Chaos 30, 103122 (2020); https://doi.org/10.1063/5.0022066 30, 103122

© 2020 Author(s).

Effect of constant-DI pacing on single cellpacing dynamicsCite as: Chaos 30, 103122 (2020); https://doi.org/10.1063/5.0022066Submitted: 16 July 2020 . Accepted: 02 October 2020 . Published Online: 21 October 2020

P. Parthiban , S. Newell , and E. G. Tolkacheva

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Effect of constant-DI pacing on single cell pacingdynamics

Cite as: Chaos 30, 103122 (2020); doi: 10.1063/5.0022066

Submitted: 16 July 2020 · Accepted: 2 October 2020 ·

Published Online: 21 October 2020 View Online Export Citation CrossMark

P. Parthiban, S. Newell,a) and E. G. Tolkacheva

AFFILIATIONS

Department of Biomedical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA

a)Author to whom correspondence should be addressed: newel127@umn.edu

ABSTRACT

Cardiac alternans, beat-to-beat alternations in action potential duration, is a precursor to fatal arrhythmias such as ventricular fibrillation.Previous research has shown that voltage driven alternans can be suppressed by application of a constant diastolic interval (DI) pacingprotocol. However, the effect of constant-DI pacing on cardiac cell dynamics and its interaction with the intracellular calcium cycle remainsto be determined. Therefore, we aimed to examine the effects of constant-DI pacing on the dynamical behavior of a single-cell numericalmodel of cardiac action potential and the influence of voltage–calcium (V–Ca) coupling on it. Single cell dynamics were analyzed in thevicinity of the bifurcation point using a hybrid pacing protocol, a combination of constant-basic cycle length (BCL) and constant-DI pacing.We demonstrated that in a small region beneath the bifurcation point, constant-DI pacing caused the cardiac cell to remain alternans-freeafter switching to the constant-BCL pacing, thus introducing a region of bistability (RB). The size of the RB increased with stronger V–Cacoupling and was diminished with weaker V–Ca coupling. Overall, our findings demonstrate that the application of constant-DI pacingon cardiac cells with strong V–Ca coupling may induce permanent changes to cardiac cell dynamics increasing the utility of constant-DIpacing.

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Cardiac alternans, or the beat-to-beat alternation in action poten-tial duration (APD), is a precursor to fatal arrhythmias such asventricular fibrillation (VF). Previous studies have establisheda credible link between alternans and initiation of ventriculararrhythmias.1–4 Hence, it is hypothesized that the eliminationof alternans could prevent VF and consequent arrhythmias inthe heart. Several attempts have been made over the past twodecades to suppress alternans in various in silico, in vitro, and ex

vivo models.5–12 Pacing protocols with a constant diastolic inter-val (DI) have suppressed voltage driven alternans. However, theeffects of constant-DI pacing on cardiac cell dynamics remainunexplored.

I. INTRODUCTION

Cardiac alternans is often considered to be a precursor to tachy-cardia, fibrillation, and other dangerous heart arrhythmias.1–4 Thus,many strategies to eliminate alternans have been developed. Specif-ically, delayed feedback control was proposed to control periodic

systems, such as in human patients with pacing induced period-2 atrioventricular-nodal conduction alternans13 or small pieces ofin vitro paced bullfrog myocardium with APD alternans.14 Delayedfeedback control often applies various modifications to periodicstimulation, otherwise known as constant-basic cycle length (BCL)pacing,

BCL = APDn + DIn, (1)

where APDn and DIn are the APD and DI following nth stimuli.When cardiac cells are paced with a constant-BCL protocol [Eq. (1)],a normal 1:1 response occurs at long BCL [Fig. 1(a)], and alternans(2:2 response) occurs at shorter BCL, where APD alternates betweenlong and short values, APDlong and APDshort [Fig. 1(b)].15

