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Towards the Emulation of the Cardiac ConductionSystem for Pacemaker Testing

Eugene Yip, Sidharta Andalam, Partha S. Roop, Avinash Malik, Mark Trew, Weiwei Ai, and Nitish Patel

Abstract—The heart is a vital organ that relies on the orches-trated propagation of electrical stimuli to coordinate each heartbeat. Abnormalities in the heart’s electrical behaviour can bemanaged with a cardiac pacemaker. Recently, the closed-looptesting of pacemakers with an emulation (real-time simulation)of the heart has been proposed. An emulated heart wouldprovide realistic reactions to the pacemaker as if it were a realheart. This enables developers to interrogate their pacemakerdesign without having to engage in costly or lengthy clinicaltrials. Many high-fidelity heart models have been developed, butare too computationally intensive to be simulated in real-time.Heart models, designed specifically for the closed-loop testing ofpacemakers, are too abstract to be useful in the testing of physicalpacemakers.

In the context of pacemaker testing, this paper presentsa more computationally efficient heart model that generatesrealistic continuous-time electrical signals. The heart model iscomposed of cardiac cells that are connected by paths. Significantimprovements were made to an existing cardiac cell model tostabilise its activation behaviour and to an existing path modelto capture the behaviour of continuous electrical propagation.We provide simulation results that show our ability to faithfullymodel complex re-entrant circuits (that cause arrhythmia) thatexisting heart models can not.

Index Terms—cardiac, electrophysiology, emulation, hybrid,automata, modelling.

I. I NTRODUCTION

The human heart is a vital organ and is responsible forpumping blood around the body to other vital organs. Patientscan develop abnormal cardiac behaviour, such as bradycardia(slow heart rate). Cardiac pacemakers can treat bradycardia bymonitoring the patient’s heart and delivering electrical stimulito the heart when needed. Pacemakers are life-critical medicaldevices that must be certified against stringent safety stan-dards, such as IEC 60601-1 [1]. Certification is a costly andtime consuming process, yet 1,210 computer-related recallsfor medical devices were reported to the US Food and DrugAdministration between 2006 and 2011 [2].

Pacemakers must be validated by clinical trials as part of thecertification process. This requires the pacemaker to be testedin closed-loop with a patient’s heart. Since clinical trials arethe only times when a pacemaker is tested on a real heart, theyprovide a glimpse of how well the pacemaker performs in thereal world. Clinical trials are usually performed late in theproduct development phase, because they are costly and time

E. Yip was and S. Andalam, P. S. Roop, A. Malik, W. Ai, and N. Patel arewith the Department of Electrical and Computer Engineering, the Universityof Auckland, New Zealand. M. Trew is with the Auckland BioengineeringInstitute, the University of Auckland, New Zealand.

E-mails:{eyip002, wai484}@aucklanduni.ac.nz and{sid.andalam, p.roop,avinash.malik, nd.patel, m.trew}@auckland.ac.nz

consuming to manage. Thus, issues found during a clinicaltrial may be costly and time consuming to fix and a newclinical trial may be required to re-evaluate the pacemaker.Some limitations of clinical trials include: potentially smallsample of patients that are not representative of the generalpopulation, difficulty in recruiting patients with specificheartconditions, difficulty in interrogating a patient’s heart to betterunderstand design issues, and inherent risk to the patients.

Recently, the emulation of the heart has been proposed tofacilitate the closed-loop testing of pacemakers [3]. Emulationis the real-time simulation of a heart model that can react toa pacemaker’s electrical shocks and also output the heart’selectrical activities for the pacemaker to sense. High-fidelityheart models provide realistic behaviour but are computation-ally intensive [4], [5], thus, precluding them from emulation.The following benefits can be gained if high-fidelity heartmodels can be emulated: cheaper and quicker testing thanwith clinical trials, earlier testing of pacemakers in closed-loop in the development phase and outside of clinics, greatertesting coverage by emulating a range of heart conditions,better understanding of design issues by interrogating theemulated heart (e.g., replaying problematic test cases), andhaving minimal risk to the patients. We envision the use ofemulated hearts alongside clinical trials to help accelerate thecertification process.

In the context of testing cardiac pacemakers, a heart modelshould possess the following properties:

• Abstraction: The model focusses on the important as-pects by ignoring irrelevant details. For example, thecardiac conduction system is the most important aspectbecause it is responsible for coordinating the heart’selectrical activities. Irrelevant details may include hemo-dynamics (e.g., blood flow), mechanics (e.g., musclemovement), and chemistry (e.g., cellular reactions).

