Resea ch A icle
Impac o Dec eased T ansmu al Conduc ion Veloci y on he
Func ion o he Human Le Ven icle: A Simula ion S udy
Jiří Va e ka ,
1
Jiří Moud ,
2
Pe Lokaj,
3
Jiří Bu ša ,
1
and Michal Pásek
2,4
1
Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Facul y o Mechanical Enginee ing, B no Uni e si y o Technology,
B no, Czech Republic
2
Depa men o Physiology, Facul y o Medicine, Masa yk Uni e si y, B no, Czech Republic
3
Depa men o In e nal Medicine and Ca diology, Uni e si y Hospi al B no, B no, Czech Republic
4
Ins i u e o The momechanics, Czech Academy o Science, P ague, Czech Republic
Co espondence should be add essed o Michal Pásek; [email p o ec ed]
Recei ed 29 Oc obe 2019; Re ised 14 Feb ua y 2020; Accep ed 24 Feb ua y 2020; Published 4 Ap il 2020
Academic Edi o : Kimimasa Tobi a
Copy igh © 2020 Jiří Va e ka e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License,
which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
This s udy in es iga es he impac o educed ansmu al conduc ion eloci y (TCV) on ou pu pa ame e s o he human hea . In a
heal hy hea , he TCV con ibu es o synch oniza ion o he onse o con ac ion in indi idual laye s o he le en icle (LV).
Howe e , i is unclea whe he he clinically obse ed dec ease o TCV con ibu es significan ly o a educ ion o LV
con ac ili y. The applied h ee-dimensional fini e elemen model o iso olumic con ac ion o he human LV inco po a es
ansmu al g adien s in elec omechanical delay and myocy e sho ening eloci y and e alua es he impac o TCV educ ion on
p essu e ise (namely, ðdP/d Þmax) and on iso olumic con ac ion du a ion (IVCD) in a heal hy LV. The model ou pu s a e
u he exploi ed in he lumped “Windkessel”model o he human ca dio ascula sys em (based on elec ohyd odynamic
analogy o espec i e diffe en ial equa ions) o simula e he impac o changes o ðdP/d Þmax and IVCD on chosen sys emic
pa ame e s (ejec ion ac ion, LV powe , ca diac ou pu , and blood p essu e). The simula ions ha e shown ha a 50% dec ease
in TCV p olongs subs an ially he iso olumic con ac ion, decele a es sligh ly he LV p essu e ise, inc eases he LV ene gy
consump ion, and educes he LV powe . These nega i e effec s inc ease p og essi ely wi h u he educ ion o TCV. In
conclusion, hese esul s sugges ha he pumping efficacy o he human LV dec eases wi h lowe TCV due o a highe ene gy
consump ion and lowe LV powe . Al hough he changes induced by he clinically ele an educ ion o TCV a e no c i ical o
a heal hy hea , hey may ep esen an impo an ac o limi ing he hea unc ion unde disease condi ions.
1. In oduc ion
Ca diac conduc ion eloci y (CV), he speed wi h which an
elec ical impulse p opaga es h ough he ca diac issue, is
one o he mos impo an elec ophysiological cha ac e is-
ics o hea muscle. In compa ison wi h no mal hea s, he
myoca dial CV was ound o be significan ly educed in dis-
eased animal and human hea s [1–4]. The educ ion o CV
was shown o inc ease he isk o een an ac i i ies ha can
lead o ca diac a hy hmias ( o e iew, see King e al. [5]).
In human ca diac muscle, he CV consis s o wo compo-
nen s, he longi udinal (be ween 60 and 70 cm/s [4]) and he
ans e sal (TCV, a ound 50 cm/s [4]). The ansmu al
dec ease o elec omechanical delay (EMD) om endoca -
dium o epica dium (EMD g adien ~2.1 ms/mm [6]) helps,
in combina ion wi h TCV, o synch onize he onse o con-
ac ion in indi idual laye s o he le en icle (LV) [6].
Howe e , he e a e, o ou bes knowledge, no published
expe imen al esul s on he impac o TCV educ ion on
he en icle con ac ili y. Thus, i is unclea whe he he
clinically obse ed dec ease o TCV and he co esponding
ansmu al desynch oniza ion o LV con ac ion con ibu es
o a educ ion o LV con ac ili y o whe he i a he ep e-
sen s a consequence o pa hological changes a a cellula le el
wi hou any significan effec on he LV unc ion.
An a emp o quan i y he effec o CV educ ion on
mechanical esponse o he mammalian hea and basic
hemodynamic pa ame e s was unde aken ecen ly by
Hindawi
BioMed Resea ch In e na ional
Volume 2020, A icle ID 2867865, 11 pages
h ps://doi.o g/10.1155/2020/2867865
Yunia i and Lim [7]. In hei simula ions using an in e-
g a ed elec omechanical model o he LV, he CV co ela ed
wi h ca diac pumping efficacy. While a dec ease o CV om
70 o 30 cm/s induced a ela i e educ ion o ejec ion ac-
ion (EF) and s oke wo k by ~7 and 12%, espec i ely, he
ATP consump ion inc eased by ~7%. Howe e , he model
was o mula ed o canine hea and did no inco po a e
ansmu al diffe ences ei he in EMD o in myocy e sho -
ening eloci y (MSV) obse ed by Co dei o e al. [8].