Recently, we and others proposed and demonstrated the effec-tiveness of constant-DI pacing to eliminate alternans and establishstable 1:1 rhythms in single-cell and one-dimensional models.6,8,10

However, the formation of alternans of cardiac action potentialsunder constant-DI pacing has been demonstrated in the presenceof altered intracellular Ca2+ cycling.6 Specifically, at rapid heartrates (low BCL), the capacity of cardiac cells to cycle intracellu-lar Ca2+ is exceeded. Thus, sarcoplasmic reticulum (SR) calcium

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FIG. 1. (a) Normal 1:1 response and (b) alternans duringperiodic pacing [Eq. (1)].

accumulates and the non-linear relationship between SR calciumload and SR calcium release becomes steep. This results in Ca tran-sient alternans, leading to a cascade of abnormal Ca2+-sensitivecurrents that cause APD alternans.6,16,17 Based on these findings,alternans was classified into two types: voltage (V)-driven and cal-cium (Ca)-driven alternans, whose mechanisms are associated withdirect instabilities in transmembrane V and abnormal intracellularCa2+ cycling, respectively. This is related to the fact that Ca and Vare bi-directionally coupled in the cell through a variety of intra-cellular mechanisms. The most notable of these mechanisms is thesodium–calcium exchanger (NCX). Other mechanisms to considerinclude the L-type calcium current (ICa,L) and Ca-induced inacti-vation of ICa,L. Due to the coupling of V and Ca, the developmentof alternans in the Ca cycle can cause alternans in the V cycle andvice versa. It has been suggested earlier6 that constant-DI pacing canonly suppress V-driven, but not Ca-driven alternans. However, theeffect of constant-DI pacing on cardiac cell dynamics has never beeninvestigated.

In this paper, we determined the effect of constant-DI pac-ing on the dynamic behavior of isolated cardiac cells close to thebifurcation by using numerical simulations of a physiological modelof cardiac action potential. We also investigated the effect of volt-age–calcium (V–Ca) coupling strength on the alternans formationand single cell dynamics.

II. METHODS

A. Mathematical model

A Mahajan–Shiferaw (MS) rabbit ventricular single-cellmodel18 that incorporates a well-described intracellular Ca2+ cycle19

at rapid heart rates was used. The NCX conductance (gNaCa) wasadjusted to alter the V–Ca coupling strength as shown in Table I.The slope of the SR calcium release function (u) was used to provideeither V-driven (u = 9.5 s−1) or Ca-driven (u = 11.3 s−1) alternans, asdescribed in Refs. 18 and 19.

TABLE I. The NCX conductance (gNaCa) for each V–Ca coupling strength

(taken from Ref. 18).

V–Ca coupling Extra strong Strong Normal Weak

gNaCa (uM/s) 0.75 0.80 0.84 0.88

B. Hybrid pacing protocol

In order to investigate whether constant-DI pacing altered thedynamical behavior of the paced cardiac cell, we implemented ahybrid pacing protocol consisting of four sequences:

(1) First, constant-BCL pacing with BCL = BCL1 was applied to thecell [see Eq. (1) and Fig. 2(a), red arrows] for 5000 stimuli [bluein Figs. 2(c) and 2(d)] to reach a steady state. Steady state APDand DI (APDSS and DISS) were calculated as the average APDand DI of the last 10 stimuli.

(2) Second, constant-DI pacing [see Fig. 2(b), red arrows] wasapplied for 5000 stimuli [black in Figs. 2(c) and 2(d)]. In orderto do that, the end of each APD was measured in real time, andstimuli were applied after a predetermined DIconst, which wascalculated as

DIconst = BCL1 − APDSS. (2)

Steady state APD and DI (APDSS and DISS) was calculatedagain, and an equivalent BCL (BCLeq) was computed as belowto compare constant-BCL and constant-DI pacing,

BCLeq = DIconst + APDSS. (3)

(3) Third, constant-BCP pacing with perturbation (δ) was appliedfor 100 stimuli [green in Figs. 2(c) and 2(d)] at BCL2 immedi-ately after the constant-DI sequence,

BCL2 = BCL1 + δ. (4)

Figures 2(c) and 2(d) illustrate the hybrid pacing protocolwithout perturbation (δ = 0) and with perturbation (δ 6= 0),respectively. The presence of perturbation (δ 6= 0) allowed us toevaluate the reversibility of changes to the dynamical behaviorof the cardiac cell caused by constant-DI pacing.

