Impact of Decreased Transmural Conduction Velocity on the Function of the Human Left Ventricle: A Simulation Study

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