In he p esen s udy, ou ecen ly published h ee-
dimensional fini e elemen (FE) model o iso olumic con-
ac ion (IVC) o he human LV [6] inco po a ing ans-
mu al g adien s in EMD and MSV was used o examine he
effec o changes in TCV on dynamics o LV p essu e ise
dPV/d and IVC du a ion (IVCD) in a heal hy human hea .
In he second s ep, we used ou lumped model o he human
ca dio ascula sys em o simula e he impac o he obse ed
changes in ðdPV/d Þmax and IVCD on ca dio ascula hemo-
dynamics and a e ial p essu e.
2. Me hods
2.1. Model o he Human Le Ven icle. The impac o
dec ease in TCV on IVCD and ðdPV/d Þmax was in es iga ed
using a h ee-dimensional FE model o he human LV c e-
a ed ecen ly (in comme cial FE so wa e ANSYS®) o simu-
la e he iso olumic phase o LV sys ole. The model is based
on simplified ellipsoidal geome y meshed wi h hexahed al
quad a ic solid elemen s. Passi e beha iou o myoca dium
(conside ed as pu ely elas ic) was desc ibed wi h a ans-
e sely iso opic s ain ene gy densi y unc ion de e mining
he cons i u i e ela ion be ween s esses and (elas ic) s ains.
Ac i e con ac ion o myocy es was modelled using special
ein o cing elemen s wi h unidi ec ional s iffness which we e
c ea ed wi hin he unde lying solid mesh. Thei ac i e en-
sion was gene a ed using a simple app oach based on fic i-
ious he mal s ains. By g adually dec easing a fic i ious
empe a u e o he ein o cing elemen s (wi h a ce ain coe -
ficien o he mal expansion), nega i e he mal s ains a e
de eloped and na u ally coun e balanced by posi i e elas ic
s ains; consequen ly, ension in he fib e di ec ion is gene -
a ed. In o de o eflec he LV fib e a chi ec u e, he o ien a-
ion o hese elemen s was changed g adually ac oss he wall
be ween +60
°
and -60
°
(wi h espec o ci cum e en ial di ec-
ion) on he endoca dial and he epica dial su aces, espec-
i ely [9]. Blood inside he LV ca i y was modelled as
incomp essible liquid. In he con ol simula ion, he elec i-
cal ac i a ion o LV myoca dium was modelled unde he
assump ion o simul aneous ac i a ion o he whole endoca -
dial su ace and subsequen endoca dium- o-epica dium
p opaga ion a a cons an TCV o 47 cm/s [1, 4]. Elec ical
ac i a ion ime o each elemen was calcula ed as a a io o
he dis ance o he elemen om he endoca dial su ace
and o he TCV alue (unde con ol condi ions). The same
calcula ion was applied wi h dec eased TCV in he simula-
ions o pa hological condi ions. T ansmu ally he e oge-
neous alues o EMD and MSV we e p esc ibed in all
simula ions ollowing Co dei o e al. [8]. As he con ac ile
elemen s gene a e ension, he in a en icula p essu e ises
un il he sys emic dias olic blood p essu e (80 mmHg) is
eached. By dec easing he TCV (while keeping he o he
pa ame e s unchanged), diffe en ime-p essu e cu es we e
calcula ed o a ious le els o myoca dial conduc i i y.
ðdPV/d Þmax and IVCD we e e alua ed om hese cu es
o each case. Besides he p essu e da a, wall s ess in he
di ec ion o fib es was assessed, as well as he o al s ain
ene gy accumula ed in he LV walls in he end o he IVC
(SEIVC) which eflec s i s ene ge ic demands. Fo u he
de ails ega ding he FE model and simula ion condi ions,
he eade is e e ed o ou p e ious pape [6].
The basic explo a ion o he impac o dec ease in TCV
on IVCD and ðdPV/d Þmax was done by compa ing he model
ou pu s in con ol condi ions and unde TCV educed o
50% (as obse ed by Tagga e al. [4] in pa ien s a e 3
minu es o ischemia).
2.2. Model o he Ca dio ascula Sys em. To simula e he
impac o he obse ed changes in ðdPV/d Þmax and IVCD
on ca dio ascula hemodynamics and a e ial p essu e, we
educed and modified ou p e iously de eloped Windkessel
(WK) model [10] desc ibing he in e ac ion o he hea wi h
he ascula sys em. The educed e sion o he WK model
inco po a es only he unc ions o he LV and le a ium
(LA) ha a e necessa y o he simula ion o effec s in es i-
ga ed in his s udy. The elec ical equi alen scheme o he
model is illus a ed in Figu e 1. In his model, he unc ion
o a io en icula and ao ic al es is ep esen ed by he
ma ks o diodes (DAV,Da) wi h in insic esis ances (RDAV,
RDa) and he esis ance o essels agains blood flow by he
ma ks o a esis o (Ra,Rp,R ). Dis ensibili y o he indi id-
ual ypes o essels ( hei iscoelas ic compliance [11]) is ep-
esen ed by he ma ks o a capaci o (Ca1,Ca2,C )in
combina ion wi h esis o s (Ra1,Ra2), and he ine ia o blood
is symbolised by he induc o (L). The olume o blood
pumped epea edly by he LV in o he a e ial sys em c ea es
cha ac e is ic changes o a e ial p essu e known as pulse
wa es. P opaga ion o hese wa es along a e ies and he
p essu e g adien be ween a e ial and enous sys em unde -
lay he blood ci cula ion. The unc ion o he LV is based on
wo impo an mechanisms influencing he ime cou se o
blood p essu e de elopmen , he F ank–S a ling mechanism,
and he law o Laplace. Thus, he model in ol es all key
e en s affec ing sys emic blood ci cula ion and ep esen s a
mo e elabo a ed sys em han hose published p e iously
(see e iews by Zhou e al. [12] and Wes e ho e al. [13]).