(4) Last, constant-BCL pacing at BCL = BCL1 was applied for 4900stimuli [red in Figs. 2(c) and 2(d)] to return the system to asteady state.

C. Analysis

At the end of each pacing sequence, steady state APDSS wascalculated at the 90% repolarization level. Bifurcation diagramswere constructed by plotting APDSS as a function of BCL (duringconstant-BCL pacing) or BCLeq (during constant-DI pacing). Dur-ing alternans, APDlong and APDshort [see Fig. 1(b)] were calculatedas the respective means of odd and even beats from the last ten

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FIG. 2. (a) Constant-BCL pacing: stimuli (red arrow) areapplied at BCL1 [see Eq. (1)]. (b) Constant-DI pacing: stimuli(red arrow) are applied at the end of a predetermined DIconst[see Eq. (2)]. (c) Hybrid pacing without perturbation: 5000 stim-uli are applied at BCL1 (blue) and DIconst (black); then 100stimuli are applied at BCL2 = BCL1 (green), i.e., δ = 0ms inEq. (4); finally, 4900 stimuli are applied at BCL1 (red). (d)Hybrid pacing with perturbation: Similar to (c), but BCL2 = BCL1−40ms (green), i.e., δ = −40ms in Eq. (4).

FIG. 3. Bifurcation diagrams constructed with normal V–Ca coupling for constant-BCL [(a) and (c)] or constant-DI [(b) and (d)] pacing in the case of V-driven [(a) and (b)]and Ca-driven [(c) and (d)] alternans. Arrows indicate the onset of alternans BCLonset = 213.5 ms (a), 241ms (c), and 223ms (d). No alternans is present in (b).

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FIG. 4. (a) Overlap of bifurcation diagrams from hybrid pacing without perturbation: constant-BCL pacing at BCL1 before (blue) and after (red) constant-DI pacing. Notethe formation of alternans at BCLonset = 213.5 ms for constant-BCL pacing before but not after constant-DI pacing. (b) Response of the cardiac cell model to hybrid pacingwithout perturbation (i.e., δ = 0) at three regions that were chosen as shown in (a) depending on the value of BCL1 with respect to BCLonset : (1) BCL1 � BCLonset , (2)BCL1 ∼ BCLonset , and (3) BCL1 � BCLonset .

steady state responses. The magnitude of alternans was computedas follows:

1APDa = |APDlong − APDshort|, (5)

and 1APDa = 5 ms was set as the alternans threshold. The onset ofalternans was defined as the smallest value of BCL (BCLonset) thatresulted in a 1:1 response before the transition into alternans.

III. RESULTS

A. Constant-DI pacing only suppresses V-driven

alternans

The cardiac cell with either V-driven or Ca-driven alternanswas paced using constant-BCL or constant-DI protocols, and corre-sponding bifurcation diagrams were then constructed. As expected,constant-BCL pacing resulted in the formation of alternans, both

TABLE II. Dynamical responses of the cardiac cell for different regions identified

in Fig. 4.

