
wse
cond
s.T
rans
vers
erel
axat
ion(
orsp
in-s
pinr
elax
atio
n)is
rela
tedt
oth
ein
tera
ctio
nof
indi
vidu
alsp
ins
caus
ingl
ocal
varia
tions
inth
est
atic
mag
netic
field
o å .The
seva
riatio
nsle
adto
fluct
uatio
nsin
the
prec
essi
onalf
requ
ency
ofth
ein
divi
dual
nu-
clei
.T
his
resu
ltsin
agr
adua
land
rand
omde
phas
ingo
fth
esp
insa
ndhe
ncei
na
deca
yin
the
mag
nitu
deof
the
tran
sver
seco
mpo
nent
ofth
em
agne
tizat
ionv
ec-
tor.
Fig
ure
A.5
show
sth
ede
phas
ingo
fsp
ins
afte
ran
90
z RF
puls
e.T
hetim
eva
riatio
nof
the
tran
sver
sem
agne
tizat
ionv
ecto
rdur
ing
rela
xatio
nis
give
nby
q äsòq å×ØÙÚ
,w
here
E î isthe
tran
sver
sere
laxa
tion
time
cons
tant
.
E î isal
way
ssm
alle
rthan
E ) andi
sty
pica
llyin
the
rang
eofa
few
mill
isec
onds
toa
few
seco
ndsin
biol
ogic
altis
sue.
A.1
Bas
icso
fMR
I15
7
M M0
Mz
T 1tim
e
M M0
Mxy
T 2tim
e
Fig
ureA
.4:T
hetim
eva
riatio
noft
hele
ngth
ofth
em
agne
tizat
ionv
ecto
rdur
ing
re-
laxa
tion.
Top:
long
itudi
nalr
elax
atio
n;bo
ttom
:tra
nsve
rser
elax
atio
n.R
elax
atio
nco
nsta
nts
E ) and
E î aresh
own.
yy
xx
Fig
ure
A.5
:D
epha
sing
ofsp
insa
ftera
90
z puls
e.Le
ft:im
med
iate
lyaf
tert
heR
Fpu
lse;
right
:so
met
ime
afte
rthe
RF
puls
e,w
hen
the
spin
sare
outo
fph
asea
ndth
em
agne
tizat
ionv
ecto
rhas
deca
yed.
158
Mag
netic
reso
nanc
eimag
ing
1
2de
cay
T*
time
2Tde
cay
FID
Fig
ure
A.6
:T
ime
varia
tion
ofth
eF
IDw
ithth
e
Eù î and
E î decay
show
n.
The
rela
xatio
ntim
eco
nsta
ntsd
epen
don
the
fact
that
diff
eren
ttis
sues
have
diff
eren
tcou
rses
ofre
laxa
tion.
The
sedi
ffer
ence
sare
the
basi
sfor
the
MR
Isc
an-
ning
. The
tran
sver
seco
mpo
nent
q äs ind
uces
acu
rren
t(a
sign
al)w
hich
isca
lled
the
Fre
eInd
uctio
nD
ecay
(FID
).T
heco
ilus
edto
crea
teth
e
o ) field
(with
itsax
isin
the
Uv ó
plan
e)ca
nal
sobe
used
tosa
mpl
ethe
sign
alin
the
Uv ó
plan
eafte
rthe
exci
tatio
npu
lse
iste
rmin
ated
.Inth
esa
mpl
ing,
ampl
itude
,freq
uenc
y,ph
asea
ndde
caya
rede
tect
ed.T
heam
plitu
deof
the
MR
sign
alde
cays
expo
nent
ially
with
atim
eco
nsta
nt
Eù î ,due
tobo
thsp
in-s
pin(
tran
sver
se)re
laxa
tiona
ndm
agne
ticfie
ldin
hom
ogen
eitie
s.
Eù î issm
alle
rtha
n
E î .Figur
eA
.6sh
ows
the
time
varia
tion
ofth
eF
IDw
ithbo
thth
e
Eù î andth
e
E î decay
ssho
wn.
A.1
.3P
ulse
sequ
ence
s
Itis
impo
rtan
tto
bear
inm
ind
that
the
MR
sign
alde
pend
son
both
E ) and
E î ,bu
tit
ispo
ssib
le,th
roug
hthe
useo
fpul
sese
quen
ces,to
obta
inan
MR
sign
alth
atde
pend
sprim
arily
onei
ther
E ) or
E î .A
puls
eseq
uenc
eisa
seto
fRF
puls
esus
edin
conj
unct
ionw
itha
sign
alsa
m-
plin
g.T
here
are
seve
ralp
ulse
sequ
ence
savai
labl
e.T
hein
divi
dual
sequ
ence
sare
char
acte
rized
bygi
ving
anM
Rsi
gnal
who
sein
tens
ityde
pend
sprim
arily
onei
ther
E ) or
E î .This
isca
lled
a
E ) or
E î weigh
tedM
Rsi
gnal
.For
aE ) w
eigh
teds
igna
lth
esa
tura
tionr
ecov
ery
orin
vers
ionr
ecov
ery
sequ
ence
sarea
vaila
ble,
and
fora
E î
A.1
Bas
icso
fMR
I15
9
wei
ghte
dsig
nalth
esp
inec
hose
quen
ceis
avai
labl
e.In
this
sect
iont
hesp
inec
hose
quen
cesw
illbe
pres
ente
dasa
nex
ampl
eofa
puls
eseq
uenc
e.
Spi
nec
ho
The
FID
sign
alfo
llow
sa
Eù î decay
caus
edby
both
spin
-spi
nrel
axat
ion(
E î decay
)an
dby
mag
netic
field
inho
mog
enei
ties,s
eeF
igur
eA.6
.Thu
sthe
FID
sign
alal
one
cann
otbe
used
tom
easu
retr
ue
E î relaxa
tion.
How
ever
,spi
n-sp
inre
laxa
tioni
sa
rand
oman
dirr
ever
sibl
epr
oces
s,w
hile
field
inho
mog
enei
tiesa
rest
atic
and
can
beac
coun
tedf
or,
and
this
isex
ploi
ted
inth
esp
inec
hopu
lse
sequ
ence
.Thi
sse
quen
ceai
ms
atel
imin
atin
gth
ein
fluen
ceof
the
field
inho
mog
enei
ties,l
eavi
ngon
lyth
etr
ue
E î relaxa
tion.
The
sequ
ence
star
tsw
itha
90z R
Fpu
lse
that
flips
the
mag
netiz
atio
nvec
tor
into
the
Uv ó
plan
e.Im
med
iate
lyaf
ter,
the
rela
xatio
npro
cess
sets
in:
the
tran
s-ve
rse
com
pone
nt
q äs de
cays
follo
win
gth
e
Eù î deca
y.T
heef
fect
ofth
est
atic
mag
netic
field
inho
mog
enei
tiesca
uses
the
indi
vidu
alsp
inst
opr
eces
sout
ofph
ase
(dep
hase
),see
Fig
ure
A.7
b–c.
At
æ ò%" (T
Eis
the
echo
time)
a18
0
z puls
eis
appl
ied.
Thi
spu
lse
inve
rts
all
the
spin
s,w
hich
isse
enas
am
irror
ing
inth
e
Uv ó
plan
e,se
eFig
ure
A.7
c–d.
Afte
rth
ispu
lse,
the
field
inho
mog
enei
tiess
tilli
n-
o
puls
e
t = T
E18
0
t = T
E
t = T
E/2−
t = T
E/2+
0 <
t <
TE
/2
TE
/2 <
t <
TE
f)
b) e)
c) d)
t = 0
a)
Fig
ure
A.7
:T
hesp
inec
hose
quen
ce:de
phas
ingo
fspi
nsdu
eto
the
field
inho
mo-
gene
ities
and
the
elim
inat
ion
ofth
eef
fect
ofth
eseb
ya
180
z puls
e.
160
Mag
netic
reso
nanc
eimag
ing
fluen
ceth
esp
ins,
butn
owit
caus
esth
emto
reph
aseb
ecau
seof
the
inve
rsio
n,se
eF
igur
eA
.7d–
f.A
ttim
e
æ ò th
esp
insa
reon
ceag
ain
inph
asea
ndth
eM
Rsi
gnal
can
besa
mpl
ed.T
heec
hosi
gnal
(MR
sign
alat
æ ò )
has
asm
alle
ram
plitu
deth
anth
esi
gnal
at
æ ò
,due
toth
esp
in-s
pinr
elax
atio
n.T
hesp
in-s
pin
rela
xatio
nis
nota
ffect
edby
the
inve
rsio
npu
lse
beca
useit
isra
ndom
and
irre-
vers
ible
.The
sign
alde
cayf
rom
æ ò
to
æ ò is
then
am
easu
reof
the
true
E î
rela
xatio
n.If
subs
eque
nt180
z puls
esar
eap
plie
d,at
æ ò% " û+% " û!% " û
and
the
sign
alsa
mpl
edat
E ò û" û+ û th
esa
mpl
edsi
gnal
will
con-
tinue
tode
caya
ccor
ding
toth
etr
ue
E î relaxa
tion
and
the
E î relaxa
tionc
anbe
dete
rmin
edw
ithhi
gher
accu
racy.
Fig
ure
A.8
show
sth
etim
eva
riatio
nof
the
sig-
nala
mpl
itude
durin
ga
spin
echo
sequ
ence
.A
part
icul
arvo
lum
ebe
ing
scan
nedis
ofte
nco
mpo
sedo
fse
vera
ltis
sues
and
the
sam
pled
MR
sign
alis
asu
mof
the
sign
alsf
rom
the
indi
vidu
altis
sues
.Eac
htis
sueh
asdi
ffer
entr
elax
atio
ntim
eco
nsta
ntsa
ndth
isis
expl
oite
din
the
puls
ese-
quen
ces,e
.g.i
nth
ech
oice
ofT
Ein
the
spin
echo
sequ
ence
.The
optim
alT
Eis
atim
eva
lue
such
that
the
diff
eren
cebe
twee
nthe
sign
alam
plitu
deso
fth
ein
di-
vidu
altis
sues
isla
rge.
Sig
nals
ampl
inga
tthe
optim
altim
ere
sults
inan
imag
e
1
1/2T
E3/
2TE
2Tde
cay
2de
cay
T*
FID
time
2TE
TE
Fig
ure
A.8
:T
ime
varia
tion
ofth
eM
Rsi
gnal
ampl
itude
durin
ga
spin
echo
se-
quen
ces.
180
z puls
esar
eap
plie
d,at
æ ò% " û+% " û!% " û a
ndth
esi
gnal
issa
mpl
edat
E ò û" û+ û .
A.2
Imag
ing
met
hods
161
M
M
time
tissu
e w
ith lo
w T
cont
rast
2
tissu
e w
ith h
igh
T 2
xy
optim
al T
E
Fig
ure
A.9
:T
rans
vers
erel
axat
ionc
urve
for
two
tissu
esw
ithdi
ffer
ent
E î values
.In
asp
inec
hose
quen
ceth
eop
timal
echo
time
isat
æ.Ù whe
reth
edi
ffer
ence
betw
eens
igna
lam
plitu
desis
larg
e.
whe
reth
ere
isa
larg
eco
ntra
stbe
twee
nthe
indi
vidu
altis
sues
whe
nth
esi
gnal
istr
ansf
orm
edin
toan
imag
e(se
eSec
tionA
.2).
Fig
ure
A.9
show
sth
etr
ansv
erse
re-
laxa
tionc
urve
sfor
two
diff
eren
ttiss
ues(
with
diff
eren
t
E î values
)with
the
optim
alT
Ein
dica
ted.
A.2
Imag
ing
met
hods
The
situ
atio
nde
scrib
edin
the
prev
ious
sect
ion
isth
atw
here
the
entir
evo
lum
eun
dero
bser
vatio
nis
exci
ted.