2.2.1. Implemen a ion o he F ank–S a ling Mechanism. The
F ank–S a ling mechanism defines he ela ion be ween end-
dias olic olume (VVed) and s eng h o ca diac muscle con-
ac ion; i was implemen ed in o he model by means o he
ollowing 3
d
and 2
nd
o de polynomial equa ions:
PVed =aVV3
Ved,ð1Þ
PVi max =PVi max,M−bVVVed −VVed,M
ðÞ
2,ð2Þ
whe e PVed and PVi max, espec i ely, s and o end-dias olic
en icula p essu e and he iso olumic maximum
2 BioMed Resea ch In e na ional
en icula p essu e ( ha could be achie ed du ing pe sis ing
IVC a a gi en end-dias olic olume VVed), and PVi max,M ep-
esen s he peak alue o PVi max (275mmHg) achie able a
VVed o 200 ml (VVed,M). The ela ed poin s (VVed,M,PVed,M)
and (VVed,M,PVi max,M) (see Figu e 2) we e hen used o com-
pu e pa ame e s aVand bV om he ela ions:
aV=PVed,M
V3
Ved,M
,
bV=PVi max,M
V2
Ved,M
:
ð3Þ
2.2.2. Implemen a ion o he Law o Laplace. LV is simplified in
his model o a sphe ical shape wi h inne adius and wall
hickness h. Consis en ly wi h he law o Laplace, he in e nal
p essu e PVinduced by no mal s ess σVin he wall o he
model ( o mula ed below) was compu ed as
PV=σVAV,ð4Þ
whe e, om he condi ion o o ce equilib a ion, i ollows
ha
AV=2h
+h

2
:ð5Þ
The app oxima ion o he LV by a sphe e allows us o
exp ess he olume o LV ca i y as
VV=4
3π 3,ð6Þ
and he olume o LV wall (hea muscle) as
Vm=4
3π +h
ðÞ
3−4
3π 3=4
3π3 2h+3 h2+h3

:ð7Þ
By combining equa ions (6) and (7), we ob ain a cubic
equa ion:
Vm
VV
=3h
+3 h

2
+h

3
:ð8Þ
The eal oo h/ in equa ion (8) can be hen exp essed as
h
=Vm
VV
+1

1/3
−1, ð9Þ
which allows us o o mula e AVas a unc ion o Vmand VV
in he o m
AV=2 Vm
VV
+1

1/3
−1
"#
+Vm
VV
+1

1/3
−1
"#
2
:ð10Þ
As Vmis cons an du ing he whole hea cycle (muscles
consis o 95% o incomp essible wa e ) and VVdec eases
a e he opening o he ao ic al e, he inc ease o AV esul -
ing om equa ion (10) con ibu es o he ise o PVdu ing he
ejec ion phase.
2.2.3. Implemen a ion o Muscle Con ac ion and Relaxa ion.
Fo he ma hema ical o mula ion o he muscle con ac ion
and elaxa ion du ing one ca diac cycle, we used he ollow-
ing unc ion in he model:
Vc = exp −abs − Vmax
ðÞ
kV1
½
iV1 −abs − Vmax
ðÞ
kV2
½
iV2
no
,
ð11Þ
whe e cons an s kV1 and kV2 and exponen s iV1 and iV2 con-
ol he con ac ion/ elaxa ion a e and Vmax is he ime om
he o igin o he exci a ion (in he sinoa ial node) o he
maximal con ac ion o LV. The de elopmen o s ess σV
in he en icle wall du ing he ca diac cycle was desc ibed
by he ollowing equa ion:
σV=aVV3
V
AV
+PVi max,M−bVVVed −VVed,M
ðÞ
2−aVV3
V
AV
VcKVc Ve,
ð12Þ
whe e KVc ep esen s a coefficien o con ac ili y (1 in con-
ol condi ions) which eflec s he le el o neu al ac i i y
and fi ness o he hea and Ve is a unc ion ha educes
Ra1 Ra2
R
Rp
Ra
DAV DaL
PAPVPa c
Ca1
QaC
Ca2
LA LV P
Pa
Figu e 1: Elec ical equi alen scheme o he model o le hea and sys emic ci cula ion. The indi idual symbols in he scheme s and o he
le a ium and en icle (LA, LV); a io en icula and ao ic al es (DAV,Da); ine ia o blood (L); esis ance agains he blood flow in ao a,
in pe iphe al essels, and in he e minal pa o he enous sys em (Ra,Rp,R ); iscoelas ic compliance o he ini ial segmen o ao ic a ch
(Ca1,Ra1) and o he emaining ao a (Ca2,Ra2); and elas ic compliance o he e minal pa o he enous sys em (C ). The symbols PA,PV,
Pa c,Pa, and P s and o he p essu es in he le a ium, le en icle, ao ic a ch, ao a, and enous sys em. Q
a
ep esen s blood flow in ao a.