Region Constant BCL1 Constant DI Constant BCL1

(1) BCL1 � BCLonset 2:2 1:1 2:2(2) BCL1 ∼ BCLonset 2:2 1:1 1:1(3) BCL1 � BCLonset 1:1 1:1 1:1

V-driven [Fig. 3(a), BCLonset = 213.5 ms] and Ca-driven [Fig. 3(c),BCLonset = 241 ms]. The application of constant-DI pacing only sup-pressed V-driven but not Ca-driven alternans [Figs. 3(b) and 3(d)],in agreement with previous studies.8,10 Nevertheless, constant-DIpacing decreased the BCLonset for Ca-driven alternans, from 241 msto 223 ms. The model was set to the V-driven configuration for theremainder of the study because alternans was not eliminated duringthe Ca-driven configuration.

B. Constant-DI pacing introduces a region of

bistability (RB)

In order to investigate the effect of constant-DI pacing on thedynamic responses of the cardiac cell, we applied the hybrid pac-ing protocol without perturbation (δ = 0, i.e., BCL1 → DIconst →

BCL1). Figure 4(a) shows the two overlapping bifurcation diagrams:before (blue) and after (red) application of constant-DI pacing.Note that application of constant-DI pacing shifted BCLonset to alower value [see Fig. 4(a)]. Three different regions for constant-BCL pacing following constant-DI pacing were identified based onproximity to the BCLonset prior to constant-DI pacing [marked inFig. 4(a)]: (1) region 1, where BCL1 � BCLonset, (2) region 2, whereBCL1 ∼ BCLonset, and (3) region 3, where BCL1 � BCLonset. Rep-resentative values of BCL1 were chosen from each region − (1)BCL1 = 208 ms, (2) BCL1 = 213.5 ms, and (3) BCL1 = 219 ms.

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FIG. 5. The effect of V–Ca coupling (see Table I) on the bifurcation diagram of acardiac cell under constant-BCL pacing. The BCLonset (arrows) shifts to lower val-ues of BCL as V–Ca coupling increases from weak (BCLonset = 218ms) to normal(BCLonset = 213.5 ms) to strong (BCLonset = 211ms).

The responses of the cell to hybrid pacing are shown in Fig. 4(b)for the three different regions. When BCL1 � BCLonset, the cardiaccell exhibits alternans at the end of the first constant-BCL segment.Alternans is suppressed when constant-DI pacing is applied butreappears upon application of the second constant-BCL segment.In contrast, when BCL1 ∼ BCLonset, the alternans at the end of thefirst constant-BCL segment is suppressed by applying constant-DIpacing, and subsequent constant-BCL pacing fails to re-initiate it.Such behavior suggests the presence of a region of bistability (RB),i.e., a range of BCLs in which the system can have both an alter-nans and a 1:1 response during constant-BCL pacing. Our resultsalso demonstrate that constant-DI pacing affects the response of thesystem to constant-BCL pacing. For BCL1 � BCLonset, the systemexhibits a stable 1:1 response independent of the pacing sequence.Therefore, our results indicate that constant-DI pacing stabilizes thedynamics of cardiac cells by eliminating alternans. These results aresummarized in Table II.

C. V–Ca coupling strength affects the RB

Next, we sought to investigate the effect of V–Ca couplingstrength on BCLonset. The overlapping bifurcation diagrams forthe down-sweep constant-BCL protocol with weak (green), normal

(blue), and strong (red) V–Ca coupling (see Table I) are shown inFig. 5. Note that increasing (decreasing) V–Ca coupling strengthdecreased (increased) BCLonset to 211 ms (218 ms) from the normalvalue of BCLonset = 213.5 ms.

We then applied the hybrid pacing without perturbation(δ = 0; i.e., BCL1 → DIconst → BCL1) at varying levels of V–Cacoupling (see Table I). Figures 6(a) and 6(b) show the changein the bifurcation diagrams obtained during constant-BCL pacingbefore (blue) and after (red) constant-DI pacing for strong and weakV–Ca coupling, respectively. Note the presence of a RB = 3 ms forstrong V–Ca coupling due to the different values of BCLonset beforeand after constant-DI pacing. This is not the case for weak V–Cacoupling, where no RB is observed [see Fig. 6(b)]. Figure 6(c) com-pares RBs for different V–Ca coupling strengths. Note that the RBincreases with increasing V–Ca coupling. Figure 6(d) demonstratesthe relative shift in BCLonset as a function of V–Ca coupling strength.As the V–Ca coupling increases, the difference between BCLonset

before and after application of constant-DI protocol increases, thusincreasing RB. Overall, this suggests that increasing V–Ca couplingfacilitates changes in the cell dynamics that are triggered by theapplication of constant-DI pacing.