• Accuracy: The model faithfully represents the cardiacconduction system and demonstrates realistic behaviours.A high-fidelity model provides an accurate reflection ofreality but requires high computational power. A lowerfidelity model requires less computational power but atthe risk of providing an inaccurate reflection of reality.

• Prediction: The model can answer questions about a realheart, such as “How does the heart respond when settingX of the pacemaker is used?”

• Inexpensiveness:The model should be cheaper and fasterto construct and use the emulated heart than to conducta clinical trial.

The heart models of Chen et al. [6], Jiang et al. [7], and

2

Mery and Singh [8] consider just the emergent features ofthe cardiac conduction system, which is composed of millionsof cells. They model the conduction system as a static, two-dimensional, sparse network of cardiac cells. Jiang et al. [3]also developed a hardware prototype that emulates the cardiacconduction system as discrete events. When the logic of apacemaker’s software is tested in closed-loop with a heartmodel, it may be sufficient to use a heart model that producesand responds to discrete events [7], [8]. However, when aphysical implementation of the pacemaker is tested in closed-loop with a heart model, it is necessary to use a heart modelthat produces and responds to continuous-time signals. Thisis because the physical pacemaker expects a real heart as itsenvironment. The heart model of Chen et al. [6] simulatesthe conduction system as continuous-time signals. However,the signals are too abstract and bear little resemblance withreality. Thus, the model lacks the accuracy and, therefore,thepredictive power.

A. Contributions

This paper reviews the state-of-the-art heart models [3], [6]–[9] that have been designed specifically for the closed-looptesting of cardiac pacemakers. Without introducing significantcomputational complexity, we propose significant improve-ments to the modelling of the cardiac conduction system tocreate a heart model that produces realistic continuous-timeelectrical signals that a pacemaker would sense. Our modelfaithfully models forward and backward conduction, which isessential in the modelling of complex re-entrant circuits [10]–[12] that cause arrhythmia (abnormal heart rate). Our primarycontributions are:

• We develop a continuous-time model of the conductionsystem as a two-dimensional network of cardiac cells.Each cell produces an accurate continuous-time signalthat represents its electrical activities. These signals arepropagated continuously along the paths between thecells. Complex conduction behaviours, such as arrhyth-mias caused by re-entrant circuits, can be reproducedfaithfully by our model. Our heart model is easily cus-tomised by modifying various parameters of the conduc-tion system.

• Each cardiac cell in our heart model is based on thehybrid automaton developed by Ye et al. [13]. We havegreatly improved the design of the hybrid automatonto overcome the following limitations: the cell becomesunstable when it is stimulated in quick succession, andthe cell is too sensitive to electrical stimulation fromits neighbours. Our improvements are elaborated in Sec-tion IV.

• Each path in our heart model is modelled with timedautomata. The path model is inspired by that of Jianget al. [7] that was designed to propagate discrete eventsrather than continuous-time signals. Our path model iselaborated in Section V.

• We demonstrate in Section VII that a MathWorksR©

Simulink R© and StateflowR© implementation of our heartmodel can simulate a wide range of heart conditions withrealistic results.

Fig. 1. Schematic of the heart and conduction system.

B. Paper Layout

Section II provides a background to the cardiac conductionsystem and the important features of electrical activitiesthatare sensed by a pacemaker. Section III reviews the state-of-the-art heart models for closed-loop testing of pacemakers.Section IV reviews the computationally efficient hybrid au-tomata model of cardiac cells developed by Ye et al. [13].We identify the limitations encountered with Ye et al.’s modelduring simulation and how we corrected them. Section Vdescribes our path model that handles the propagation ofcontinuous-time signals. In Section VI, we create our proposedheart model by composing instances of our cardiac cell andpath models into a network that replicates the conductionpathway. Section VII evaluates the capabilities of our proposedheart model with the recent heart model of Chen et al. [6].Section VIII concludes this paper and discusses future workfor improving the proposed heart model.

II. BACKGROUND

The heart pumps blood around the body in a rhythmicmanner. Figure 1 is a schematic of the heart and showsits four chambers: the right and left atriums and ventricles.The right atrium and ventricle are responsible for pumpingdeoxygenated blood through the lungs, while the left atriumand ventricle are responsible for pumping oxygenated bloodthrough the body. The contractions of the chambers are coordi-nated by electrical stimuli that propagate throughout the heart’sconduction system. The conduction pathways are shown inFigure 1 as solid black lines with dots at important locations.The names of these locations are labelled with an acronymand their full forms are given in Table I.