The alues o indi idual pa ame e s a e specified in Table 1.
3BioMed Resea ch In e na ional
σVdu ing he ejec ion along wi h he dec ease o VVand,
hence, s e ch o muscle fib es. This unc ion was o mula ed
o ensu e he physiological ime cou se o PV[15] and alues
o dias olic and sys olic a e ial p essu es du ing a s eady
ca diac cycle unde con ol condi ions [16]. I s ma hema i-
cal o m is
Ve =1−1
Ke
−ln VV
VVed

ie
,ð13Þ
and nume ical alues o pa ame e s Keand iea e specified in
Table 1.
An analogical app oach as used o he o mula ion o LV
unc ion was applied o desc ibe he unc ion o LA. How-
e e , because a es he con ibu ion o LA o he pe o -
mance o no mal le hea is small [17], he desc ip ion o
LA was simplified. The ela ion be ween LA p essu e (PA)
and olume (VA) du ing he LA filling was o mula ed by
means o 5
h
o de polynomial equa ion:
PA=aAV5
A,ð14Þ
whe e
aA=PA,M
V5
A,M
,
VA,M= 100 ml,
PA,M= 30 mmHg:
ð15Þ
The de elopmen o PAdu ing he whole ca diac cycle
was hen desc ibed by he equa ion:
PA=aAV5
A+ Ac7:5−bAVA−VA,M
ðÞ
2

,ð16Þ
whe e bA=0:00075 mmHg/ml2and Ac ep esen LA con-
ac ion defined by he e m:
Ac = exp −abs − Amax
ðÞ
kA
½
iA
no
:ð17Þ
He e, cons an kAand exponen iAcon ol he con ac-
ion/ elaxa ion a e o LA and Amax is he ime om he o i-
gin o he exci a ion (in he sinoa ial node) o he maximal
con ac ion o LA.
The pa ame e s o he WK model (see Table 1) we e
ecu si ely op imised by he leas squa e me hod using no -
malised diffe ences be ween he model ou pu s and he
equi ed alues (see Table 2—s anda d) o make he model
capable o mimic he physiological p ope ies o he human
ca dio ascula sys em. The co e o he model consis ing
300
250
200
150
100
50
0
020
40 60 80 100
Volume (ml)
P essu e (mmHg)
120 140 160 180 200
PVed
PVi max
(VVed,M’ PVed,M)
(VVed,M’ PVi max,M)
Figu e 2: P essu e- olume diag am showing he passi e end-dias olic p essu e- olume cu e (PVed e sus VVed) and iso olumic maxima
cu e (PVi max e sus VVed) o mula ed in he WK model o eflec alues in he human LV [14]. The poin s (VVed,M,PVed,M) and (VVed,M,
PVi max,M) ep esen alues a heo e ically maximal dias olic filling.
Table 1: Pa ame e s o he WK model.
RDAV 0.012 mmHg·s/ml∗KVc 1
RDa 0.025 mmHg·s/ml∗kV1 5.68722
Ra1 0.05 mmHg·s/ml kV2 5.2270
Ra2 0.026 mmHg·s/ml iV1 2.0224
Ra0.0001 mmHg·s/ml iV2 9.11538
Rp1 mmHg·s/ml Vmax 0.3568 s
R 0.01 mmHg·s/ml Ke1.355
Ca1 0.08 ml/mmHg ie0.35
Ca2 1.3 ml/mmHg kA24
C 70 ml/mmHg iA7
L0.0003 mmHg·s
2
/ml Amax 0.12 s
∗Valid only o open s a e. Unde closed s a e (when PV>PAo Pa c >PV),
he co esponding esis ance (RDAV o RDa
)
is se o 10
4
mmHg·s/ml.
4 BioMed Resea ch In e na ional
om 6 diffe en ial and wo algeb aic equa ions is p esen ed
in he appendix. The model was implemen ed in he com-
pu a ional sys em MATLAB14A-Simulink (Ma hWo ks,
Inc.). The nume ical compu a ion o he sys em o diffe -
en ial equa ions was pe o med using sol e ODE-45 (wi h
absolu e and ela i e e o s se o 10
-4
and 5·10
-6
, espec-
i ely). To ob ain s eady cycles unde con ol condi ions
o dec eased TCV, he model was un o 60 s o equi a-
len eal ime; p olonga ion o he simula ion ime o
120 s did no change he model ou pu alues by mo e
han 0.01%. The s abili y o he model was es ed by un-
ning he model a pa ame e s changed by 30 and 50%,
specifically hose ela ed o ca diac con ac ili y (K c),
physical p ope ies o he al es (RDAV
,
RDa), and o he
essels (Ra1,Ra2,Ra,Rp,R ,Ca1,Ca2,C ,L). In all he
cases, he model con e ged and s eady cycles we e achie ed
wi hin 60 s.