D. Reversibility of a bistable response is variable

Finally, we investigated if the RB triggered by constant-DI pac-ing can be reversed by perturbing the system. To achieve this, weapplied the hybrid protocol with perturbation (δ 6= 0; i.e., BCL1 →

DIconst → BCL2 → BCL1). In Fig. 7, 1APD (APDSS − APDlong

and APDSS − APDshort) is plotted as a function of the value of δ

for representative BCLs chosen from the RB for normal (blue) andstrong V–Ca coupling (red) (BCL1 = 213.5 ms and BCL1 = 209 ms,respectively). Note that a RB did not exist for weak V–Ca couplingand, therefore, was not plotted. Changes in the cell dynamics werereversed (i.e., the cell returned to alternans) when the value of δ

reached a critical value (δcrit); see Table III. The inset in Fig. 7 fur-ther indicates that the transition from a 1:1 response to alternansis qualitatively different for different V–Ca coupling: normal V–Cacoupling resulted in a gradual increase in 1APD close to BCLonset

FIG. 6. Overlap of bifurcation diagrams from hybrid pac-ing without perturbation (i.e., δ = 0): constant-BCL pacing atBCL1 before (blue) and after (red) constant-DI pacing for (a)strong V–Ca coupling and (b) weak V–Ca coupling. RB is indi-cated by the dashed box in A. (c) RB as a function of differentV–Ca coupling strengths. (d) The shift of BCLonset before (blue)and after (red) constant-DI pacing for varying V–Ca couplingstrengths.

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FIG. 7. 1APD as a function of δ when the hybrid pacing protocol with perturba-tion was applied for the case of strong (red) and normal (blue) V–Ca coupling.The inset shows the region of transition from 1:1 into alternans for the cardiac cellwith normal V–Ca coupling.

before a sharp increase in 1APD resulting in alternans, while strongV–Ca coupling abruptly transitioned into alternans.

Next, the reversibility of the dynamic responses is explored forthree different BCL1 (208, 209, and 210 ms) within the RB for strongV–Ca coupling in Fig. 6(a). Figure 8(a) demonstrates three overlap-ping bifurcation diagrams for the BCL1, and Fig. 8(b) demonstrates1APD as a function of δ. Moving BCL1 closer to BCLonset (211 ms)resulted in the δcrit increasing from 3.6 ms (BCL1 = 208 ms) to 5.5 ms(BCL1 = 209 ms). At BCL1 = 210 ms, the system did not exhibit alter-nans (green) for any value of δ. Also, the sharpness in transition toalternans increased as BCL1 moved closer to BCLonset (comparingyellow and red traces).

IV. DISCUSSION

In this study, we first reiterated that constant-DI pacing onlysuppresses V-driven alternans.8,10 We then investigated the responseof the cardiac cell model to constant-DI pacing in the V-drivenand demonstrated that: (1) Constant-DI pacing introduces a RB in

TABLE III. δcrit required for the system to exhibit alternans after application

of constant-DI pacing during strong V–Ca coupling. The respective maximum value

of 1APDa given for each BCL1.

BCL1 (ms) δcrit (ms) max(1APDa) (ms)

208 −3.6 24.9209 −5.5 20210 . . . 0

the model. (2) The RB increases at stronger V–Ca coupling andis diminished at weaker V–Ca coupling. That is to say, increas-ing V–Ca coupling facilitates changes in the cell dynamics that aretriggered by constant-DI pacing. (3) The RB can be reversed byperturbing the system. The magnitude of perturbation required toreverse the RB increases as V–Ca coupling increases and/or thedifference between BCL1 and BCLonset increases.