A. Cardiac Cycle

This section describes the major actions of the heart duringone cardiac cycle (one heart beat) with the help of Figure 2.In the first phase of the cardiac cycle, Figure 2a, the sinoa-trial (SA) node generates an electrical stimulus that spreadsquickly throughout the right and left atriums. This causes

3

TABLE IFULL NAMES OF NODES ALONG THE CONDUCTION PATHWAYS.

AV Atrioventricular LVS Left ventricular septum

BB Bachmann’s bundle OC Os cordis

BH Bundle of His RA Right atrium

CS Coronary sinus RBB Right bundle branch

CT Crista terminalis RV Right ventricle

FP Fast path RVA Right ventricular apex

LA Left atrium RVS Right ventricular septum

LBB Left bundle branch SA Sinoatrial

LV Left ventricle SP Slow path

LVA Left ventricular apex

(a) Phase 1 (b) Phase 2

(c) Phase 3 (d) Phase 4

Fig. 2. Phases of the cardiac cycle. Adapted from http://philschatz.com/anatomy-book/contents/m46664.html#sinoatrial-sa-node.

the atriums to contract, pumping blood from the atriums intothe ventricles. In the second phase, Figure 2b, the electricalstimulus reaches the atrioventricular (AV) node and is delayedmomentarily before it continues down into the ventricles.This delay is very important because it gives the atriumsenough time to contract and fully fill the (relaxed) ventricles.In the third phase, Figure 2c, the electrical stimulus reachesthe right and left ventricular apexes and travels out to thefast conducting Purkinje fibers. This causes the right and leftventricles to contract and pump out blood. In the fourth phase,Figure 2d, the ventricles relax after pumping out all theirblood.

In a normal heart, each cardiac cycle begins from theSA node, which generates periodic electrical stimuli thatspread through the conduction system. The following sectionsdescribe the genesis of the heart’s electrical activity andhowit appears to a pacemaker. Finally, we describe some common

Fig. 3. Phases of the action potential. Adapted from [14].

arrhythmias that a heart model should aim to capture.

B. Action Potentials of Cardiac Cells

Most of the heart’s electrical activities, that a pacemakersenses, are generated by the myocytes (muscle cells) [15]. Acell’s electrical activities result from the movement of ionsacross its membrane, creating potential differences. The cell’selectrical response to an electrical stimulus is describedbyits action potential[16]. Figure 3 shows the four phases ofa typical action potential, which plots the cell’s membranepotential over time. In theresting phase, the cell is inactiveand has a resting potential of approximately−85mV . Thecell enters thestimulated phasewhen excited electrically byits neighbours or by an artificial pacemaker. The cell returnsto the resting phase if its membrane potential fails to crossthe threshold voltageVT of approximately−40mV whenthe excitation stops. Otherwise, the cell enters theupstrokephaseanddepolarisesby allowing ions to move rapidly acrossits membrane, causing its membrane potential to reach anovershoot voltageVO of approximately+45mV . Then the cellenters theplateau and early repolarisation phase. The cellcontracts and starts torepolarise, i.e., its membrane potentialstarts to return to its resting potential. When the membranepotential is less than the voltageVR of approximately−55mV ,the cell has relaxed and returned to the resting phase.

All cardiac cells can only respond to subsequent excitationsin the later portion of its action potential, called therelative re-fractory period. However, the membrane potential must crossa higher threshold voltage. Figure 4 shows a normal actionpotential at 0ms and some possible secondary excitationsbetween160 and300ms. For a secondary excitation at180ms,the resulting action potential has a lower overshoot voltageVO and a shorter action potential duration. The secondaryexcitation at300ms results in a more normal action potentialbecause the cell has rested for a longer period.

Prominent biophysical cardiac cell models, which explainthe genesis of action potentials in terms of ionic flow, in-clude Luo-Rudy [16] and Hodgkin-Huxley [18]. Althoughbiophysical models have high-fidelity, they are computation-ally intensive. Ye et al. [13], [19] create computationallyefficient cardiac cell models by considering just the emergentfeatures of the biophysical models, i.e., the action potential

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Fig. 4. Dynamic behaviour of secondary excitations. Adapted from [17].

and its dynamic response to secondary excitation. It shouldbenoted that the action potential duration of a human ventricularmyocyte is approximately twice that of an atrial myocyte.