3. Resul s
3.1. Impac o Dec eased T ansmu al Conduc ion Veloci y
on he Func ion o he Le Ven icle du ing Iso olumic
Con ac ion. To explo e he impac o dec eased TCV on
unc ion o le en icle du ing IVC, we used ou FE model
o LV and simula ed he de elopmen o in a en icula
p essu e and unde lying changes in wall s ess unde con ol
condi ions, and when TCV was slowed by 50% (see me hods
o de ailed explana ion). The esul s illus a ed in
Figu e 3(a) show ha such dec ease in TCV would cause
an inc ease o IVCD om 60 o 71 ms and a sligh educ ion
o ðdPV/d Þmax om 1780 o 1750 mmHg/s. Fo compa ison,
Figu e 3(a) includes also wo clinically measu ed no mal
p essu e aces (digi ized om li e a u e [18, 19]) which
demons a e a good ag eemen be ween ou FE model and
clinical obse a ions. Besides he changes in alues o he
pa ame e s de i ed om he p essu e aces, an inc eased
wall s ess was de ec ed in he endoca dial and midmyoca -
dial laye s o he LV a he end o IVC (Figu e 3(b)). This
ele a ion o wall s ess was eflec ed by an inc ease o S
EIVC om 441 o 466 mJ (by 6%) indica ing highe ene ge ic
demands o IVC when TCV was slowed.
3.2. Impac o Dec eased T ansmu al Conduc ion Veloci y on
Le Ven icula Pe o mance and Blood P essu e. The analy-
sis desc ibed in he p e ious sec ion indica es ha 50%
educ ion o TCV causes a significan p olonga ion o IVCD
by 18% and a small educ ion o ðdPV/d Þmax by 2%. To
inco po a e his effec in o he WK model, we inc eased he
Vmax o 0.3885 and educed KVc o 0.982. Such change con-
sis en ly esul ed in he inc ease o IVCD ( om 60 o 71ms)
and educ ion o ðdPV/d Þmax om 1783 o 1750mmHg/s
du ing he fi s cycle. The consequences o hese changes
on ca dio ascula hemodynamics and a e ial p essu e in a
s eady cycle (a e 60 s o 1.2 Hz s imula ion) a e illus a ed
in Figu e 4. The simula ions e eal ha hese changes impli-
ca e a delayed and weakened con ac ion (uppe g aph), wi h
consequences o ime dis ibu ion and magni ude o LV and
a e ial p essu es (middle g aph), and o LV powe (WLV)
compu ed om he a ea o he loop in he PV–VVdiag am
(bo om g aph). The quan i a i e analysis o his effec sum-
ma ized in Table 2 shows a educ ion o EF, ca diac ou pu
(CO), and WLV by ~2, 2, and 4%, espec i ely, and he con-
sequen dec ease o he sys olic and dias olic a e ial p es-
su es (Pa,sand Pa,d) by 2 and 1%, espec i ely.
To assess he ins an aneous impac o TCV educ ion on
IVCD, ðdPV/d Þmax, and SEIVC in g ea e de ail, we epea ed
he simula ions wi h he FE model using TCV alues be ween
100 and 10%. The esul s p esen ed in Figu e 5 show ha a
educ ion o TCV om 100 o 50% caused a nea ly linea
inc ease o IVCD and dec ease o ðdPV/d Þmax. Howe e , u -
he educ ion o TCV below 50% caused a highly nonlinea
change o bo h o hese con ac ili y indexes. On he o he
hand, he s ain ene gy exhibi ed app oxima ely linea
dependence on TCV in he whole ange o he explo ed
alues. Adjus ing he WK model o alues o IVCD and
ðdPV/d Þmax ha we e ob ained by he FE model a TCV
educed o 30, 20, and 10% o he con ol alue esul ed in
a educ ion o WLV and CO, espec i ely, by ~7, 20, and
41% and by ~4, 10, and 23%. Consequen ly, he Pa,sand
Pa,ddec eased, espec i ely, by ~4, 10, and 22% and by ~3,
9, and 20% gi ing alues Pa,s/Pa,do 120/78, 113/73, and
98/64.
To sum up, hese simula ions sugges ha he isola ed
impac o TCV on LV pe o mance is a he small when
TCV is educed om 100 o 50% o i s con ol alue bu ha
i inc eases p og essi ely unde u he educ ion o TCV.
4. Discussion
Ca diac CV is a pa ame e de e mining he eloci y o depo-
la iza ion wa e p opaga ion h ough he myoca dium. As
he exci a ion is apidly dis ibu ed o he whole inne endo-
ca dial laye by he ca diac conduc ion sys em and ex ensi e
Table 2: Pa ame e s ep esen ing ca dio ascula hemodynamics
and LV pe o mance in a s eady cycle ob ained om li e a u e
(S anda d), om he model unde con ol condi ions (Con ol),
and a TCV dec eased o 50% (50% TCV).