Sin
ceth
esa
mpl
edM
Rsi
gnal
isa
resu
ltoft
here
lax-
atio
npr
oces
sinth
een
tire
volu
me,
itco
ntai
nsno
spat
ialin
form
atio
n.In
orde
rto
cons
truc
tan
MR
imag
ethe
sign
alha
sto
besp
atia
llyen
code
d.T
hesp
atia
lenc
od-
ing
isac
hiev
edus
ing
the
fact
that
the
Larm
orfr
eque
ncy
depe
ndso
nth
em
agne
ticfie
ldst
reng
th.I
fth
em
agne
ticfie
ldva
riesw
ithpo
sitio
nin
aco
ntro
lled
way
,the
reso
nanc
efreq
uenc
yof
the
indi
vidu
alsp
insv
arie
sacc
ordi
ngly.
Thu
sspi
nsca
nbe
pick
edou
tand
exci
ted
byth
eirp
ositi
onin
spac
e.In
anM
Rsc
anne
rthre
eset
sofc
oils
are
pres
ent,t
hem
ain
coils
,whi
chcr
eate
the
stat
icm
agne
ticfie
ld
o å ,thehi
ghfr
eque
ncy
coils
,whi
chem
itth
eR
Fpu
lses
ands
ampl
ethe
MR
sign
al,a
ndfin
ally
the
grad
ient
coils
,whi
char
euse
dto
cont
rol
the
spat
ialv
aria
tion
ofth
em
agne
ticfie
ld.
The
grad
ient
coils
crea
tea
mag
netic
162
Mag
netic
reso
nanc
eimag
ing
field
whi
chis
para
llelto
the
mai
nfie
ld
o å .Ifw
eas
sum
e,as
prev
ious
ly,t
hatt
hem
ain
field
ispa
ralle
ltoth
e
r ó axis
we
have
o tòo å äUsv tr
whe
re
ä ,
s and
t are
mag
netic
field
grad
ient
sinth
eU ó
,v ó
and
r ó direc
tions
,re
spec
tivel
y.T
hegr
adie
ntfie
ldis
wea
kcom
pare
dto
the
stat
icm
agne
ticfie
ld,b
e-in
gty
pica
llyin
the
rang
e1–1
0mT
/m(A
nder
sen,
1997
).T
heth
reeg
radi
entfi
elds
are
ofte
nre
ferr
edto
asth
esl
ice
sele
ctio
ngra
dien
t,the
read
-out
orfr
eque
ncy
en-
codi
nggr
adie
nt,a
ndth
eph
asee
ncod
ingg
radi
ent,b
ecau
seof
thei
rfun
ctio
nsin
anM
RI
sequ
ence
.
A.2
.1S
lice
sele
ctio
ngra
dien
t
Ifa
grad
ient
istu
rned
onw
hile
the
volu
me
isbe
ing
expo
sedt
oan
RF
puls
econ
-ta
inin
ga
give
nra
nge
offr
eque
ncie
s,on
lyth
esl
ice
whi
chco
ntai
nssp
inw
itha
Larm
orfr
eque
ncy
that
mat
ches
the
freq
uenc
iesin
the
puls
ewill
beex
cite
d.T
heth
ickn
esso
fa
slic
eis
dete
rmin
edby
the
band
wid
thof
the
RF
puls
e(i.
e.th
era
ngeo
ffr
eque
ncie
sinth
epu
lse)
.The
slic
ese
lect
iong
radi
entfi
eld
ispe
r-pe
ndic
ular
toth
esl
ice
dire
ctio
nan
dca
nbe
chos
enin
any
give
ndi
rect
ion.
The
MR
imag
eis
cons
truc
tedb
ya
Four
iera
naly
siso
fth
esa
mpl
edM
Rsi
gnal
.E.g
.if
we
use
asl
ice
sele
ctio
ngra
dien
tpar
alle
lto
the
r ó axis
inco
mbi
natio
nwith
asp
inec
hose
quen
ceof
puls
es,th
eFo
urie
rtra
nsfo
rmof
the
MR
sign
alw
illbe
apr
ojec
tiono
fa
E î weigh
teds
igna
lfrom
the
entir
ese
lect
edsl
ice
onto
the
r ó axis
.B
utth
erei
sno
spat
ialre
solu
tiono
fthe
sele
cted
slic
e.T
heFo
urie
rcon
nect
ionb
etw
eenM
Rim
agea
ndR
Fpu
lses
(thr
ough
the
sam
-pl
edsi
gnal
)infl
uenc
esth
ech
oice
ofR
Fpu
lse.
Whe
na
shar
pre
ctan
gula
rslic
e
tran
sfor
mF
ourie
r
Fig
ure
A.1
0:A
sinc
wav
epu
lsea
ndits
Four
iert
rans
form
:the
step
func
tion.
The
sinc
wav
eis
ade
cayi
ngsi
new
ave:
ì æí ò ì æí%ì æí
A.2
Imag
ing
met
hods
163
profi
leis
need
edth
eR
Fpu
lse
shou
ldbe
ade
cayi
ngsi
ncw
ave,
sinc
eth
esi
ncw
ave
and
the
step
func
tion
are
each
othe
rsFo
urie
rtra
nsfo
rms,
seeF
igur
eA
.10.
The
mag
netic
field
influ
enci
ngth
esp
ins
insi
deth
ese
lect
edsl
ice
will
vary
slig
htly
beca
useo
fth
egr
adie
nt.T
hefr
eque
ncy
ofth
ein
divi
dual
spin
swill
then
vary
acco
rdin
gly,
and
thus
they
will
deph
ase.
Toac
hiev
eth
em
axim
alsi
gnal
the
spin
sare
reph
ased
byap
plyi
nga
mag
netic
field
grad
ient
with
the
oppo
site
pola
rity
ofth
esl
ice
sele
ctio
ngra
dien
t.Thi
sis
calle
da
reph
asin
ggra
dien
t.W
ithth
esl
ice
sele
ctio
ngr
adie
nt,a
part
icul
arsl
ice
tobe
exci
ted
has
been
pick
edou
tbut
noin
form
atio
non
how
that
slic
eis
spat
ially
reso
lved
isob
tain
ed.
A.2
.2P
hase
enco
ding
grad
ient
Afte
rthe
slic
ese
lect
iona
ndre
phas
ingg
radi
ents
have
been
appl
ieda
long
with
the
RF
puls
eall
spin
sin
the
sele
cted
slic
epr
eces
satt
hesa
mef
requ
ency
andi
nph
ase.
Ifa
seco
ndgr
adie
nt,th
eph
asee
ncod
ingg
radi
ent,i
sth
entu
rned
on,e
.g.i
nth
e
v ó
dire
ctio
n,th
ere
sona
ncefr
eque
ncy
ofth
esp
insc
hang
esal
ong
thev ó
axis
and
the
rate
ofpr
eces
sion
chan
gesa
ccor
ding
ly.W
hen
the
phas
eenc
odin
ggra
dien
tisth
entu
rned
off,
the
spin
sonc
eaga
inpr
eces
satt
hesa
mef
requ
ency
,but
now
thei
rph
ases
are
nolo
nger
the
sam
e.T
hein
divi
dual
spin
shav
ea
phas
etha
tdep
ends
onth
eirp
ositi
onal
ong
thev ó
axis
.
A.2
.3R
ead-
outg
radi
ent
Ifa
third
grad
ient
,the
read
-out
orfr
eque
ncy
grad
ient
e.g.
inth
e
U ó dire
ctio
n,is
appl
ied
whi
leth
eM
Rsi
gnal
isbe
ing
sam
pled
,the
freq
uenc
yof
the
emitt
edsi
gnal
varie
sacc
ordi
ngto
itspo
sitio
nal
ong
the
U ó axi
s.T
hesi
gnal
ata
spec
ific
freq
uenc
yal
ongt
he
U ó axi
sis
asu
mof
alls
pins
ina
row
alon
gthev
ó axis
(whe
reth
erei
sa
phas
evar
iatio
n).
The
sign
alis
sam
pled
at
ä
poin
tsin
the
read
-out
(or
U ó )di
rect
ion.
The
mea
sure
men
tisre
peat
ed
s times
ford
iffer
entv
alue
soft
heph
asee
ncod
ing(v ó
)gr
adie
nt,th
usge
nera
tinga
mat
rixof
äm s s
igna
lpoi
nts.
Per
form
inga
2DFo
urie
rtra
nsfo
rmof
the
mat
rixyi
elds
a2D
spec
trum
who
sein
tens
ityis
ava
lue
ofth
esi
gnal
dist
ribut
ion
inth
eim
agin
gsl
ice.
The
typi
calfi
nalM
Rim
age
isa
grey
-sca
lere
pres
enta
tiono
fth
esp
ectr
um.F
igur
eA
.11
show
san
MR
spec
trum
and
itsco
rres
pond
ingg
rey
scal
eim
age.
The
imag
emat
rixde
term
ines
the
num
bero
fpix
els
inth
eM
Rim
age,
and
the
colo
urof
each
pixe
lrep
rese
ntsth
ein
tens
ityof
the
MR
sign
alfr
omth
eco
rres
pond
-in
gvo
lum
eel
emen
t,orv
oxel
.The
volu
me
ofa
voxe
lis
dete
rmin
edby
the
mat
rixsi
zedi
vide
dby
the
field
ofvi
ew(a
ctua
lsiz
eof
imag
ein
mm
)mul
tiplie
dby
the
164
Mag
netic
reso
nanc
eimag
ing
Fig
ure
A.1
1:M
Rsp
ectr
uman
dco
rres
pond
ingg
rey
scal
eim
age.
slic
eth
ickn
ess.
The
MR
sign
alfr
oma
voxe
lis
the
aver
ageo
fthe
sign
alfr
omal
lth
enu
clei
inth
atvo
xel.
The
abov
epr
oced
uref
orob
tain
ing
anM
RI
isju
stan
exam
ple,
and
alte
rna-
tive
proc
edur
esar
eav
aila
ble.
An
MR
Ico
uld
e.g.
also
beob
tain
edas
avo
lum
eim
age/
3Dim
age.
Ina
volu
me
imag
eth
een
tire
volu
me
isex
cite
d(i.
e.no
slic
ese
lect
iond
urin
gex
cita
tion)
and
toob
tain
asl
ice
sele
ctio
nase
cond
phas
eenc
od-
ing
grad
ient
isap
plie
d.T
head
vant
ages
ofth
ism
etho
dis
that
nois
eis
redu
ced,
slic
eth
ickn
essc
anbe
redu
ced,
and
spat
ialre
solu
tioni
nal
lthr
eedi
rect
ions
canb
eid
entic
al,i.
e.is
otro
picv
oxel
s.
A.2
.4Im
agin
gse
quen
ces
The
proc
esso
fcr
eatin
gan
MR
Iis
done
onth
eba
siso
fan
imag
ing
sequ
ence
s.Im
agin
gse
quen
cesde
scrib
ethe
RF
puls
es,th
eap
plic
atio
nof
grad
ient
field
san
dth
etim
ing
ofM
Rsi
gnal
aqui
sitio
n.T
hus
anim
agin
gse
quen
ceis
anex
tend
edpu
lse
sequ
ence
s,whi
chw
asde
scrib
edin
Sec
tion
A.1
.3.
An
imag
ing
sequ
ence
typi
cally
cons
ists
ofth
ree
part
s:sp
inpr
epar
atio
n,si
gnal
prod
uctio
nan
dim
age
gene
ratio
n.W
here
imag
egen
erat
ions
impl
yis
Four
iert
rans
form
atio
noft
heM
Rsi
gnal
.
Spi
npr
epar
atio
n
Insp
inpr
epar
atio
nRF
prep
ulse
sor
mag
netic
field
grad
ient
sare
empl
oyed
toch
ange
the
char
acte
ristic
soft
heM
Rsi
gnal
.Spi
npr
epar
atio
ncan
cons
isto
fpre
-pu
lses
(e.g
.an
inve
rsio
npul
se)o
rit
can
besa
tura
tiono
fpa
rtof
the
volu
me,
sat-
A.2
Imag
ing
met
hods
165
urat
ion
bein
ga
long
term
exci
tatio
n.T
here
sult
isth
atth
em
agne
tizat
ionv
ecto
rap
proa
chesz
eroa
nddu
ring
the
follo
win
gex
cita
tion
puls
eonl
yth
evo
lum
eout
side
the
satu
rate
dpar
tis
exci
ted
(And
erse
n,19
97).