A RB has not previously been observed in cardiac tis-sue models. This phenomenon may be cardiac memory relatedbecause constant-DI pacing conditionally stabilizes ionic mecha-nisms related to cardiac memory. This new finding encouragesfuture research into constant-DI pacing. Moreover, the RB wasaltered through changes in V–Ca coupling strength allowing forpotentially further stabilization benefits.

Previous studies exhibited that coupling between V and Camay lead to unstable voltage dynamics under preserved APDrestitution.19 In our study, we explored the possibility of the con-verse hypothesis where V–Ca coupling stabilizes alternating APDby creating a larger RB. In theory, under the condition of strongV–Ca coupling, the NCX is highly activated. Hence, the extrusionof Ca under rapid pacing becomes easier. This might reduce theSR calcium load and prevent instability in the intracellular Ca2+

concentration at rapid rates. This would explain why the onset ofalternans gets pushed further with increased coupling strengths andthe RB grows in size with increased V–Ca coupling. The same mech-anism might enable the persistence of a normal rhythm for a longerperiod under a strong V–Ca coupling condition.

As previously mentioned, V–Ca coupling strength is modu-lated in the MS model by altering the conductance of the NCX.In large mammals, the NCX is a major calcium extrusion pathwayand responsible for an efficient balance between calcium extrusionrequired for relaxation and availability of calcium for an SR uptake.This in turn determines the strength of the consecutive beat. Alsonote that the NCX is an electrogenic transporter that carries onepositive charge into the cell for the extrusion of every Ca2+ ion. Inthe case of strong V–Ca coupling, where we decrease the conduc-tance of NCX, the amplitude of the inward current decreases. Hence,the contribution of NCX during the early repolarization phase isblunted, repolarization happens sooner, and the APD decreases.Please see Fig. S3 in the supplementary material where this rela-tionship is clearly illustrated. From a global perspective, this allowsthe cell to relax better and have more time for local APD adap-tations in order to stabilize APD alternans under rapid pacing.Similarly, during weak V–Ca coupling when we increase the con-ductance of the NCX, the peak of the inward current increases,causing a depolarizing current while the cell is trying to repolarize.This increases APD and makes the cell prone to unstable rhythmsat rapid pacing rates (see Fig. S3 in the supplementary material).However, it must be noted that changing the NCX conductancesignificantly impacts calcium dynamics. For example, strong cou-pling increases the amplitude of Ca transients (see Fig. S3 in thesupplementary material), and a significant increase in the couplingstrength could lead to Ca alternans at the cost of stabilizing theAPD alternans. Hence, an intricate balance between V–Ca cou-pling strength and the pacing interval (BCL) is required to opti-mize the benefits of V–Ca coupling strength toward constant-DIpacing.

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FIG. 8. (a) Fig. 6(a) with three BCL1 within the RB (Table III)clearly labeled. (b) The value of alternans1APD as a functionof δ during the hybrid pacing protocol with perturbation for eachBCL1 listed in Table III.

Several limitations of our study need to be regarded. Ouranalysis was conducted on numerical simulations, and there is nocurrent experimental evidence available. Hence, the results need tobe validated in experimental models. However, experimental imple-mentation of constant-DI pacing remains to be a challenge. We alsostudied the effects of constant-DI pacing on a single rabbit ventric-ular cell model. Our results for Figs. 3 and 4 were replicated on theten Tusscher and Panfilov ventricular epicardial single-cell model(see Figs. S1 and S2 in the supplementary material) allowing us toconfirm that our results are not limited to rabbit ventricular cellmodels. However, our results may not be present in all models as thedynamics vary across models due to differences in ionic modelingand calcium cycling. Tissue models may also have different dynam-ics due to cell coupling, conduction velocity restitution, and otherhigher-order tissue capacities. Moreover, the formulation of V–Cacoupling dynamics in the primary model studied is single-facetedand simplifies the real physiological situation of the complex inter-play of Ca2+ currents. Also, we have not analyzed the consequence ofour analysis on various other Ca2+-sensitive ionic currents (such asCa2+-sensitive Cl− and K+ currents). However, the nature of actionpotentials under all tested coupling strengths reflected physiologicmorphology validating the relevance of our findings.