C. Action Potentials and the Electrogram

A key function of any pacemaker is to sense the heart’selectrical activity, by using one or more electrodes attachedto the inside of the heart wall. The electrical activity ofthe cardiac cells in the electrode’s immediate vicinity aresensed most strongly. A recording of the sensed activities iscalled an electrogram (EGM) [20]. Figure 5a shows threeaction potentials and their corresponding EGMs. To helpunderstand the EGM, Figure 5b shows that the EGM deflectsup and down whenever an electrical wavefront passes underthe electrodes [15]. The faster that the wavefront passes, thesteeper the deflection. In Figure 5a, two distinct deflections canbe seen in each EGM and they correspond with the upstrokeand resting phases of their respective action potential.

A heart model that produces distorted action potentials willalso produce distorted EGMs. Such distorted EGMs cannot beused to reliably test a pacemaker’s ability to discern the timingof important cardiac activities. Moreover, the predictivepowerof a heart model is compromised when the action potentialsare distorted. For example, a heart model with accurate actionpotentials might predict that a cell goes into its upstroke phasebecause its neighbours’ voltages are high enough. However,a heart model with distorted action potentials might insteadpredict that the cell returns to its resting phase because itsneighbours’ voltages are too low. Thus, arrhythmia would bepredicted incorrectly.

D. Common Arrhythmias

Arrhythmias can be caused by abnormalities in the gener-ation and propagation of action potentials through the con-duction system. The abnormalities may be due to congenitaldefects, side-effects of medication, or cell death. The following

(a) Three action potentials (APs) from different regions ofthe heart andtheir corresponding electrograms (EGMs). Adapted from [21].

(b) EGM deflections due to a travelling electrical wave-front. Adapted from [15].

Fig. 5. Electrograms (EGMs).

are some common arrhythmias [11], [12], [20] that a heartmodel for pacemaker testing should aim to capture:

• Heart block: This occurs when electrical stimuli has dif-ficulty propagating through the AV node. The propagationof the stimuli may be delayed for longer than usual ormay be prevented from propagating altogether.

• AV node re-entrant tachycardia: This occurs whena re-entry circuit forms around the AV node, causingtachycardia.

• Bundle branch block: This occurs when electrical stim-uli travels slower or not at all down one of the bundlebranches.

• Wolff-Parkinson-White syndrome: This occurs whenthere is an extra conduction pathway between the atriumsand ventricles. The extra pathway allows electrical stimulito bypass the AV node and create a feedback loopbetween the atriums and ventricles.

• Long Q-T syndrome: This occurs when the repolarisa-tion of the ventricles is delayed, i.e., their action potentialdurations are longer than usual.

• VA conduction: This occurs when electrical stimuli fromthe ventricles conduct backwards through the conductionpathways and into the atriums.

Pacemakers can also cause arrhythmias when they areunable to correctly sense the timing of the heart’s electricalactivities. For example, pacemaker-mediated tachycardiaiscaused by the pacemaker inadvertently conducting electricalstimuli from the ventricles back to the atriums. Pacemakersthat deliver electrical stimuli that are not synchronised with

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TABLE IIQUALITATIVE COMPARISON OF HEART MODELS THAT ARE DESIGNED FORTESTING CARDIAC MEDICAL DEVICES. AP = ACTION POTENTIAL. HA =

HYBRID AUTOMATA . TA = TIMED AUTOMATA .

Reality Less Abstract← Heart Models → More Abstract

Real Heart [22] Hi-Fi [23], [24] UoA Oxford [6] UPenn [3], [7] LORIA [8] MES [9]

CellModel

Continuous APsfrom biophysicalprocesses [25]

Continuous APsfrom biophysical

models [25]

Continuous APsfrom improvedStony Brook

HA [13]

Continuous APsfrom simplified

Stony BrookHA [19]

Discrete APsfrom TA

Discrete APsfrom logico-mathematics

Continuous AVsignal generators

mimic wholeheart electrical

activityPathModel

Continuouspropagations

from biophysicalprocesses [10]

Continuouspropagations

from reaction-diffusion

equations [25]

Continuouspropagationsfrom TA andcontribution

function

Continuouspropagations

fromcontribution

function

Discretepropagations

from TA

Discretepropagationsfrom cellular

automata

SpatialModel

3D tissue (layersof bundles offibers) that

deforms

3D finite-volumethat deforms 2D, static, and sparse network of cells along the conductionpathway

Black boxes ofmajor heartcomponents

the heart’s cardiac rhythm can cause the heart to fibrillate [26],i.e., twitch uncontrollably.

III. R ELATED WORK

The electrophysiology of the heart has been well re-searched [10], [27], resulting in the proposal of many theories.These theories are validated by creating high-fidelity wholeheart models [23], [24] and ascertaining if they can reproduceexperimental observations, i.e., the models are accurate,re-alistic, and predictive. These high-fidelity models are usefulin predicting the prognosis of patient-specific heart condi-tions [28] and in assisting with interventional cardiology[29].