Thr
ough
the
use
ofan
inve
rsio
npr
epul
sein
the
spin
echo
sequ
ence
itis
poss
ible
toge
nera
teim
ages
that
are
E )
wei
ghte
dand
not
E î weigh
teda
sast
anda
rdsp
inec
hose
quen
ce.
Sig
nalp
rodu
ctio
n
Sig
nalp
rodu
ctio
ncon
sist
sofg
ener
atin
gthe
MR
sign
alan
den
codi
ngit
with
spa-
tial
info
rmat
ion.
Thi
sca
nbe
done
inth
ree
way
s:F
ID,
spin
echo
and
grad
ient
echo
.A
FID
gene
rate
dsig
nali
sa
sam
plin
gof
the
FID
sign
al.
Thi
sgi
ves
anim
agew
here
imag
eint
ensi
tyis
prop
ortio
nalto
the
nete
quili
briu
mm
agne
tizat
ion
vect
or.
The
spin
echo
gene
rate
dsig
nalis
aco
mbi
natio
nof
the
spin
echo
puls
ese
quen
cede
scrib
edin
Sec
tionA
.1.3
and
the
the
MR
Ipr
oced
ured
escr
ibed
inS
ec-
tions
A.2
.1-A
.2.3
.T
hepu
lse
sequ
ence
isre
peat
ed
s tim
esto
obta
inth
efu
llsp
atia
linfo
rmat
ion.
The
time
betw
eens
ucce
ssive
exci
tatio
npu
lses
isca
lled
the
repe
titio
ntim
eT
R.T
hust
hefu
llsc
antim
eis
sm TR
.The
grad
ient
echo
gene
r-at
edsi
gnal
isba
sedo
na
com
bina
tiono
fthe
grad
ient
echo
sequ
ence
and
the
MR
Ipr
oced
ured
escr
ibed
abov
e(S
ectio
nsA
.2.1
-A.2
.3).
The
grad
ient
echo
sequ
ence
isde
scrib
edin
the
follo
win
gse
ctio
n.
Gra
dien
tech
ose
quen
ce
Inth
egr
adie
ntec
hose
quen
cea
mag
netic
field
grad
ient
isus
edto
crea
tean
echo
,as
oppo
sedt
oth
eus
eofa
180
z puls
ein
the
spin
echo
sequ
ence
.In
the
grad
ient
echo
sequ
ence
agr
adie
ntis
appl
ied
inth
ere
ad-o
utdi
rect
ion,
e.g.
the
U ó dire
ctio
n,fo
ra
limite
dtim
edi
rect
lyaf
tert
heex
cita
tion
puls
e.T
his
mak
esth
esp
inpr
eces
satd
iffer
entr
ates
alon
gth
ere
ad-o
utdi
rect
ion,
resu
lting
ina
deph
asin
gofs
pins
.The
grad
ient
isth
enre
vers
ed,w
hich
caus
essp
into
reph
ase
and
crea
tean
echo
.F
igur
eA
.12
sket
ches
the
grad
ient
echo
sequ
ence
.An
exci
tatio
npu
lse
with
flip
angl
e
Ý isapp
lied
atth
esa
met
imes
asth
esl
ice
sele
ctio
ngra
dien
tin
the
r ó direc
tion.
Imm
edia
tely
follo
win
gth
esl
ice
sele
ctio
ngra
dien
tisth
ere
phas
ing
grad
ient
inth
esl
ice
sele
ctio
ndire
ctio
n,se
eSec
tionA
.2.1
.The
nth
eph
asee
ncod
-in
ggr
adie
ntis
turn
edon
(inth
isca
sein
thev ó
dire
ctio
n)at
the
sam
etim
eas
the
echo
grad
ient
inth
ere
ad-o
utdi
rect
ion
(U ó dire
ctio
n).T
heec
hogr
adie
ntco
nsis
tsof
two
part
swith
reve
rsed
pola
rity
(asd
escr
ibed
abov
e).A
fter
the
echo
grad
ient
the
echo
iscr
eate
dand
the
sign
alis
sam
pled
inth
ere
ad-o
utdi
rect
ion
atth
esa
me
time
asth
ere
ad-o
utgr
adie
ntis
appl
ied.
Thi
sse
quen
ceis
repe
ated
s times
with
diff
eren
tval
ueso
fthe
phas
eenc
odin
ggra
dien
t.Thi
sis
illus
trat
edin
the
figur
eby
166
Mag
netic
reso
nanc
eimag
ing
Gy
Gx
Gz
αpu
lse
αpu
lse
echo
grad
ient
echo
grad
ient
TR
Sig
nal
RF
slic
ese
lect
ion
reph
asin
g
phas
een
codi
ng
grad
ient
read
−ou
t
TE
Fig
ure
A.1
2:T
hegr
adie
ntec
hose
quen
ce.
seve
rall
ines
draw
nat
the
phas
eenc
odin
ggra
dien
t.T
heec
hotim
e,T
E,a
ndth
ere
petit
iont
ime,
TR
,are
indi
cate
d.T
head
vant
ages
ofth
egr
adie
ntec
hose
quen
ce,co
mpa
redt
oth
esp
inec
hose
quen
ce,ar
eth
epo
ssib
ility
ofsm
alle
rpul
sefli
pan
gles
,sho
rter
echo
times
and
shor
terr
epet
ition
times
,yie
ldin
ga
muc
hsh
orte
rtota
lim
agin
gtim
e.In
the
spin
echo
sequ
ence
the
repe
titio
ntim
eha
sto
belo
ngto
assu
reth
atth
elo
ngitu
dina
lrel
axat
ioni
sco
mpl
eteb
efor
ethe
next
exci
tatio
npu
lsei
sap
plie
d.If
the
long
itudi
nalr
elax
atio
nis
notc
ompl
etet
heef
fect
ive
mag
netiz
atio
nvec
tor
flipp
edin
toth
e
Uv ó
plan
eis
smal
ler.
Afte
rse
vera
lexc
itatio
nsth
em
agne
tiza-
tion
vect
orde
cays
tow
ard
zero
and
the
volu
me
beco
mes
satu
rate
d(re
fert
oS
ec-
A.2
Imag
ing
met
hods
167
tion
A.2
.4ab
ove)
.T
here
petit
ion
time
coul
dbe
redu
cedb
yus
ing
anex
cita
tion
puls
ewith
asm
alle
rflip
angl
etha
n90
z (e.g
.15-
45
z ),m
akin
gth
etim
eto
rest
ore
long
itudi
nalr
elax
atio
nsm
alle
r.B
utev
enin
the
case
ofa
puls
ew
itha
low
flip
angl
ea
larg
epa
rtof
the
netm
agne
tizat
ionis
inth
elo
ngitu
dina
ldire
ctio
n.W
itha
180
z inve
rsio
npul
seth
isw
ould
also
affe
ctth
elo
ngitu
dina
lmag
netiz
atio
nand
noto
nly
the
tran
sver
sem
agne
tizat
ion.
Inth
egr
adie
ntec
hose
quen
ceit
ispo
ssib
leto
use
puls
esw
ithsh
orte
rflip
angl
esbe
caus
ethe
echo
iscr
eate
dby
agr
adie
ntra
ther
than
anin
vers
ion
puls
e.Lo
wer
flip
angl
esle
adto
shor
terr
ecov
ery
time
ofth
elo
ngitu
dina
lmag
netiz
atio
n(p
rior
toth
ene
xtex
cita
tion
puls
e).
Thi
ssh
orte
nsth
eT
Ran
dth
ereb
ythe
tota
lim
agin
gtim
e.T
his
isth
epr
imar
yar
gum
entfo
rus
ing
agr
adie
ntec
hose
quen
cera
ther
than
asp
inec
hose
quen
ce.
A.2
.5M
otio
nar
tifac
ts
Mot
ion
resu
ltsin
ghos
tingo
rbl
urrin
gof
the
MR
imag
e.In
the
pres
entc
ontex
tm
otio
nus
ually
mea
nspa
tient
mov
emen
ts,re
spira
tory
orca
rdia
cmot
ion
orm
otio
ndu
eto
swal
low
ing.
Dep
endi
ngon
the
type
ofM
Rim
ageb
eing
reco
rded
diff
eren
tm
otio
nsar
eun
wan
ted.
E.g
.if
the
brai
nis
bein
gm
appe
dcar
diac
orre
spira
tory
mot
ion
isno
tapr
oble
m.
Sin
ceth
isth
esis
isco
ncer
nedw
ithca
rdia
cMR
Iw
ede
scrib
ethe
proc
edur
esfo
rel
imin
atin
gth
eef
fect
ofpa
tient
mov
emen
ts,a
sw
ell
asca
rdia
cand
resp
irato
rym
otio
n.
Sig
nala
vera
ging
:B
ecau
seof
the
rand
omna
ture
ofpa
tient
mov
emen
tthe
effe
ctca
nbe
redu
cedb
yav
erag
ingt
heM
Rsi
gnal
over
seve
rals
ampl
ings
.The
draw
back
isth
atth
eto
tali
mag
ing
time
isin
crea
sed.
Car
diac
trig
gerin
g:T
heef
fect
ofca
rdia
cmot
ion
can
beel
imin
ated
byco
uplin
gth
eaq
uisi
tiont
oa
trig
gers
igna
l,in
the
card
iacc
aset
heE
CG
(ele
ctro
card
io-
gram
)sig
nal.
The
hear
tcyc
leis
assu
med
perio
dica
ndby
trig
gerin
gon
e.g.
the
Rpe
akof
the
EC
G,a
quis
ition
isdo
neat
the
sam
etim
ein
the
hear
tcyc
leev
ery
time
and
the
effe
ctof
card
iacm
otio
nis
min
imiz
ed.T
hedr
awba
ckis
that
tota
lim
agin
gtim
eis
subs
tant
ially
incr
ease
d.
Res
pira
tory
trig
gerin
g:T
his
issi
mila
rto
card
iact
rigge
ringe
xcep
ttha
ther
ethe
trig
gerin
gis
rela
tedt
oa
sign
alin
dica
ting
expi
ratio
n.
Res
pira
tory
gatin
g:A
quis
ition
isdo
new
hile
the
resp
irato
rym
otio
nis
min
imal
.T
his
isan
effe
ctiv
epr
oced
ureb
utin
crea
sesth
eto
tali
mag
ing
time
subs
tan-
tially
.
168
Mag
netic
reso
nanc
eimag
ing
Bre
ath
hold
:In
ash
orta
quis
ition
sequ
ence
the
aqui
sitio
ncan
bedo
new
hile
the
patie
ntho
lds
his/
herb
reat
h,th
ereb
yelim
inat
ing
the
effe
ctof
resp
irato
rym
otio
n.
A.3
Vel
ocity
mea
sure
men
ts
All
the
MR
imag
esde
scrib
edpr
evio
usly
are
grey
scal
eim
ages
ofth
ein
tens
ityof
the
MR
sign
al.