S anda d Con ol 50% TCV
Pa,s 120 mmHg 125 mmHg 122 mmHg
Pa,d 80 mmHg 80 mmHg 79 mmHg
dPV/d
ðÞ
max 1780 mmHg/s 1783 mmHg/s 1751 mmHg/s
VV,ed 120 ml 114 ml 114 ml
VV,es 40 ml 36 ml 37 ml
IVCD 60 ml 60 ms 71 ms
EPD 210 ml 211 ms 213 ms
EF 67% 69% 67%
CO 5600 ml/min 5653 ml/min 5538 ml/min
WLV 1.5 W 1.51 W 1.45 W
Pa,s: sys olic p essu e in he ao a; Pa,d: dias olic p essu e in he ao a; VV,ed:
end-dias olic olume in he LV; VV,es: end-sys olic olume in he LV; EPD:
du a ion o ejec ion phase; EF: ejec ion ac ion; CO: ca diac ou pu ; WLV:
powe o he LV. The s anda d alues o pa ame e s we e aken om [6, 21].
5BioMed Resea ch In e na ional

ne o Pu kinje fib es in human LV [22], he c i ical ac o
esponsible o he p opaga ion o exci a ion h ough he
en icula wall is TCV. Al hough he CV and he ela ed
TCV ha e been obse ed o dec ease in diseased human
hea s [1, 2, 4], i is no clea how much his dec ease con-
ibu es o he educ ion o LV con ac ili y. To answe his
ques ion, we used ou p e iously published FE model o
human LV and pe o med simula ions showing he effec
o slowed TCV on ðdPV/d Þmax and IVCD. Subsequen ly,
he impac o he changes o ðdPV/d Þmax and IVCD—in-
duced in he FE model by he lowe TCV—on he ca dio as-
cula hemodynamics and he a e ial p essu e was simula ed
using a modified e sion o ou lumped model o sys emic
ca dio ascula ci cui .
4.1. Causes o Slowed T ansmu al Conduc ion Veloci y in he
Ca diac Le Ven icle. In p inciple, he TCV is de e mined
by he a e o local depola isa ion o ca diomyocy es and by
he a e o exci a ion p opaga ion be ween hem (in ans e -
sal di ec ion). These wo de e minan s o TCV a e closely
ela ed o he ampli ude o as Na
+
cu en (INa) in en ic-
ula myocy es, o hei memb ane capaci ance (Cm), and o
hei ans e sal esis ance (R ) con olled by ca diac gap
junc ions (mainly o med by connexin43, Cx43) which eal-
ize he cell- o-cell couplings. Changes o hese h ee ac o s
unde lying slowed TCV ha e been obse ed in a a ie y o
pa hophysiological condi ions. Fi s ly, INa is educed by he
impai ed unc ion o Na
+
channels ha a ise clinically du ing
hea ailu e, ischemia, achyca dia, o as a consequence o
ea men wi h class I an ia hy hmic d ugs [5]. Such educ-
ion may be also induced by Na
+
channel mu a ions ha
occu in Lenèg e disease, B ugada synd ome, sick sinus syn-
d ome, and a ial fib illa ion [5, 23]. Secondly, Cmis usually
subs an ially inc eased unde en icula hype ophy which
eflec s he hickening and elonga ion o en icula myo-
cy es. This change is known o be induced by hype ension
[24], al ula disease (mi al al e egu gi a ion o ao ic
al e s enosis [25, 26]), congeni al hea disease (such as pa -
en duc us a e iosus o coa c a ion o he ao a [27, 28]), and
a p ima y disease o he myoca dium which di ec ly cause
hype ophy (hype ophic ca diomyopa hy [29]). Finally,
R may inc ease due o gap junc ion decoupling ollowing
ischemia, fib o ic change o he hea issue (e.g., a e myo-
ca dial in a c ion) [30], o as a esul o mu a ions o genes
encoding gap junc ion p o ein connexions [31]. Besides,
down egula ion and dephospho yla ion o Cx43 ha e been
epo ed o con ibu e o an inc ease o R and hus o a
slowe p opaga ion o exci a ion h ough he LV wall in ail-
ing hea s [1, 3].
4.2. Effec o Slowed T ansmu al Conduc ion Veloci y on he
Func ion o he Ca dio ascula Sys em. The simula ions on
he FE model sugges ha he isola ed educ ion o TCV
esul s in a p olonga ion o IVCD and dec ease o
ðdPV/d Þmax. To explo e he effec o hese wo changes on
he LV pe o mance and ca dio ascula hemodynamic, he
modified o m o ou WK model [10] was used (see
Figu e 1). An analysis o he WK model pa ame e s showed
ha he abo e effec obse ed in he FE model could be ep-
lica ed mos effec i ely by an inc ease o Vmax which con ols
he ime o ac i a ion o LV con ac ion (equa ion (11)), and
by a dec ease o coefficien KVc which de e mines he s eng h
0 1020304050607080
Time (ms)
0
2
4
6
8
10
12
P essu e (kPa)
Con ol
50% TCV
Manolas (2015)
Cu iss (1975)
(dP/d )max = 1780 mmHg/s
IVC ime = 60 ms
(dP/d )max = 1750 mmHg/s
IVC ime = 71 ms
(a)
0 20406080100
Wall dep h (%)
40
45
50
55
60
65
Cauchy s ess (kPa)
Con ol
50% TCV
(b)
Figu e 3: (a) Effec o a 50% dec ease o TCV on p essu e ise in he LV du ing IVC. P essu e de elopmen ob ained in he con ol simula ion
is compa ed wi h wo no mal p essu e aces (in g ey) digi ized om li e a u e [18, 19]. (b) Effec o a 50% dec ease o TCV on dis ibu ion o
s esses (in he di ec ion o myofib es) ac oss he LV wall ( om endoca dium—0% o epica dium—100%) in he end o IVC.