On the other hand, abstract heart models have been de-veloped with the goal of enabling the closed-loop testingof cardiac pacemakers. Table II provides a qualitative com-parison of existing heart models. The abstract heart modelsfrom Oxford [6], UPenn [3], [7], LORIA [8], and MES [9]are designed for testing the pacemaker logic. To enable theformal verification of the pacemaker logic, Oxford, UPenn,and LORIA use hybrid automata (HA) or timed automata (TA)to develop formal models of the cardiac conduction system.UPenn and LORIA model the transitions between the resting,upstroke, and early refractory phases of the action potential asdiscrete events on a continuous timeline. These discrete eventsare propagated between cells and the propagation is eithersuccessful or unsuccessful. These abstractions result in heartmodels that may produce more behaviours than is possible bya real heart, i.e., an over-approximation. Thus, all problemsdetected during closed-loop testing must be validated againsta more concrete heart model [7].

The heart model from Oxford [6] is more concrete thanthose from UPenn and LORIA because Oxford models theaction potentials as continuous signals. Oxford uses a simpli-fied model of the cardiac cell that Ye et al. [19] developedwith hybrid automata. Oxford incorporates ag(~v) functioninto the cell model to capture the continuous electrical ac-tivity that a cell receives from its neighbours. However, theg(~v) function does not consider the directional behaviour ofelectrical propagation due to the refractory period of cardiaccells. Noting these limitations, Sections IV and V describeour

Fig. 6. Comparison of the action potentials (APs) produced by Luo-Rudy [16],Stony Brook [13], Oxford [6], and our improved version (UoA). Note thatthe AP of UoA overlaps that of Stony Brook’s because they are identical.

improvements to the cell and path models. Our heart modelcan faithfully simulate complex re-entrant circuits without asignificant increase in computational complexity.

IV. CARDIAC CELL MODEL

The Oxford heart model [6] uses a simplified version of theisolated cardiac cell model developed by Stony Brook [19].The Stony Brook model is itself a simplification of the Luo-Rudy model [16] because it only models the action potentialand its dynamic response to secondary excitation. The StonyBrook model uses three piecewise-continuous variables, calledvx, vy, and vz, to capture different features of the actionpotential. The sum of these three variables produces theaction potential. The Oxford heart model discards thevy andvz variables and retains just thevx variable. Unfortunately,Figure 6 shows that the resulting action potential is no longerrealistic and this diminishes the predictive power of Oxford’sheart model (Section II-C). Thus, for our heart model, weretain all three variables and we use an updated Stony Brookcardiac cell model [13]. This section describes the Stony Brookmodel in more detail and our improvements that overcomesome of the model’s limitations.

6

q0 : Resting & FR

vx = α0

xvx

vy = α0

yvy

vz = α0

zvz

v = vx − vy + vz

{v ≤ VR}

q1 : Stimulated

vx = α1

xvx + βxVin

vy = α1

yvy + βyVin

vz = α1

zvz + βzVin

v = vx − vy + vz

{v ≤ VT }

q2 : Upstroke

vx = α2

xvx

vy = α2

yvy

vz = α2

zvz

v = vx − vy + vz

{v ≤ VO − 80.1√θ}

q3 : Plateau & ER

vx = α3

xvx

vy = α3

yf(θ)vy

vz = α3

zvz

v = vx − vy + vz

{v ≥ VR}

[Vin 6= 0]v′x = vxv′y = vyv′z = vz

θ′ = v/VR

[Vin ≤ 0 ∧ v < VT ]v′x = vxv′y = vyv′z = vz

[v ≥ VT ]v′x = vxv′y = vyv′z = vz

[v ≥ VO − 80.1√θ]

v′x = vxv′y = vyv′z = vz

[v ≤ VR]v′x = vxv′y = vyv′z = vz

Fig. 7. Stony Brook cardiac cell model [13].

TABLE IIICOEFFICIENTS AND CONSTANTS IN THESTONY BROOK CARDIAC CELL

MODEL [13].