The
yar
eof
ten
refe
rred
toas
mag
nitu
deor
mod
ulus
imag
es.I
tis
also
poss
ible
toob
tain
info
rmat
ion
onth
eve
loci
tyof
the
tissu
e(e.
g.bl
ood)
inea
chvo
xeli
nan
MR
I.T
heph
aseo
fspi
nsin
bloo
dar
ealte
redb
yth
ebl
ood
flow
,aph
enom
enon
calle
da
phas
eshi
ft.T
his
can
beex
ploi
ted
ina
phas
evel
ocity
map
whe
reth
eph
aseo
fth
esp
insa
reco
rrel
ated
toth
eve
loci
tyof
the
mov
ing
spin
s.W
hen
am
ovin
gsp
inis
subj
ecte
dto
abi
pola
rgra
dien
tthe
spin
hasa
phas
eat
the
term
inat
iono
fth
egr
adie
ntw
hich
ispr
opor
tiona
ltoth
eve
loci
tyof
the
spin
.T
his
isop
pose
dto
ast
atic
spin
,whi
chha
sno
netp
hase
atth
ete
rmin
atio
nof
abi
pola
rgra
dien
t.S
oby
appl
ying
abi
pola
rgra
dien
tinan
imag
ese
quen
ceit
ispo
ssib
leto
reco
rdth
eve
loci
tyof
the
bloo
din
each
voxe
l.T
hebi
pola
rgra
dien
tca
nbe
appl
ied
inan
ych
osen
dire
ctio
n,or
inal
lthr
eesp
atia
ldire
ctio
nsto
obta
inth
reed
imen
sion
alve
loci
tyin
form
atio
n.A
phas
eshi
ftca
nha
veot
herc
ause
stha
nflo
wve
loci
ty,
e.g.
mag
netic
field
inho
mog
enei
ties.T
hus
itis
nece
ssar
ytore
cord
aflo
w-c
ompe
nsat
edba
ckgr
ound
imag
e.T
hrou
ghth
eus
eof
flow
-com
pens
ated
grad
ient
sit
ispo
ssib
leto
reco
rda
phas
eimag
e,w
here
phas
eeffe
ctss
tem
only
from
unw
ante
ddev
iatio
ns.T
his
flow
-co
mpe
nsat
edim
agec
anth
enbe
subt
ract
edfr
omth
eph
asev
eloc
itym
ap,le
avin
ga
phas
eim
agew
hich
isth
ere
sulto
nly
offlo
wve
loci
typh
ases
hifts
.P
hase
velo
city
map
ping
requ
ireso
btai
ning
seve
ralim
ages
pers
lice:
onei
mag
epe
rvel
ocity
com
pone
ntan
don
eve
loci
ty-c
ompe
nsat
edim
age.
Thu
sit
incr
ease
sth
eto
tals
can
time.
Fig
ure
A.1
3sh
ows
thre
eim
ages
from
ave
loci
tyen
code
dca
rdia
cMR
I:th
em
odul
usim
age,
the
velo
city
imag
e(w
ithve
loci
tyve
ctor
sfro
mth
ein
-pla
neco
mpo
nent
s)and
the
velo
city
imag
esu
perim
pose
don
the
mod
ulus
imag
e.T
heve
loci
tyim
agei
sof
limite
dva
lue
onits
own,
buti
tis
very
usef
ulw
hen
itis
supe
rimpo
sedo
nth
em
odul
usim
age.
Thi
sco
mbi
natio
nis
ave
ryst
rong
imag
ing
tech
niqu
ewhe
ne.
g.th
eca
rdia
cfun
ctio
nis
eval
uate
d.
A.3
Vel
ocity
mea
sure
men
ts16
9
Fig
ure
A.1
3:Im
ages
from
aca
rdia
cM
RI
with
velo
city
enco
ding
.To
p:th
em
odul
usim
age,
mid
dle:
the
velo
city
imag
ew
ithin
-pla
neve
loci
tyco
mpo
nent
ssh
own
asve
ctor
s,bo
ttom
:vel
ocity
imag
esup
erim
pose
don
mod
ulus
imag
e.T
hean
atom
yoft
hehe
arts
eeno
nth
eM
RI
isex
plai
nedi
nF
igur
e6.
5.
Bib
liogr
aph
y
Ade
ler,
P.T.
and
Jaco
bsen
,J.M
.(20
01).
Blo
odflo
win
the
hear
t,in
J.T.
Otte
sen,
M.S
.Olu
fsen
and
J.La
rsen
(eds
),A
pp
lied
Ma
the
ma
tica
lMo
de
lsin
Hu
ma
nP
hys
iolog
y,to
bepu
blis
hedb
yS
IAM
,cha
pter
3.
And
erse
n,O
.T.(
1997
).B
illed
dann
else
med
NM
R(in
Dan
ish)
,Me
dic
insk
Ele
k-tr
on
ik(in
Da
nis
h),
Inst
itutf
orIn
form
atio
nste
knol
ogi,T
echn
ical
Uni
vers
ityof
Den
mar
k,ch
apte
r11.
Bey
er,R
.P.(
1992
).A
com
puta
tiona
lmod
elof
the
coch
leau
sing
the
imm
erse
dbo
unda
rym
etho
d,J.
Co
mp
ut.P
hys
.98(
1):1
45–1
62.
Bey
er,R
.P.a
ndLe
Veq
ue,R
.J.(
1992
).A
naly
siso
faon
e-di
men
sion
almod
elfo
rth
eim
mer
sedb
ound
arym
etho
d,S
IAM
J.N
um
er.A
na
l.29
(2):
332–
364.
Bor
gers
,C.a
ndP
eski
n,C
.S.(
1987
).A
lagr
angi
anfr
actio
nals
tepm
etho
dfor
the
inco
mpr
essi
bleN
avie
r-S
toke
seq
uatio
nson
ape
riodi
cdo
mai
n.,J
.Co
mp
ut.
Ph
ys.7
0(2)
:397
–438
.
Cap
pello
,A.,
Gnu
di,G
.an
dLa
mbe
rti,
C.
(199
5).
Iden
tifica
tion
ofth
eth
ree-
elem
entw
indk
esse
lmod
elin
corp
orat
inga
pres
sure
-dep
ende
ntcom
plia
nce,
An
na
lso
fBio
me
dic
alE
ng
ine
erin
g23(
2):1
64–1
77.
Cho
rin,A
.J.
(196
8).N
umer
ical
solu
tion
ofth
eN
avie
r-S
toke
seq
uatio
ns,M
ath
.C
om
p.2
2:74
5–76
2.
Cho
rin,
A.
J.(1
969)
.O
nth
eco
nver
genc
eof
disc
rete
appr
oxim
atio
nsto
the
Nav
ier-
Sto
kese
quat
ions
,Ma
th.C
om
p.2
3:34
1–35
3.
Dav
is,P
.(19
95).
The
arto
fthe
hear
tval
ve,S
IAM
New
s28
(5):
1an
d13
.
Des
popo
ulos
,A.
and
Silb
erna
gl,S
.(19
91).
Co
lor
Atla
so
fPh
ysio
logy,
4th
edn,
Thi
eme,
Stu
ttgar
t.
172
Bib
liogr
aph
y
Din
g,T.
and
Sch
oeph
oers
ter
,R.T
.(19
97).
Eva
luat
ion
ofgl
obal
left
vent
ricul
arfu
nctio
nba
sedo
nsi
mul
ated
flow
dyna
mic
scom
pute
dfro
mre
gion
alw
all
mot
ion,
AS
ME
,Bio
en
gin
ee
rin
gDiv
isio
n(P
ub
lica
tion
)BE
D35
:193
–194
.
Fun
g,Y
.C.(
1993
).B
iom
ech
an
ics.
Me
cha
nic
alP
rop
ert
ieso
fLiv
ing
Tiss
ue
s,2n
ded
n,S
prin
ger-V
erla
g,N
ewY
ork.
Gan
ong,
W.
F.(1
993)
.R
evie
wo
fM
ed
ica
lPh
ysio
logy,
16th
edn,
Pre
ntic
e-H
all
Inte
rnat
iona
l,Lon
don.
Geo
rgia
dis,
J.G
.,W
ang,
M.
and
Pasi
poul
arid
es,A
.(1
992)
.Com
puta
tiona
lfluid
dyna
mic
sof
left
vent
ricul
arej
ectio
n,A
nn
als
of
Bio
me
dic
alE
ng
ine
erin
g20
(1):
81–9
7.
Gon
zale
z,E.a
ndS
choe
phoe
rste
r,R
.T.(
1996
).A
sim
ulat
iono
fthr
ee-d
imen
sion
alsy
stol
icflo
wdy
nam
icsi
na
sphe
rical
vent
ricle
:E
ffect
sof
abno
rmal
wal
lm
otio
n,A
nn
als
ofB
iom
ed
ica
lEn
gin
ee
rin
g24:
48–5
7.
Gre
enbe
rg,S
.,M
cQue
en,D
.M.a
ndP
eski
n,C
.S.(
1987
).T
hree
-dim
ensi
onalfl
uid
dyna
mic
sin
atw
o-di
men
sion
alam
ount
ofce
ntra
lmem
ory,
inA
.J.
Cho
rinan
dA
.J.
Maj
da(e
ds),
Wa
veM
otio
n:
Th
eo
ry,M
od
elli
ng
an
dC
om
pu
tatio
n.
Pro
cee
din
gso
faC
on
fere
nce
inH
on
oro
fth
e6
0th
Birth
da
yo
fPe
terD
.La
x.,
Spr
inge
r-Ver
lag,
New
Yor
k,pp
.85–
146.
Gro
en,J
.P.a
ndD
ijk,
P.v.
(198
7).D
esig
noffl
owad
just
ableg
radi
entw
avef
orm
s,in
J.T.
E.d
.Kle
ef(e
d.),
So
cie
tyo
fM
agn
etic
Re
son
an
cein
Me
dic
ine,
6th
an
nu
alm
ee
ting,
p.86
8.
Guy
ton,
A.
C.(
1991
).Te
xtb
oo
kofM
ed
ica
lPh
ysio
logy,
8th
edn,
W.B
.Sau
nder
sC
ompa
ny,P
hila
delp
hia,
US
A.
Hou
lind,
K.
(200
1a).
Priv
ate
com
mun
icat
ion.
Hou
lind,
K.
(200
1b).
Ea
rly
dia
sto
licle
ftve
ntr
icu
larb
loo
dflo
wp
atte
rns.
Ass
e-
me
nts
by
mag
ne
ticre
son
an
ceve
loci
tym
ap
pin
g.,P
hDth
esis
,Dep
t.of
Car
-di
otho
raci
cand
Vas
cula
rSur
gery
T,A
arhu
sUni
vers
ityH
ospi
tal.
Hou
lind,
K.,
Ped
erse
n,E.M
.,O
yre,
S.,
Kim
,W
.Y
.,W
alke
r,P.
G.,
Ege
blad
,H.
and
Yog
anat
han,A
.P.
(199
4).L
eftv
entr
icul
arbl
ood
flow
patte
rnsa
sses
sed
bym
agne
ticre
sona
ncev
eloc
itym
appi
ngin
patie
ntsw
ithis
chem
iche
artd
is-
ease
,Am
.J.o
fNo
nin
vas.
Ca
rdio
l.8:
317–
325.
Bib
liogr
aph
y17
3
Hou
lind,
K.,
Sch
roed
er,A
.P.
,E
gebl
ad,H
.an
dP
eder
sen,E
.M.
(199
9).
Age
-de
pend
entch
ange
sin
spat
iala
ndte
mpo
ralb
lood
velo
city
dist
ribut
ion
ofea
rlyle
ftve
ntric
ular
fillin
g,M
agn
etic
Re
son
an
ceIm
agin
g17
(6):
859–
868.
Hou
lind,
K.,
Sch
roed
er,A
.P.
,Stø
dkild
e-Jø
rgens
en,H
.,Pa
ulse
n,P.
K.,
Ege
blad
,H
.an
dP
eder
sen,E
.M
.(2
001)
.In
trav
entr
icul
ardi
sper
sion
and
tem
pora
lde
layo
fea
rlyle
ftve
ntric
ular
fillin
gaf
tera
cute
myo
card
ialin
farc
tion.
Ass
-es
men
tby
mag
netic
reso
nanc
evel
ocity
map
ping
,inH
oulin
d(2
001b
).
Jaco
bsen
,J.M
.(1
999)
.P
eski
n’s2D
mod
elof
ase
ctio
nof
the
left
side
ofth
ehe
art:
Des
crip
tion,
Inve
stig
atio
nand
Mod
ifica
tion.
Unp
ublis
hedr
epor
t.
Jaco
bsen
,J.M
.,A
dele
r,P.
T.,
Kim
,W
.Y
.,H
oulin
d,K
.,P
eder
sen,E
.M
.an
dLa
rsen
,J.(
2001
).E
valu
atio
nofa
2Dm
odel
ofth
ele
ftsi
deof
ahu
man
hear
tag
ains
tmag
netic
reso
nanc
evel
ocity
map
ping
,Ca
rdio
vasc
ula
rEn
gin
ee
rin
g1(
2):5
9–76
.