6 BioMed Resea ch In e na ional
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
0,0
0,2
0,4
0,6
0,8
1,0
Vc x KVc
(s)
𝛥 Vmax
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
0
20
40
60
80
100
120
140
PV, PA, Pa (mmHg)
(s)
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140
VV (ml)
PV (mmHg)
Figu e 4: Simula ion o unc ion o sys emic ca dio ascula ci cui du ing a s eady cycle a es ing hea a e (72 bea s/min, s imula ion
in e al 0.8333 s) in con ol condi ions (black lines) and a e inco po a ion o changes (inc ease o Vmax and dec ease o KVc) esul ing in
p olonga ion o IVCD and educ ion o ðdPV/d Þmax (g ey lines); hese changes we e ob ained by means o he FE model a e dec ease o
TCV o 50%. The aces in he uppe g aph ep esen he ime cou ses o LV con ac ion, and Δ Vmax ep esen s a delay o he maximum
con ac ion unde he educed TCV agains con ol condi ions. The middle g aph shows he ime cou se o de elopmen o PV(solid), PA
(do ed), and Pa(dashed) unde bo h explo ed condi ions; he small black e ical lines ma k he beginning and end o IVC. The bo om
g aph shows he co esponding PV–VVdiag ams wi h hei loop a ea ep esen ing he LV s oke wo k; o compa ison wi h PV–VV
diag am measu ed in no mal human LV see Figu e 12.2 in [20].
7BioMed Resea ch In e na ional
o ca diac muscle con ac ion (equa ion (12)). Implemen a ion
o he effec s o 50% educ ion o TCV (i.e., ~18% inc ease o
IVCD and ~2% dec ease o ðdPV/d Þmax, see Figu e 3(a)) in
he WK model affec ed i s beha iou only mode a ely: EF
and CO dec eased by 2%, WLV by 4%, and he effec on Pa,s
and Pa,dwassmall.Howe e ,i isimpo an oemphasize ha
in ac he e alua ed impac s on bo h SEIVC and WLV sum up.
While he FE model shows an inc ease o SEIVC by 6%, he WK
model shows a dec ease o WLV by 4% unde hese condi ions.
I means ha du ing con ac ion, he LV consumes mo e
ene gy o de elop wall s ess bu i s con ac ile powe declines.
Consequen ly, he esul ing efficiency o he hea con ac ion
dec eases app oxima ely by 10%; clea ly, his is only ue when
heinc easeo ene gyconsump iondu ing heejec ionphase
(no included in he FE model) is p opo ional o ha du ing
IVC. In any case, such a dec ease o efficiency o hea con ac-
ion may be significan o he efficiency o blood supply, espe-
cially in combina ion wi h some o he pa hologies impai ing
he LV unc ion. The simula ions also p edic ha he abo e
desc ibed effec s o lowe TCV would inc ease subs an ially i
TCV d opped unde 50%. The eason o his inc ease was a
con inuous ise o SEIVC (Figu e 5) and a p og essi e educ-
ion o WLV (see Sec ion 3.2). Hence, he educ ion o TCV o
30, 20, and 10% o con ol alue esul ed in a dec ease o con-
ac ion efficacy o LV by 16, 29, and nea ly 50%, espec i ely.
The desc ibed effec s a e ully consis en wi h he ecen
wo k by Yunia i and Lim [7] p esen ing simula ions based
on an elec omechanical model o canine hea coupled
wi h a lumped model o ci cula o y sys em. Thei esul s
also showed an inc ease in he elec ical ac i a ion ime
(equi alen o IVCD) and in end-sys olic olume wi h
educ ion o he CV, while sys olic p essu e, s oke olume,
and s oke wo k dec eased a he mode a ely. All hese en-
dencies co espond o hose depic ed in Figu es 3 and 4 and
sugges ha clinically ele an educ ion o TCV does no
affec c i ically he unc ion o he ca dio ascula sys em
unde no mal condi ions.
4.3. Clinical Implica ions. The educ ion o CV is usually mi -
o ed by he inc eased du a ion o QRS complex in ECG
eco ds. QRS p olonga ion (>120 ms) is a significan p edic-
o o LV sys olic dys unc ion in pa ien s wi h hea ailu e
[32] and is known o be accompanied by highe p opensi y
o a hy hmias [33, 34]. On he o he hand, because QRS
can be affec ed by diso de s o ca diac elec ical conduc ion
sys em (e.g., by le bundle b anch block), he p olonged
QRS does no necessa ily mean ha in a en icula CV is
slowed down. To unambiguously diffe en ia e be ween he
causes o QRS p olonga ion, new diagnos ic me hods allow-
ing o moni o LV ac i a ion pa e n [35, 36] would be e y
020
40 60 80 100
TCV (%)
60
80
100
120
140
IVCD (ms)
(a)
020
40 60 80 100
TCV (%)
1300
1400
1500
1600
1700
1800
(dP /d )max (mmHg/s)
(b)
020
40 60 80 100
TCV (%)
440
460
480
500
520
SEIVC (mJ)
(c)
Figu e 5: Impac o TCV educ ion on h ee indexes cha ac e izing le en icle con ac ili y and ene ge ic demands o IVC in he model:
(a) IVCD, (b) ðdPV/d Þmax, and (c) s ain ene gy. 100% TCV ep esen s a no mal human LV.