Future researchers should look into Ca-alternans in the pres-ence of varying V–Ca coupling strengths. In this study, we do notanalyze Ca-alternans since we are primarily concerned about themechanism of suppressing V-alternans. However, it has been estab-lished that the restitution of the calcium transient is independentlyinfluenced by other factors besides the DI. Hence, the possibility ofCa-alternans under cases of monotonic APD restitution and conse-quence of V–Ca coupling changes on these Ca-alternans need to bepursued in future studies. A detailed analysis of the nature of bifur-cation under different coupling strengths needs to be conducted.Type association (smooth vs border collision bifurcation) with cou-pling related changes could open new avenues to understandingalternans propagation.

We reinstate that cellular alternans has been proven to be thebasis of T-wave alternans, which is clinically associated with VF.4

Furthermore, instabilities in the Ca2+ cycle have shown to aggravateventricular tachycardia into fibrillation.16 Hence, the importanceof pacing strategies to eliminate alternans is paramount and theexploration of such techniques needs to account for interconnectedvoltage and calcium dynamics. Our study proves that the intricatebalance between constant-DI pacing and V–Ca coupling has thepotential to permanently remodel the cardiac cell. Thus, it has a

strong potential even for short term periods of pacing. Future studiesin this direction will definitely provide better insight into alternanssuppression.

V. CONCLUSION

In this study, we examined the effect of constant-DI pacingon cardiac cell dynamics. For this, we designed a hybrid pacingprotocol with the capability to switch between constant-BCL andconstant-DI pacing. We compared the state of the system beforeand immediately after the constant-DI pacing sequence. We intro-duced short perturbations during the transition to briefly disturbthe constant-DI condition and test the robustness of the inducedchanges. The model of the cardiac cell used for this study allowsone to dynamically adjust the V–Ca coupling strength and ana-lyze the effect of the calcium cycle on changes in APD. We showthat in a region close to the bifurcation point, constant-DI remod-els the cardiac cell. In the absence of perturbations in this region,the cell does not revert to alternans even after the constant-DI con-dition is removed. Increasing the V–Ca coupling strength increasesthe size of this region. At higher coupling strengths, the changes inAPD are irreversible despite large perturbations. Our study is lim-ited to single-cell analysis and confined changes to V–Ca couplingsince we alter only the NCX activity. Future research should con-sider investigating invasive changes to the intracellular Ca2+ cycleby exploring changes to other Ca transporters and expanding theanalysis to tissue level studies. Regardless, our results emphasizethat the interaction between constant-DI pacing and V–Ca couplingprovides insight into alternans propagation and elimination.

SUPPLEMENTARY MATERIAL

See the supplementary material for results on the ten Tusscherand Panfilov ventricular epicardial single-cell model.20

AUTHORS’ CONTRIBUTIONS

P.P. and S.N. contributed equally to this work.

ACKNOWLEDGMENTS

This study was funded by the National Science Foundation(NSF) DCSD (No. 1662250) as well as IEM UMN seed grants. Theoriginal model code for the rabbit ventricular cell was provided by

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Y. Shiferaw. The original model code for the ten Tusscher and Pan-filov ventricular epicardial single-cell model was downloaded fromK. ten Tusscher’s source code website.

DATA AVAILABILITY

The data that support the findings of this study are availablewithin the article.

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