Jako
bsen
,B.
and
Nis
s,K
.(2
000)
.B
ehan
dlin
gaf
impu
lsve
dki
lder
ogdr
æn
iC
.S.P
eski
ns2D
-hje
rtem
odel(
inD
anis
h),T
ech
nic
alR
ep
ort
,Te
kstN
r3
88,
Ros
kild
eUni
vers
ity.
Jone
s,T.
N.a
ndM
etax
as,D
.N.(
1998
).Pa
tient
-spe
cific
anal
ysis
ofle
ftve
ntric
-ul
arbl
ood
flow
,in
W.
M.
Wel
ls,A
.C
olch
este
rand
S.D
elp
(eds
),M
ed
ica
lIm
age
Co
mp
utin
gan
dC
om
pu
er-A
ssis
tedI
nte
rve
ntio
n–M
ICC
AI’
98
.Pro
-ce
ed
ing
s.,V
ol.1
498o
fLe
ctu
reN
ote
sin
Co
mp
ute
rSci
en
ce,S
prin
gerV
erla
g,pp
.156
–166
.
Kim
,W
.Y.,
Wal
ker,
P.G
.,P
eder
sen,E
.M.,
Pou
lsen
,J.K
.,O
yre,
S.,
Hou
lind,
K.
andY
ogan
atha
n,A.P
.(19
95).
Left
vent
ricul
arbl
ood
flow
patte
rnsin
norm
alsu
bjec
ts:A
quan
titat
ivean
alys
isby
thre
e-di
men
sion
almag
netic
reso
nanc
eve
loci
tym
appi
ng,J
.Am
.Co
ll.C
ard
iol.
26(1
):22
4–23
8.
Kim
,Y.H
.(20
01).
Priv
ate
com
mun
icat
ion.
Lai,
M.-
C.a
ndP
eski
n,C
.S.(
2000
).A
nim
mer
sedb
ound
arym
etho
dwith
form
alse
cond
orde
racc
urac
yan
dre
duce
dnum
eric
alvi
scos
ity,
J.C
om
pu
t.P
hys
.16
0:70
5–71
9.
Lem
mon
,J.D
.an
dY
ogan
atha
n,A.
P.(1
998)
.T
hree
-dim
ensi
onalc
ardi
acm
od-
els
with
fluid
-str
uctu
rein
tera
ctio
n:A
pplic
atio
nof
the
imm
erse
dbou
ndar
ym
etho
datp
hysi
olog
icR
eyno
lds
num
bers
,AS
ME
,Bio
en
gin
ee
rin
gDiv
isio
n(P
ub
lica
tion
)BE
D39
:73–
74.
174
Bib
liogr
aph
y
Lem
mon
,J.D
.and
Yog
anat
han,A
.P.
(200
0a).
Com
puta
tiona
lmod
elin
gof
left
hear
tdia
stol
icfu
nctio
n:E
xam
inat
iono
fven
tric
ular
dysf
unct
ion,
Jou
rna
lof
Bio
me
cha
nic
alE
ng
ine
erin
g122
(4):
297–
303.
Lem
mon
,J.D
.and
Yog
anat
han,A
.P.(
2000
b).T
hree
-dim
ensi
onalc
ompu
tatio
nal
mod
elof
left
hear
tdia
stol
icfu
nctio
nw
ithflu
id-s
truc
ture
inte
ract
ion,
Jou
rna
lo
fBio
me
cha
nic
alE
ng
ine
erin
g122
(2):
109–
117.
May
o,A
.A
.an
dP
eski
n,C
.S.(
1993
).A
nim
plic
itnu
mer
ical
met
hodf
orflu
iddy
nam
icsp
robl
emsw
ithim
mer
sede
last
icbo
unda
ries,i
nA
.Y
.C
heer
and
C.P
.van
Dam
(eds
),Flu
idD
yna
mic
sin
Bio
log
y,V
ol.1
41,A
mer
ican
Mat
h-em
atic
alS
ocie
ty,P
rovi
denc
e,R
hode
Isla
nd,p
p.26
1–27
7.
McC
rack
en,M
.F.
and
Pes
kin,
C.
S.(
1980
).A
vort
exm
etho
dfo
rbl
ood
flow
thro
ughh
eart
valv
es,J
.Co
mp
ut.P
hys
.35:
183–
205.
McQ
ueen
,D.
M.
and
Pes
kin,
C.S
.(19
83).
Com
pute
r-ass
iste
ddes
igno
fpi
vot-
ing
disc
pros
thet
icm
itral
valv
es,J
ou
rna
lof
Th
ora
cic
an
dC
ard
iova
scu
lar
Su
rge
ry86
:126
–135
.
McQ
ueen
,D.M
.and
Pes
kin,
C.S
.(19
85).
Com
pute
r-ass
iste
ddes
igno
gbu
tterfl
ybi
leafl
etva
lves
for
the
mitr
alpo
sitio
n,S
can
d.J
.T
ho
r.C
ard
iova
sc.S
urg
.19
(2):
139–
148.
McQ
ueen
,D.
M.
and
Pes
kin,
C.S
.(19
89).
Ath
ree-
dim
ensi
onalc
ompu
tatio
nal
met
hodf
orbl
ood
flow
inth
ehe
art.I
I.C
ontr
actil
eFib
ers,
J.C
om
pu
t.Ph
ys.
89:2
89–2
97.
McQ
ueen
,D.
M.
and
Pes
kin,
C.S
.(19
97).
Sha
red-
mem
oryp
aral
lelv
ecto
rim
-pl
emen
tatio
nof
the
imm
erse
dbo
unda
rym
etho
dfo
rth
eco
mpu
tatio
nof
bloo
dflo
win
the
beat
ing
mam
mal
ianh
eart
,Jo
urn
alo
fS
up
erco
mp
utin
g11
(3):
213–
236.
McQ
ueen
,D.
M.
and
Pes
kin,
C.S
.(20
00).
Hea
rtsi
mul
atio
nby
anim
mer
sed
boun
dary
met
hodw
ithfo
rmal
seco
nd-o
rdera
ccur
acy
andr
educ
ednu
mer
ical
visc
osity
,P
roce
ed
ing
sof
ICTA
M2
00
0,K
luw
erA
cade
mic
Pub
lishe
rs.T
oap
pear.
McQ
ueen
,D.M
.,P
eski
n,C
.S.a
ndY
ellin
,E.E
.(19
82).
Flu
iddy
nam
icso
fth
em
itral
valv
e:P
hysi
olog
ical
aspe
ctso
fam
athe
mat
icalm
odel
,Am
.J.P
hys
iol.
242:
H10
95–H
1110
.
Bib
liogr
aph
y17
5
Mei
sner
,J.S
.,M
cQue
en,D
.M.,
Ishi
da,Y
.,V
ette
r,H
.O.,
Bor
tolo
tti,U
.,S
trom
,J.
A.,
Fra
ter,
R.W
.M
.,P
eski
n,C
.S.a
ndY
ellin
,E
.L.
(198
5).
Effe
cts
oftim
ing
ofat
rials
ysto
leon
LVfil
ling
and
mitr
alva
lve
clos
ure:
com
pute
rand
dog
stud
ies,
Am
.J.P
hys
iol.2
49:H
604–
H61
9.
Net
ter,F
.H.(
1991
).H
ea
rt,V
ol.5
ofT
heC
iba
Co
llect
iono
fMe
dic
alIl
lust
ratio
ns,
CIB
A.
Nis
hiha
ra,K
.,M
ikam
i,T.
,Ta
kats
uji,
H.,
Ono
zuka
,H.,
Sai
to,N
.,Y
amad
a,S
.,U
rasa
wa,
K.
and
Kita
bata
ke,A
.(2
000)
.Use
fuln
esso
fea
rlydi
asto
licflo
wpr
opag
atio
nvel
ocity
mea
sure
dby
colo
rM-m
ode
Dop
pler
tech
niqu
efor
the
asse
smen
tofle
ftve
ntric
ular
dias
tolic
func
tion
inpa
tient
swith
hype
rtro
phic
card
iom
yopa
thy,J
.Am
.So
c.E
cho
card
iog
r.13
:801
–808
.
Noo
rder
graa
f,A
.(1
978)
.C
ircu
lato
ryS
yste
mD
yna
mic
s,A
cade
mic
Pre
ss,S
anD
iego
.
Oht
e,N
.,N
arita
,H.,
Has
him
oto,
T.,
Hay
ano,
J.,A
kita
,S
.and
Kur
okaw
a,K
.(1
999)
.Dif
fere
ntia
tiono
fab
norm
alre
laxa
tionp
atte
rnw
ithag
ing
from
ab-
norm
alre
laxa
tion
patte
rnw
ithco
rona
ryar
tery
dise
asei
ntr
ansm
itral
flow
with
the
use
oftis
sue
Dop
pler
imag
ing
ofth
em
itral
annu
lus,
J.A
m.S
oc.
Ech
oca
rdio
gr.
12:6
29–6
35.
Olu
fsen
,M.,
Ade
ler,
P.T.
,Dan
iels
en,M
.an
dLa
rsen
,J.(
2001
).C
ardi
ovas
cula
ran
dpu
lmon
aryp
hysi
olog
yand
anat
omy,i
nJ.
T.O
ttese
n,M
.S.O
lufs
enan
dJ.
Lars
en(e
ds),
Ap
plie
dM
ath
em
atic
alM
od
els
inH
um
an
Ph
ysio
logy,
tobe
publ
ishe
dby
SIA
M,c
hapt
er2.
Park
,J.,
Met
axas
,D.
N.
and
Axe
l,L.
(199
6).
Ana
lysi
sof
left
vent
ricul
arw
all
mot
ion
base
don
volu
met
ricde
form
able
mod
elsa
ndm
risp
amm
,Me
dic
al
Imag
eA
na
lysi
s1(1
):53
–71.
Pes
kin,
C.S
.(19
72a)
.F
low
Patte
rns
aro
un
dH
ea
rtVa
lve
s,P
hDth
esis
,Alb
ert
Ein
stei
nCol
lege
ofM
edic
ine.
Pes
kin,
C.S
.(19
72b)
.Flo
wpa
ttern
saro
undh
eart
valv
es:A
num
eric
alm
etho
d,J.
Co
mp
ut.P
hys
.10:
252–
271.
Pes
kin,
C.S
.(19
75).
Mat
hem
atic
alas
pect
sofh
eart
phys
iolo
gy,C
oura
ntIn
stitu
teof
Mat
hem
atic
alS
cien
ces,N
ewY
ork
Uni
vers
ity,L
ectu
reN
otes
.
Pes
kin,
C.S
.(19
77).
Num
eric
alan
alys
isof
bloo
dflo
win
the
hear
t,J.
Co
mp
ut.
Ph
ys.2
5:22
0–25
2.
176
Bib
liogr
aph
y
Pes
kin,
C.S
.(19
81).
Mat
hem
atic
alas
pect
sofp
hysi
olog
y,in
F.C
.Hop
pens
tead
t(e
d.),
Ma
the
ma
tica
lAsp
ect
sofP
hys
iolog
y.L
ect
ure
sin
Ap
plie
dM
ath
em
at-
ics,
Vol.
19,
Am
eric
anM
athe
mat
ical
Soc
iety,
Pro
vide
nce,
Rho
deIs
land
,pp
.69–
93.
Pes
kin,
C.S
.(19
89).
Fib
erar
chite
ctur
eoft
hele
ftve
ntric
ular
wal
l:A
nas
ympt
otic
anal
ysis
,Co
mm
un
.Pu
reA
pp
l.M
ath
.XLI
I:7
9–11
3.
Pes
kin,
C.S
.(19
98).
Inco
mpr
essi
blefl
uid
dyna
mic
swith
dist
ribut
edso
urce
sand
sink
s.U
npub
lishe
dnot
e.
Pes
kin,
C.S
.(20
01).
Priv
ate
com
mun
icat
ion.