8 BioMed Resea ch In e na ional
help ul in clinical p ac ice. The possibili y o di ec ly iden i y
a educed TCV and knowledge o i s ela ion o ca diac con-
ac ion efficiency migh be an impulse o he de elopmen
o new and mo e effec i e he apies a ge ed o no maliza ion
o in a en icula sp ead o exci a ion in pa ien s wi h ca -
diac disease. Besides epe usion o hea issue, his could
in ol e also a po en ia ion o up egula ion o some mem-
b ane anspo e s (e.g., sodium channels o gap junc ion
channels) which could lead o no maliza ion o cellula exci -
abili y and in e cellula elec ical conduc ance. A u u e mo e
elabo a ed e sion o he model inco po a ing cell- o-cell
elec ical in e ac ion could be also help ul o mapping o
a hy hmogenic subs a e in he myoca dium in pa ien s
wi h a he edi a y ca diac disease such as B ugada synd ome.
4.4. Limi a ions o he Model. The FE model used in his
s udy is based on an idealized (ellipsoidal) geome y o he
LV. I also employs a simplified elec ical ac i a ion pa e n
aking in o conside a ion he p opaga ion o he elec ical
signal only in he ansmu al di ec ion; consequen ly, he
en i e endoca dium is ac i a ed simul aneously. Ne e he-
less, assuming ha p opaga ion o depola isa ion a ound
he LV ca i y is much as e han in he ansmu al di ec ion
[37], his ep esen s a easonable app oxima ion. Also, he
passi e mechanical beha iou o human myoca dium is
o ho opic [38] a he han ans e sely iso opic as applied
in ou model; hus, u he imp o emen could be achie ed
by employing an o ho opic hype elas ic model, e.g., ha
p oposed by Holzap el and Ogden [39]. Finally, besides a
d ama ic ansmu al a ia ion, mode a e changes in fib e
di ec ion ha e been obse ed also in ci cum e en ial di ec-
ion and be ween base and apex [9]. These mino a ia ions
a e no included in ou model. We belie e he men ioned
limi a ions may change he esul s quan i a i ely bu wi hou
a significan impac on he d awn conclusions.
5. Conclusions
On he basis o combina ion o wo compu a ional models,
FE model o le en icle and WK model o ca dio ascula
hemodynamics, he p esen ed s udy sugges s ha he pump-
ing efficacy o human hea dec eases wi h lowe TCV due o
a highe ene gy consump ion and lowe LV powe . Al hough
he obse ed changes induced by he clinically ele an
educ ion o TCV a e no c i ical o heal hy hea , hey
may ep esen an impo an ac o limi ing ca diac unc ion
when combined wi h o he pa hologies impai ing con ac il-
i y o he LV. As nume ous hea pa hologies a e associa ed
wi h TCV educ ion, u he explo a ion o he impac o
TCV on he con ac ili y o diseased hea s is needed.
Appendix
Di e en ial Equa ions o he WK Model
P essu e induced by he elas ic componen o ao ic a ch
dPCa1
d =PV−Pa c
ðÞ
/RDa −Qa
Ca1
,ðA:1Þ
whe e
Pa c =PV
Ra1
RDa +Ra1
+PCa1
RDa
RDa +Ra1
−Qa
Ra1RDa
RDa +Ra1
:ðA:2Þ
P essu e induced by he elas ic componen o he ao a
dPCa2
d =Qa−Pa−P
ðÞ
/Rp
Ca2
,ðA:3Þ
whe e
Pa=P
Ra2
Rp+Ra2
+PCa2
Rp
Rp+Ra2
+Qa
RpRa2
Rp+Ra2
:ðA:4Þ
P essu e induced by he elas ic componen o he enous
sys em
dP
d =Pa−P
ðÞ
/Rp−P −PA
ðÞ
/R
C
:ðA:5Þ
Blood flow h ough he ao a
dQa
d =Pa c −Pa−RaQa
ðÞ
L:ðA:6Þ
Volume o he le a ium
dVA
d =P −PA
R
−PA−PV
RDAV
:ðA:7Þ
Volume o he le en icle
dVV
d =PA−PV
RDAV
−PV−Pa c
RDa
:ðA:8Þ
Da a A ailabili y
The da ase s gene a ed and analysed du ing he cu en s udy
a e a ailable om he co esponding au ho upon eques .
Con lic s o In e es
The au ho s decla e ha he e is no conflic o in e es
ega ding he publica ion o his pape .
Acknowledgmen s
This wo k was suppo ed h ough NETME CENTRE
PLUS (LO1202) by financial means om he Minis y o
Educa ion, You h and Spo s unde he “Na ional Sus ain-
abili y P og amme I”and h ough ins i u ional suppo
RVO: 61388998.
Re e ences
[1] A. V. Glukho , V. V. Fedo o , P. W. Kalish e al., “Conduc ion
emodeling in human end-s age nonischemic le en icula
9BioMed Resea ch In e na ional