α0x = −0.0087 α1

x = −0.0236 α2x = −0.0069 α3

x = −0.0332

α0y = −0.1909 α1

y = −0.0455 α2y = 0.0759 α3

y = 0.0280

α0z = −0.1904 α1

z = −0.0129 α2z = 6.8265 α3

z = 0.0020

VR = 30 βx = 0.7772

VT = 44.5 βy = 0.0589

VO = 131.1 βz = 0.2766

A. Stony Brook Cardiac Cell Model

The Stony Brook model [13] models the time course ofthe action potential with thehybrid automaton(HA) shownin Figure 7. The four phases of the action potential, describedin Section II-B, are represented as four locations in the HA.In each location, the membrane potential is defined by thevariable v as the sum of the variablesvx, vy , and vz. Therates at which the variablesvx, vy, andvz change are definedby their derivativesvx, vy, andvz , respectively. The values ofthe coefficients and constants are given in Table III. Note thatthe Stony Brook model offsets all the voltages such that theresting potential is at0mV .

By default, the HA always starts in locationq0, which isthe resting phase of the action potential. It stays inq0 as longas the invariantv ≤ VR is true. The HA transitions fromlocationq0 to q1 when the voltageVin around the cell is greaterthan 0mV . During this transition, the last values ofvx, vy,and vz when leavingq0 are used to set their initial valueswhen enteringq1. Recall from Figure 4 that the amplitudeand duration of the action potential depends on how long thecell had been in the resting phase when it is stimulated. Thisisapproximated by assuming that the closer the cell’s membranepotential is to0mV , the longer the cell has been in locationq0.Thus, the time that the cell had been in the resting phase isapproximated by normalising the membrane potentialv againstthe voltageVR, i.e., θ = v/VR. In location q1, the rate at

Fig. 8. Action potentials shorten unnaturally for the StonyBrook model.Stimulated at200ms intervals

0 200 400 600 800 1,000 1,2000

50

100

150

* * * * * *

Time (ms)

v(m

V)

Fig. 9. Action potentials for the UoA model when stimulated at 200msintervals.

which the cell’s membrane potential increases depends on thestrength ofVin. The HA transitions back to locationq0 if thevoltageVin around the cell fails to stimulate the cell abovethe threshold voltageVT . However, if the cell’s membranevoltagev is stimulated aboveVT , then the HA transitions tolocation q2. The amplitude of the cell’s membrane potentialis calculated asVO − 80.1

√θ, i.e., it depends on how long

the cell had been in the resting phase. The HA transitions tolocation q3 and the cell’s membrane potential starts to drop.The rate of repolarisation is determined by the functionf(θ)and depends on how long the cell had been in the restingphase:

f(θ) = 0.29e62.89θ + 0.70e−10.99θ (1)

A higher value forf(θ) means a faster rate of repolarisation.Once the cell’s membrane voltagev drops below the restingvoltageVR, the HA transitions back toq0.

B. Improvement: Stabilising the Action Potential

The bottom plot in Figure 8 shows the action potentials pro-duced by the Stony Brook model when stimulated at200msintervals. We can see that the duration of the action potentialsshorten towards zero over time. This is unnatural because theaction potentials should settle to a constant duration whenstimulated at a constant interval [30]. The top three plots inFigure 8 show the values ofvx, vy, andvz , respectively, overtime. Note thatvx describes the initial voltage drop of the

7

q0 : Resting & FR

vx = α0

xvx

vy = α0

yvy

vz = α0

zvz

v = vx − vy + vz

{v ≤ VT }

q1 : Stimulated

vx = α1

xvx + βxh(~v)

vy = α1

yvy + βyh(~v)

vz = α1

zvz + βzh(~v)

v = vx − vy + vz

{v ≤ VT }

q2 : Upstroke

vx = α2

xvx

vy = α2

yvy

vz = α2

zvz

v = vx − vy + vz

{v ≤ VO − 80.1√θ}

q3 : Plateau & ER

vx = α3

xf(θ)vx

vy = α3

yf(θ)vy

vz = α3

zvz

v = vx − vy + vz

{v ≥ VR}

[h(~v) > VT ]v′x = vxv′y = vyv′z = vz

θ′ = v/VR

[h(~v) ≤ 0 ∧ v < VT ]v′x = vxv′y = vyv′z = vz

[v ≥ VT ]v′x = vxv′y = vyv′z = vz

[v ≥ VO − 80.1√θ]

v′x = vxv′y = vyv′z = vz

[v ≤ VR]v′x = vxv′y = vyv′z = vz

Fig. 10. Improved cardiac cell model (UoA).

plateau phase. We can see that thevx variable takes too long todecrease in locationq0. Thus, each time the cell is stimulated,it enters locationq1 with a slightly higher value forvx. Thiscauses the values ofθ and f(θ) to increase over time. Anincreasing value off(θ) causes a faster rate of repolarisationand, hence, shorter action potential durations.