Pes
kin,
C.S
.and
McQ
ueen
,D.M
.(1
980)
.Mod
elin
gpr
osth
etic
hear
tval
vesf
ornu
mer
ical
anal
ysiso
fblo
odflo
win
the
hear
t,J.C
om
pu
t.Ph
ys.3
7:11
3–13
2.
Pes
kin,
C.S
.and
McQ
ueen
,D.M
.(1
989)
.AT
hree
-Dim
ensi
onalc
ompu
tatio
nal
met
hodf
orbl
ood
flow
inth
ehe
art,J
.Co
mp
ut.P
hys
.81:
372–
405.
Pes
kin,
C.S
.and
McQ
ueen
,D.M
.(1
992)
.Car
diac
fluid
dyna
mic
s,C
ritic
alR
e-
view
sin
Bio
me
dic
alE
ng
ine
erin
g20(
5,6)
:451
–459
.
Pes
kin,
C.S
.and
McQ
ueen
,D.M
.(19
93a)
.Car
diac
fluid
dyna
mic
s,in
T.C
.Pilk
-in
gton
etal
.(ed
.),H
igh
-pe
rfo
rma
nce
Co
mp
utin
gin
Bio
me
dic
alR
ese
arch
,C
RC
Pre
ss.
Pes
kin,
C.S
.and
McQ
ueen
,D.M
.(19
93b)
.Com
puta
tiona
lbiofl
uid
dyna
mic
s,in
A.
Y.C
heer
and
C.P
.van
Dam
(eds
),F
luid
Dyn
am
icsi
nB
iolo
gy,
Vol
.141
,A
mer
ican
Mat
hem
atic
alS
ocie
ty,P
rovi
denc
e,R
hode
Isla
nd,p
p.16
1–18
6.
Pes
kin,
C.S
.and
McQ
ueen
,D.M
.(1
994)
.M
echa
nica
lequ
ilibr
ium
dete
rmin
esth
efr
acta
lfibe
rar
chite
ctur
eof
aort
iche
artv
alve
leafl
ets,
Am
.J.
Ph
ysio
l.26
6:H
319–
H32
8.
Pes
kin,
C.S
.and
McQ
ueen
,D.
M.
(199
5).
Age
nera
lmet
hodf
orth
eco
mpu
ter
sim
ulat
ion
ofbi
olog
ical
syst
emsi
nter
actin
gwith
fluid
s,in
C.P
.E
lling
ton
andT
.J.P
edley
(eds
),Bio
log
ica
lFlu
idD
yna
mic
s:P
roce
ed
ing
sofa
me
etin
gh
eld
att
he
Un
ive
rsity
ofL
ee
ds,
UK
,4
-8Ju
ly1
99
4,V
ol.4
9of
Sym
po
siao
fth
eS
oci
ety
for
Exp
erim
en
talB
iolo
gy,
Cam
brid
ge,U
K,p
p.26
5–27
6.
Pes
kin,
C.S
.and
McQ
ueen
,D.
M.
(199
6).
Flu
iddy
nam
icso
fth
ehe
arta
ndits
valv
es,in
H.O
thm
er,F
.R.A
dler
,M.
A.
Lew
isan
dJ.
C.D
allo
n(e
ds),
Ca
seS
tud
iesin
Ma
the
ma
tica
lMo
de
ling
-E
colo
gy,
Ph
ysio
logy,
an
dC
ell
Bio
log
y,P
rent
ice-
Hal
l,Inc
.,E
nglew
ood
Clif
fs,N
ewJe
rsey,
chap
ter1
4,pp
.309
–337
.
Bib
liogr
aph
y17
7
Pes
kin,
C.
S.a
ndP
rintz
,B.
F.(1
993)
.Im
prov
edvo
lum
eco
nser
vatio
nin
the
com
puta
tiono
fflow
sw
ithim
mer
sedb
ound
arie
s,J.C
om
pu
t.Ph
ys.1
05:3
3–46
.
Phi
lips
(n.d
.).B
asi
cPrin
cip
leso
fMR
Imag
ing,
Phi
lips
Med
ical
Sys
tem
.
Rid
eout
,V.(
1991
).M
ath
em
atic
ala
nd
Co
mp
ute
rMo
de
ling
ofP
hys
iolog
ica
lSys
-te
ms,
Pre
ntic
eHal
l,B
ioph
ysic
sand
Bio
engi
neer
ingS
erie
s.
Sab
er,N
.R
.,G
osm
an,A
.D
.,W
ood,
N.
B.,
Kiln
er,
P.J.
,Cha
rrie
r,C
.L.
and
Firm
in,
D.
N.
(200
1).
Com
puta
tiona
lflow
mod
elin
gof
the
left
vent
ricle
base
don
invi
voM
RI
data
:Ini
tial
expe
rienc
e,A
nn
als
of
Bim
ed
ica
lEn
gi-
ne
erin
g29(
4):2
75–2
83.
Sch
laik
jer,M
.and
Kol
enda
,T.(
1997
).M
RI,
den
grun
dlgg
ende
teor
i(in
Dan
ish)
,S
tud
en
trep
ort,
Tech
nica
lUni
vers
ityof
Den
mar
k.U
npub
lishe
d.
Sch
oeph
oers
ter
,R.T
.,S
ilva,
C.L
.an
dR
ay,G
.(19
93).
Fin
itean
alyt
icm
odel
for
left
vent
ricul
arsy
stol
icflo
wdy
nam
ics,
Jou
rna
lofE
ng
ine
erin
gMe
cha
nic
s11
9(4)
:733
–747
.
Sch
oeph
oers
ter
,R.T
.,S
ilva,
C.L
.an
dR
ay,
G.(
1994
).E
valu
atio
nof
left
ven-
tric
ular
func
tion
base
don
sim
ulat
edsy
stol
icflo
wdy
nam
icsc
ompu
tedf
rom
regi
onal
wal
lmot
ion,
J.B
iom
ech
an
ics2
7(2)
:125
–136
.
Sch
roed
er,A
.P.(
1999
).M
agn
etic
reso
na
nce
imag
ing
for
the
stu
dyo
fle
ftve
ntr
ic-
ula
rre
mo
de
linga
fte
racu
tem
yoca
rdia
lin
farc
tion,
PhD
thes
is,D
epar
tmen
tof
Car
diol
ogy,
Ske
jby
Hos
pita
l,Uni
vers
ityof
Aar
hus.
St.
Goa
r,F.
G.,
Mas
uyam
a,T.,
Ald
erm
an,E
.L.a
ndP
opp,
R.L
.(19
91).
Left
ven-
tric
ular
dias
tolic
dysf
unct
ioni
nen
d-st
aged
ilate
dca
rdio
myo
path
y:Sim
ulta
-ne
ousD
oppl
erec
hoca
rdio
grap
hyand
hem
odyn
amice
valu
atio
n,J.
Am
.So
c.E
cho
card
iog
r.4:
349–
360.
Ste
ine,
K.,
Stu
gaar
d,M
.an
dS
mis
eth,
O.(
1999
).M
echn
ism
sof
reta
rded
apic
alfil
ling
inac
utei
sche
mic
left
vent
ricul
arfa
ilure
,Cir
cula
tion
99:2
048–
2054
.
Sto
ckie
,J.M
.an
dW
etto
n,B
.T.
R.(
1995
).S
tabi
lity
anal
ysis
for
the
imm
erse
dfib
erpr
oble
m,S
IAM
J.A
pp
l.M
ath
.55(
6):1
577–
1591
.
Str
eete
r,Jr
.,D
.D
.,P
ower
s,W
.E
.,R
oss,
M.
A.
and
Torr
ent-
Gua
sp,F.
(197
8).
Thr
ee-d
imen
sion
alfibe
rorie
ntat
ioni
nth
em
amm
alia
nlef
tven
tric
lew
all,
inJ.
Baa
n,A
.N
oord
ergr
aafa
ndJ.
Rai
nes(
eds)
,Ca
rdio
vasc
ula
rSys
tem
Dy-
na
mic
s,M
ITP
ress
,Cam
brid
ge,M
A,
pp.7
3–84
.
178
Bib
liogr
aph
y
Str
eete
r,Jr
.,D
.D
.,S
potn
iz,H
.M
.,Pa
tel,
D.
P.,
Ros
s,Jr
.,J.
and
Son
nenb
lick,
E.H
.(1
969)
.F
iber
orie
ntat
ioni
nth
eca
nine
left
vent
ricle
durin
gdi
asto
lean
dsy
stol
e,C
irc.
Re
s.24
:339
–347
.
Stu
gaar
d,M
.,B
roda
hl,U
.,To
rp,H
.an
dIh
len,
H.
(199
4a).
Abn
orm
aliti
esof
left
vent
ricul
arfil
ling
inpa
tient
swith
coro
nary
arte
rydi
seas
e:as
sesm
entby
colo
urM
-mod
eD
oppl
erte
chni
que,
Eu
rop
ea
nHe
art
Jou
rna
l15:
318–
327.
Stu
gaar
d,M
.,R
isoe
,C.,
Ihle
n,H
.an
dS
mis
eth,
O.
(199
4b).
Intr
acav
itary
fill-
ing
patte
rnin
the
faili
ngle
ftve
ntric
leas
sese
dby
colo
rM
-mod
eD
oppl
erec
hoca
rdio
grap
hy,J.
Am
.Co
ll.C
ard
iol.
24:6
63–6
70.
Stu
gaar
d,M
.,S
mis
eth,
O.
A.,
Ris
oe,
C.
and
Ihle
n,H
.(1
993)
.In
trav
entr
ic-
ular
early
dias
tolic
fillin
gdu
ring
acut
em
yoca
rdia
linfa
rctio
n,C
ircu
latio
n88
:270
5–27
13.
Tayl
or,T
.W.,
Oki
no,H
.and
Yam
aguc
hi,T
.(19
93).
The
effe
ctso
fsup
raval
vula
rao
rtic
sten
osiso
nre
alis
ticth
ree-
dim
ensi
onalle
ftve
ntric
ular
bloo
dej
ectio
n,B
iorh
eo
log
y30
(5–6
):42
9–43
4.
Tayl
or,
T.W
.,S
uga,
H.,
Got
o,Y
.,O
kino
,H.
and
Yam
aguc
hi,T
.(1
996)
.T
heef
fect
sof
card
iaci
nfar
ctio
non
real
istic
thre
e-di
men
sion
allef
tve
ntric
ular
bloo
dej
ectio
n,J.
Bio
me
ch.E
ng.
118(
1):1
06–1
10.
Tayl
or,T
.W.a
ndY
amag
uchi
,T.(
1995
a).F
low
patte
rnsin
thre
e-di
men
sion
alleft
vent
ricul
arsy
stol
ican
ddi
asto
licflo
ws
dete
rmin
edfr
omco
mpu
tatio
nalflu
iddy
nam
ics,
Bio
rhe
olo
gy
32(1
):61
–71.
Tayl
or,
T.W
.an
dY
amag
uchi
,T.
(199
5b).
Rea
listic
thre
e-di
men
sion
allef
tve
n-tr
icul
arej
ectio
ndet
erm
ined
from
com
puta
tiona
lfluid
dyna
mic
s,M
ed
.En
g.P
hys
.17(
8):6
02–6
08.
Tho
mas
,C.E
.(19
57).
The
mus
cula
rarc
hite
ctur
eoft
heve
ntric
leso
fhog
and
dog
hear
ts,T
he
Am
erica
nJo
urn
alo
fAn
ato
my1
01:1
7–57
.
Tho
rup,
P.(1
997)
.He
mo
dyn
am
icMo
de
llin
go
fth
eH
ea
rt,M
aste
r’sth
esis
,Dep
t.of
Hyd
rody
nam
icsa
ndW
ater
Res
ourc
es(I
SVA
),Te
chni
calU
nive
rsity
ofD
enm
ark(
DT
U).
Tort
ora,
G.J
.(19
99).
Prin
cip
les
ofH
um
an
An
ato
my,
8th
edn,
Add
ison
Wes
leyLo
ngm
an,In
c.,M
enlo
Park
,Cal
iforn
ia.
Bib
liogr
aph
y17
9
Tu,
C.a
ndP
eski
n,C
.S.(
1992
).S
tabi
lity
and
inst
abili
tyin
the
com
puta
tiono
fflo
ws
with
mov
ing
imm
erse
dbou
ndar
ies:
Aco
mpa
rison
ofth
ree
met
hods
,S
IAM
J.S
ci.S
tat.C
om
pu
t.13(
6):1
462–
1376
.
Ves
ier,
C.
C.
and
Yog
anat
han,
A.
P.(1
992)
.A
com
pute
rmet
hod
for
sim
ula-
tion
ofca
rdio
vasc
ular
flow
field
s:va
lidat
ion
ofap
proa
ch,J.
Co
mp
ut.P
hys
.99
(2):
271–
287.
Wal
ker,
P.G
.,C
rann
ey,G
.B.,
Grim
es,R
.Y.,
Del
ator
e,J.
,Rec
tenw
ald,
J.,P
ohos
t,G
.M.
and
Yog
anat
han,A
.P.
(199
6).
Thr
ee-d
imen
sion
alrec
onst
ruct
iono
fth
eflo
win
ahu
man
left
hear
tby
usin
gm
agne
ticre
sona
ncep
hase
velo
city
enco
ding
,An
na
lso
fBio
me
dic
alE
ng
ine
erin
g24(
1):1
39–1
47.
Wal
ker,
P.G
.,C
rann
ey,G
.B.,
Sch
eide
gger
,M.
B.,
Was
eles
ki,G
.,P
ohos
t,G.M
.an
dY
ogan
atha
n,A.
P.(1
993)
.Sem
i-aut
omat
edm
etho
dfor
nois
ered
uctio
nan
dba
ckgr
ound
phas
eerr
orre
duct
iono
fM
Rph
asev
eloc
ityda
ta,J
.Mag
n.
Re
son
.Imag
ing
3:52
1–53
0.
Wei
ss,J
.L.,
Fre
derik
sen,J
.W.a
ndW
eisf
eldt
,M.L
.(19
76).
Hem
odyn
amicd
eter
-m
inan
tsof
the
time-
coru
seof
fall
inca
nine
left
vent
ricul
arpr
essu
re,Jo
urn
al
ofC
linic
alI
nve
stig
atio
n58:
751–
760.
Yam
amot
o,K
.,M
asuy
ama,
T.,
Tano
uchi
,J.,
Nai
to,
J.,
Man
o,T.
,K
ondo
,H.,
Nag
ano,
R.,
Mas
atsu
guan
dK
amad
a,T.
(199
5).I
ntra
vent
ricul
ardi
sper
sion
ofea
rlydi
asto
licfil
ling:
Ane
wm
arke
rofl
eftv
entr
icul
ardi
asto
licdy
sfun
c-tio
n,A
m.H
ea
rt.J
.129
:291
–299
.
Yel
lin,E
.L.,
Pes
kin,
C.S
.,Y
oran
,C.,
Koe
nigs
berg,
M.,
Mat
sum
oto,
M.,
Lani
ado,
S.,
McQ
ueen
,D.M
.,S
hore
,D.a
ndF
rate
r,R.W
.M.(
1981
).M
echa
nism
sof
mitr
alva
lve
mot
ion
durin
gdi
asto
le,A
m.J
.Ph
ysio
l.241
(3):
H38
9–H
400.
Yog
anat
han,A
.P.
,Lem
mon
,J.D
.,K
im,
Y.
H.,
Levi
ne,R
.A.
and
Ves
ier,
C.C
.(1
995)
.Ath
ree-
dim
ensi
onalc
ompu
terin
vest
igat
iono
fint
rave
ntric
ular
fluid
dyna
mic
s:E
xam
inat
ioni
nto
the
initi
atio
nof
syst
olic
ante
riorm
otio
nof
the
mitr
alva
lve
leafl
ets,
J.B
iom
ech
.En
g.11
7:94
–102
.
Yog
anat
han,
A.
P.,
Lem
mon
,J.D
.,K
im,
Y.
H.,
Wal
ker,
P.G
.,Le
vine
,R
.A
.an
dV
esie
r,C
.C
.(1
994)
.A
com
puta
tiona
lstu
dyof
ath
in-w
alle
dth
ree-
dim
ensi
onal
left
vent
ricle
durin
gea
rlysy
stol
e,J.
Bio
me
ch.E
ng.
116:
307–
314.
Ph.
D.t
hese
sfro
mIM
M1.
Lars
en,R
asm
us.(
1994
).E
stim
atio
nofv
isu
alm
otio
nin
imag
ese
qu
en
ces.
xiv
+14
3pp
.
2.R
ygar
d,Je
nsM
ober
g.(1
994)
.De
sig
nan
do
ptim
iza
tiono
fflex
ible
ma
nu
-fa
ctu
rin
gsy
ste
ms.x
iii+
232
pp.
3.La
ssen
,Nie
lsC
hris
tian
Krie
ger.
(199
4).
Au
tom
ate
dde
term
ina
tiono
fcr
ysta
lorie
nta
tion
sfro
me
lect
ron
ba
cksc
atte
rin
gpa
tte
rns.
xv+
136
pp.
4.M
elga
ard,
Hen
rik.
(199
4).I
de
ntifi
catio
no
fph
ysic
alm
od
els
.xvi
i+24
6pp
.
5.W
ang,
Chu
nyan
.(1
994)
.Sto
cha
stic
diff
ere
ntia
leq
ua
tion
san
da
bio
log
i-ca
lsys
tem
.xxii
+15
3pp
.
6.N
iels
en,A
llan
Aas
bjer
g.(1
994)
.A
na
lysi
sof
reg
ula
rly
an
dirre
gu
larly
sam
ple
dsp
atia
l,mu
ltiva
ria
te,a
nd
mu
lti-t
em
po
rald
ata
.xxi
v+
213
pp.
7.E
rsbø
ll,A
nnet
teK
jær.
(199
4).O
nth
esp
atia
lan
dte
mp
ora
lco
rre
latio
ns
inex
pe
rim
en
tatio
nwith
agricu
ltura
lap
plic
atio
ns.
xviii
+34
5pp
.
8.M
ølle
r,D
orte
.(1
994)
.Me
tho
dsf
or
an
aly
sisa
nd
de
sig
nofh
ete
roge
ne
ou
ste
leco
mm
un
ica
tionn
etw
ork
s.V
olum
e1-
2,xx
xviii
+28
2pp
.,28
3-56
9pp.
9.Je
nsen
,Jen
sChr
istia
n.(1
995)
.Te
ore
tiske
oge
ksp
erim
en
telled
yna
mis
keu
nd
ers
øge
lse
rafj
ern
ba
ne
køretø
jer.v
iii+
174
pp.
10.
Kuh
lman
n,Li
onel
.(1
995)
.O
na
uto
ma
ticvi
sua
lin
spe
ctio
nofre
flect
ive
surf
ace
s.V
olum
e1,
xviii
+22
0pp
.,(V
olum
e2,
vi+
54pp
.,fo
rtro
lig).
11.
Laza
rides
,Nik
olao
s.(1
995)
.No
nlin
ea
rity
insu
pe
rcon
du
ctiv
itya
nd
Jose
ph
-so
nJu
nct
ion
s.iv
+15
4pp
.
12.
Ros
tgaa
rd,M
orte
n.(1
995)
.M
od
elli
ng,
est
ima
tiona
nd
con
trol
of
fast
sam
ple
ddyn
am
ica
lsys
tem
s.xiv
+34
8pp
.
13.
Sch
ultz
,Net
te.
(199
5).S
egm
en
tatio
nan
dcl
ass
ifica
tiono
fbio
log
ica
lob
-je
cts.
xiv
+19
4pp
.
14.
Jørg
ense
n,M
icha
elF
inn.
(199
5).N
on
line
arH
am
ilto
nia
nsy
ste
ms.x
iv+
120
pp.
15.
Bal
le,S
usan
neM
.(1
995)
.Dis
trib
ute
d-m
em
orym
atr
ixco
mp
uta
tion
s.iii
+10
1pp
.
16.
Koh
l,N
ikla
s.(1
995)
.E
xact
me
tho
dsf
or
time
con
stra
ine
dro
utin
ga
nd
rela
teds
che
du
lingp
rob
lem
s.xv
iii+
234
pp.
17.
Rog
on,T
hom
as.(
1995
).Po
rou
sme
dia
:An
aly
sis,
reco
nst
ruct
iona
nd
pe
r-co
latio
n.x
iv+
165
pp.
18.
And
erse
n,A
llan
The
odor
.(19
95).
Mo
de
llin
go
fpa
cke
ttra
ffic
with
ma
trix
an
aly
ticm
eth
od
s.xv
i+24
2pp
.
19.
Hes
thav
en,J
an.(
1995
).N
um
erica
lstu
die
sofu
nst
ea
dyc
oh
ere
nts
tru
ctu
res
an
dtr
an
spo
rtin
two
-dim
en
sio
na
lflow
s.R
isø
-R-8
35(E
N)2
03pp
.
20.
Sliv
sgaa
rd,E
vaC
harlo
tte.
(l995
).O
nth
ein
tera
ctio
nb
etw
ee
nwh
ee
lsan
dra
ilsin
railw
ay
dyn
am
ics.
viii
+19
6pp
.
21.
Har
teliu
s,K
arst
en.
(199
6).A
na
lysi
sofi
rreg
ula
rly
dis
trib
ute
dp
oin
ts.x
vi+
260
pp.
22.
Han
sen,
Anc
aD
anie
la.
(199
6).
Pre
dic
tive
con
trol
an
did
en
tifica
tion
-A
pp
lica
tion
sto
ste
erin
gdyn
am
ics.
xviii
+30
7pp
.
23.
Sad
egh,
Pay
man
.(19
96).
Exp
erim
en
tde
sig
nan
do
ptim
iza
tioni
nco
mp
lexsy
ste
ms.x
iv+
162
pp.
24.
Ska
nds,
Ulri
k.(1
996)
.Qu
an
tita
tivem
eth
od
sfo
rth
ea
na
lysi
sofe
lect
ron
mic
rosc
op
eim
age
s.xv
i+19
8pp
.
25.
Bro
-Nie
lsen
,Mor
ten.
(199
6).
Me
dic
ali
mag
ere
gis
tratio
na
nd
surg
ery
sim
ula
tion
.xxv
ii+
274
pp.
26.
Ben
dtse
n,C
laus
.(19
96).
Para
lleln
um
erica
lalg
orith
msf
orth
eso
lutio
no
fsy
ste
mso
ford
ina
ryd
iffe
ren
tiale
qu
atio
ns.
viii
+79
pp.
27.
Laur
itsen
,M
orte
nB
ach.
(199
7).
De
lta-d
om
ain
pre
dic
tive
con
trol
an
did
en
tifica
tionf
or
con
trol.
xxii
+29
2pp
.
28.
Bis
chof
f,S
vend
.(19
97).
Mo
de
llin
gco
llid
ing
-pu
lsem
od
e-lo
cked
sem
ico
n-
du
cto
rla
sers
.xxi
i+21
7pp
.
29.
Arn
bjer
g-N
iels
en,K
arst
en.
(199
7).S
tatis
tica
lan
aly
siso
furb
an
hyd
rol-
ogy
with
spe
cia
lem
ph
asi
son
rain
fall
mo
de
llin
g.In
stitu
tfor
Milj
øte
knik
,D
TU
.xiv
+16
1pp
.
30.
Jaco
bsen
,Jud
ithL.
(199
7).
Dyn
am
icm
od
elli
ng
of
pro
cess
esin
rive
rsa
ffect
edb
yp
reci
pita
tion
run
off.
xix
+21
3pp
.
31.
Som
mer
,H
elle
Mø
lgaa
rd.
(199
7).V
aria
bili
tyin
mic
rob
iolo
gic
ald
egra
-d
atio
nex
pe
rim
en
ts-A
na
lysi
san
dca
sest
ud
y.xi
v+
211
pp.
32.