To prevent the unnatural shortening of the action potential,the value ofvx needs to be closer to zero when the celltransitions from locationq3 to q0. Thus, we increase the rateat whichvx decreases towards zero by including the functionf(θ) in location q3 for vx. The improved HA is shown inFigure 10. Figure 9 shows that the action potentials producedby the improved HA have constant durations when stimulatedat 200ms intervals.

C. Improvement: Bounding the Rate of Repolarisation

In the Stony Brook model, the action potential durationapproaches zero if the cell was stimulated shortly after enteringlocationq0. This behaviour is unnatural because the minimumaction potential duration that a cardiac cell can achieve isapproximately40ms. This problem is due to the functionf(θ). When the HA returns to locationq0 from q3, the valueof v is VR. An immediate stimulation to locationq1 would setθ = VR/VR = 1. Consequently,f(1) = 5.96 × 1026, whichcauses an extremely fast rate of repolarisation.

To prevent such a fast rate of repolarisation from occurring,we limit the maximum value that functionf(θ) can return.Figure 11 shows that action potential durations of35 ms areproduced whenf(θ = 0.04) = 4.0395. Thus, we impose amaximum value of4.0395 for the functionf(θ):

f(θ) =

{

0.29e62.89θ + 0.70e−10.99θ if θ < 0.044.0395 if θ ≥ 0.04

D. Discussion

We improved the original Stony Brook cardiac cellmodel [13] to stabilise the action potentials and to impose

Fig. 11. Relationship between the action potential duration (APD) and thevalue off(θ).

a reasonable minimum action potential duration. Figure 9shows that the shape of the action potentials generated bythe improved model remains realistic. However, it is impor-tant to demonstrate that the action potential durations varyrealistically in response to the timing of secondary excitations(see Figure 4 for illustration). This dynamic behaviour canbeevaluated with a restitution curve [30]. To plot the restitutioncurve, the cardiac cell is electrically stimulated at a constanttime period, called the base cycle length (BCL). The BCLconsists of two time intervals: the action potential durationfollowed by thediastolic interval. The diastolic interval beginswhen the action potential falls to 10% of its peak amplitude.For a range of BCLs, the action potential duration from thetenth BCL is plotted against the diastolic interval from theninth BCL.

Figure 12 shows the restitution curves for the StonyBrook [13], Oxford [6], and our improved (UoA) mod-els. These models were implemented in MathWorksR©

Simulink R©/StateflowR©. The Oxford implementation was pro-vided by the original authors [6]. The restitution curve of theStony Brook model has been demonstrated [13] to behave real-istically compared to the Luo-Rudy model. However, becausethe Stony Brook model is unstable, we could only reproduceits restitution curve by taking the action potential duration anddiastolic interval of the first BCL. The restitution curve for ourimproved model shows dynamic behaviour that is very similarto Stony Brook. The difference is due to the changes describedin Section IV-B. The parameters in the cell model can be tunedto produce a variety of restitution curves and is particularlyuseful when modelling diseased cardiac cells. The Oxfordmodel, on the other hand, shows unrealistic behaviour. Theaction potential duration is either9ms or 98ms, dependingon whether the diastolic interval is greater or less than51ms.

V. CARDIAC PATH MODEL

The Oxford heart model [6] models the conduction systemas a sparse network of cells (Figure 1). The cells are connectedelectrically by a voltage contribution functiongk(~v), whichcalculates the voltage induced at cellk by its neighbours:

gk(~v) =

n∑

i=1

vi(t− δki)aki − vkdk (2)

8

Fig. 12. Electrical restitution curves comparing the dynamic behaviour ofLuo-Rudy [16], Stony Brook [13], Oxford [6], and our improved version(UoA).

Cell 1

Cell 2

gk(~v)

0

50

100

150

*Cell

1(m

V)

0 200 400 600 8000

50

100

150

Time (ms)

Cell

2(m

V)

Fig. 13. Two cardiac cells connected electrically by the Oxford voltagecontribution functiongk(~v).

For a given cell k with n connected neighbours,~v =(v1 · · · vn) is a vector of all the neighbours’ membrane poten-tials, such thatvi is the membrane potential of neighbouri.The membrane potential of cellk is vk. The time for celli’saction potential to propagate to cellk, called theconductiontime, is represented byδki. Thus, vi(t − δki)aki representsthe membrane potential of celli that reaches cellk after adelay of δki and with a gain ofaki. The strength of cellk’smembrane potential relative to its neighbours is taken intoaccount byvkdk, wheredk is a distance coefficient.