Ma,
Xin
.(1
997)
.Ad
ap
tivee
xtre
mu
mco
ntro
lan
dw
ind
turb
ine
con
trol.
xix
+29
3pp
.
33.
Ras
mus
sen,
Kim
Ørs
kov.
(199
7).N
on
line
ara
nd
sto
cha
stic
dyn
am
icso
fco
he
ren
tstr
uct
ure
s.x
+21
5pp
.
34.
Han
sen,
Lars
Hen
rik.
(199
7).
Sto
cha
stic
mo
de
llin
go
fce
ntra
lh
ea
ting
syst
em
s.xxi
i+30
1pp
.
35.
Jørg
ense
n,C
laus
.(1
997)
.Driftso
ptim
erin
gpa
kra
ftva
rme
syst
em
er
.29
0pp
.
36.
Sta
unin
g,O
le.(
1997
).A
uto
ma
ticva
lida
tion
ofn
um
erica
lso
lutio
ns.
viii
+11
6pp
.
37.
Ped
erse
n,M
orte
nW
ith.
(199
7).
Op
timiz
atio
no
fre
curre
ntn
eu
ral
ne
t-w
ork
sfo
rtim
ese
rie
smo
de
ling.
x+
322
pp.
38.
Tho
rsen
,Run
e.(1
997)
.Re
sto
ratio
no
fha
nd
fun
ctio
nin
tetra
ple
gic
su
sin
gm
yoe
lect
rica
llyco
ntro
lled
fun
ctio
na
lele
ctrica
lstim
ula
tion
oft
he
con
trol-
ling
mu
scle.
x+
154
pp.+
App
endi
x.
39.
Ros
holm
,And
ers.
(199
7).S
tatis
tica
lme
tho
dsf
or
seg
me
nta
tiona
nd
cla
s-si
fica
tion
ofi
mag
es.
xvi+
183
pp.
40.
Pet
erse
n,K
imT
ilgaa
rd.
(199
7).E
stim
atio
nofs
pe
ech
qu
alit
yin
tele
com
-m
un
ica
tions
yste
ms.x
+25
9pp
.
41.
Jens
en,C
arst
enN
ords
trø
m.(
1997
).N
on
line
ars
yste
msw
ithd
iscr
ete
an
dco
ntin
uo
use
lem
en
ts.19
5pp
.
42.
Han
sen,
Pet
erS
.K.(
1997
).S
ign
als
ub
spa
cem
eth
od
sfor
spe
ech
en
ha
nce
-m
en
t.x
+22
6pp
.
43.
Nie
lsen
,Ole
Mø
ller.
(199
8).W
ave
lets
insc
ien
tificc
om
pu
ting.
xiv
+23
2pp
.
44.
Kje
ms,
Ulri
k.(1
998)
.B
aye
sia
nsig
na
lpro
cess
inga
nd
inte
rpre
tatio
no
fb
rain
sca
ns.
iv+
129
pp.
45.
Han
sen,
Mic
hael
Pile
gaar
d.(1
998)
.Me
tah
eu
rist
icsfo
rm
ulti
ple
ob
ject
ive
com
bin
ato
ria
lop
timiz
atio
n.x
+16
3pp
.
46.
Riis
,Sø
ren
Kam
aric
.(1
998)
.Hid
de
nm
ark
ovm
od
els
an
dn
eu
raln
etw
ork
sfo
rsp
ee
chre
cog
niti
on
.x+
223
pp.
47.
Mø
rch,
Nie
lsJa
cob
San
d.(1
998)
.Am
ulti
varia
tea
pp
roa
chto
fun
ctio
na
ln
eu
rom
od
elin
g.xv
i+14
7pp
.
48.
Fry
dend
al,I
b.(
1998
.)Q
ua
lity
insp
ect
iono
fsu
ga
rbe
ets
usi
ng
visi
on
.iv
+97
pp.+
app.
49.
Lund
in,
Lars
Kris
tian.
(199
8).
Para
llel
com
pu
tatio
nof
rota
ting
flow
s.vi
ii+
106
pp.
50.
Bor
ges,
Ped
ro.(
1998
).M
ulti
crite
ria
pla
nn
ing
an
do
ptim
iza
tion
.-H
eu
ris-
tica
pp
roa
che
s.xi
v+
219
pp.
51.
Nie
lsen
,Jak
obB
irked
al.
(199
8).
New
dev
elo
pm
en
tsin
the
the
ory
of
wh
ee
l/ra
ilco
nta
ctm
ech
an
ics.
xviii
+22
3pp
.
52.
Fog
,Tor
ben.
(199
8).C
on
diti
on
mo
nito
rin
ga
nd
fau
ltd
iag
no
sisi
nm
arin
ed
iese
len
gin
es.
xii+
178
pp.
53.
Knu
dsen
,Ole
.(19
98).
Ind
ust
ria
lvis
ion
.xii
+12
9pp
.
54.
And
erse
n,Je
nsS
trod
l.(1
998)
.Sta
tistic
ala
na
lysi
sofb
iote
sts.
-A
pp
lied
toco
mp
lexp
ollu
ted
sam
ple
s.xx
+20
7pp
.
55.
Phi
lipse
n,P
eter
Als
hede
.(19
98).
Re
con
stru
ctio
nan
dre
sto
ratio
no
fPE
Tim
age
s.vi
+13
2pp
.
56.
Th
yges
en,U
ffeH
øgs
bro.
(199
8).R
obu
stp
erf
orm
an
cea
nd
dis
sip
atio
nof
sto
cha
stic
con
trols
yste
ms.1
85pp
.
57.
Hin
tz-M
adse
n,M
ads.
(199
8).A
pro
ba
bili
stic
fra
mew
ork
forc
lass
ifica
tion
ofd
erm
ato
sco
pici
mag
es.
xi+
153
pp.
58.
Sch
ram
m-N
iels
en,K
arin
a.(1
998)
.E
nvir
on
me
nta
lre
fere
nce
ma
teria
lsm
eth
od
san
dca
sest
ud
ies.
xxvi
+26
1pp
.
59.
Sky
ggeb
jerg
,Ole
.(1
999)
.Acq
uis
itio
na
nd
an
aly
siso
fco
mp
lexd
yna
mic
intr
a-a
nd
inte
rce
llula
rsig
na
ling
eve
nts
.83
pp.
60.
Jens
en,K
are
Jean
.(1
999)
.S
ign
alp
roce
ssin
gfo
rd
istr
ibu
tion
ne
two
rkm
on
itorin
g.xv
+19
9pp
.
61.
Fol
m-H
anse
n,Jø
rgen
.(19
99).
On
chro
ma
tica
nd
geo
me
tric
alc
alib
ratio
n.
xiv
+23
8pp
.
62.
Lars
en,J
espe
r.(19
99).
Para
lleliz
atio
no
fth
eve
hic
lero
utin
gp
rob
lem
with
time
win
do
ws.
xx+
266
pp.
63.
Cla
usen
,Car
lB
alsl
ev.
(199
9).
Sp
atia
lso
lito
nsin
qu
asi
-ph
ase
ma
tch
ed
stru
ktu
res.
vi+
(fler
epa
g.)
64.
Kvi
st,
Trin
e.(1
999)
.Sta
tistic
alm
od
elli
ngo
ffish
sto
cks.
xiv
+17
3pp
.
65.
And
rese
n,P
erR
øns
holt.
(199
9).S
urf
ace
-bo
un
de
dgro
wth
mo
de
linga
p-
plie
dto
hu
ma
nma
nd
ible
s.xx
ii+
125
pp.
66.
Sø
rens
en,P
erS
ette
rgre
n.(1
999)
.Sp
atia
ldis
trib
utio
nm
ap
sfo
rb
en
thic
com
mu
niti
es.
67.
And
erse
n,H
elle
.(19
99).
Sta
tistic
alm
od
els
for
sta
nd
ard
ize
dto
xici
tyst
ud
-ie
s.vi
ii+
(fler
epa
g.)
68.
And
erse
n,La
rsN
onbo
e.(1
999)
.Sig
na
lpro
cess
ingi
nth
ed
olp
hin
son
ar
syst
em
.xii
+21
4pp
.
69.
Bec
hman
n,H
enrik
.(1
999)
.Mo
de
llin
go
fwa
stew
ate
rsys
tem
s.xvi
ii+
161
pp.
70.
Nie
lsen
,Hen
rikA
albo
rg.(
1999
).Pa
ram
etr
ica
nd
no
n-p
ara
me
tric
syst
em
mo
de
llin
g.xv
iii+
209
pp.
71.
Gra
mk
ow,C
laus
.(19
99).
2D
an
d3
Do
bje
ctm
ea
surem
en
tfor
con
trola
nd
qu
alit
ya
ssu
ran
cein
the
ind
ust
ry.
xxvi
+23
6pp
.
72.
Nie
lsen
,Jan
Nyg
aard
.(1
999)
.Sto
cha
stic
mo
de
llin
go
fdyn
am
icsy
ste
ms.
xvi+
225
pp.
73.
Lars
en,A
llan.
(200
0).T
he
dyn
am
icve
hic
lero
utin
gp
rob
lem
.xvi
+18
5pp
.
74.
Hal
kjæ
r,S
øre
n.(2
000)
.Ela
stic
wa
vep
rop
aga
tion
ina
nis
otro
pic
inh
om
o-
gen
eo
usm
ate
ria
ls.x
iv+
133
pp.
75.
Lars
en,
The
isLe
th.
(200
0).
Ph
osp
ho
rusd
iffu
sio
nin
floa
tzo
ne
silic
on
crys
talg
row
th.v
iii+
119
pp.
76.
Dirs
cher
l,K
ai.
(200
0).O
nlin
eco
rre
ctio
no
fsca
nn
ingp
rob
em
icro
sco
pe
sw
ithp
ixe
lacc
ura
cy.1
46pp
.
77.
Fis
ker,
Run
e.(2
000)
.M
aki
ng
de
form
ab
lete
mp
late
mo
de
lso
pe
ratio
na
l.xx
+21
7pp
.
78.
Hul
tber
g,T
imH
elge
.(20
00).T
op
icsi
nco
mp
uta
tion
allin
ea
rop
timiz
atio
n.
xiv
+18
0pp
.
79.
And
erse
n,K
laus
Kaa
e.(2
000)
.Sto
cha
stic
mo
de
llin
go
fe
ne
rgy
syst
em
s.xi
v+
191
pp.
80.
Th
yreg
od,P
eter
.(2
001)
.Mo
de
llin
ga
nd
mo
nito
rin
gin
inje
ctio
nm
old
ing.
xvi+
132
pp.
81.
Sch
jødt
-Erik
sen,
Jens
.(2
001)
.A
rre
stin
go
fco
llap
sein
inh
om
oge
ne
ou
sa
nd
ultr
afa
stK
err
me
dia
.
82.
Ben
nets
en,J
ens
Chr
istia
n.(2
000)
.N
um
erica
lsim
ula
tion
of
turb
ule
nt
airflo
win
live
sto
ckbu
ildin
gs.
xi+
205
pp+
App
endi
x.
83.
Hø
jen-
Sø
rens
en,P
edro
A.d
.F.R
.(20
01).
Ap
pro
xim
atin
gm
eth
od
sfo
rin
-tr
act
ab
lep
rob
ab
ilist
icm
od
els
:-A
pp
lica
tion
sin
ne
uro
scie
nce.xi
+10
4pp
+A
ppen
dix.
84.
Nie
lsen
,Tor
ben
Sko
v.(2
001)
.O
n-lin
ep
red
ictio
na
nd
con
trol
inn
on
-lin
ea
rsto
cha
stic
syst
em
s.xvi
ii+
242
pp.
85.
Oje
lund
,H
enrik
.(2
001)
.M
ulti
varia
teca
libra
tion
of
che
mic
als
en
sors.
xviii
+18
4pp
.
86.
Ade
ler,
Per
nille
Tho
rup.
(200
1).H
em
od
yna
mics
imu
latio
no
fth
eh
ea
rtu
sin
ga
2D
mo
de
lan
dM
Rd
ata
.xv
+17
9pp
.