Figure 13 plots the action potentials of two cardiac cellsconnected by Oxford’s functiongk(~v). Only cell 1 receivesan external stimulus at10ms and the time delay between thecells is90ms. The expected behaviour is for cell1 to producean action potential that stimulates cell2 to produce an actionpotential. Cell2’s action potential would not propagate backto cell 1 because of the refractory feature of cardiac cells [11].However, Figure 13 shows that both cells produce a sequenceof action potentials. This is because the termvi(t − δki) inequation (2) requires cell2’s action potential to be propagatedback to cell 1 after a time delay. This incorrect positivefeedback behaviour is shown in Figure 13 as arrows betweenthe action potentials.

To properly model the propagation of action potentials alonga path, its behaviour in real cardiac tissue needs to be reviewed.

(a) Propagation from cell 1. (b) Propagation has reached cell 5.

(c) Another propagation from cell 10. (d) Propagations collide.

(e) Propagations annihilated. (f) The path model mimics the propa-gation along a chain of cells.

Fig. 14. Propagation and collision of electrical stimuli along cardiac tissue.

Figure 14a shows a chain of cardiac cells where only cell1has entered its upstroke phase. The cells’ membrane potentialsare plotted above the cells along with their corresponding HAlocation. In Figure 14a, cell1’s membrane potential starts tostimulate cell2 to enter the upstroke phase. Cell2 will thenstimulate its neighbour and so on. In Figure 14b, cell5 entersthe upstroke phase, while cells1 to 4 have entered the plateauand early repolarisation phase. Cells1 to 4 are unresponsive toany electrical stimuli applied to them. This refractory featureforces an action potential to propagate in one direction along apath. When two action potentials propagate towards each other(Figures 14c), they will collide (Figure 14d) and annihilateeach other (Figure 14e), i.e., the action potentials will notpass through each other.

A computationally efficient heart model can be createdby replacing chains of cardiac cells with paths that mimicthe propagation of action potentials (Figure 14f). The pathmodel of UPenn [7] does consider the refractory feature ofcardiac cells and can model the collision of electrical stimuli.However, only the propagation of discrete action potentialevents, rather than continuous-time signals, are modelled.Moreover, a cell can only be stimulated by the electricalactivity of one neighbour at a time. We propose a new pathmodel that handles the collision of continuous-time actionpotentials and calculates the overall voltage induced by a cell’sneighbours with a reaction-diffusion [10] function.

A. Improved Path Model

Our timed automaton(TA) for determining whether anaction potential can propagate along a path is shown inFigure 15a. To keep the path model simple, we assume thatthe path is only long enough for one complete action potentialto propagate through. That is, we assume that the duration ofthe propagating action potential is longer than its conduction

9

idle

annihilate

directioni

t = 0

waiti

t ≤ δij

relayi

t ≤ δj

directionj

t = 0

waitj

t ≤ δji

relayj

t ≤ δi

[api = q2∧apj = q2]

[api 6= q2 ∧ apj 6= q2]

last := 0

[api = q2∧apj 6= q2]

t := 0[last = 0 ∨ last = i]

[last = j

∧apj(t− δji) = q0]

[last = j

∧apj(t− δji) 6= q0]

[t ≥ δij]

starti!last := i

t := 0

[apj = q2]

[t ≥ δj]

[apj = q2∧api 6= q2]

t := 0[last = 0 ∨ last = j]

[last = i

∧api(t− δij) = q0]

[last = i

∧api(t− δij) 6= q0]

[t ≥ δji]

startj !last := j

t := 0

[api = q2]

[t ≥ δi]

(a) TA to determine which action potentials can propagate.

(b) TA to relay the action potential of celli at cell j. (c) TA to relay the action potential of cellj at cell i.

Fig. 15. Improved path model (UoA) consisting of a TA to detect collisions and two TA to relay the action potentials.

time through the path. An action potential from celli will failto propagate to cellj under the following circumstances:

• Cell j enters the upstroke phase during the conductiontime.

• A recent propagation from cellj to cell i failed tostimulate celli, i.e., a partial propagation, and the cellsin between have not returned back to their resting phase.

To recognise these circumstances, the TA requires the follow-ing information of the cardiac cells (Figure 10):

• api andapj: The phase of celli andj’s action potential.• api(t− δij): The phase of celli’s action potential when

it reaches cellj after a conduction time ofδij .• apj(t− δji): The phase of cellj’s action potential when

it reaches celli after a conduction time ofδji.In addition, the local variablelast is needed to remember thedirection of the last propagation:

last =

0 if the last propagation was annihilatedi if the last propagation started from cellij if the last propagation started from cellj

In Figure 15a, the left half of the TA determines if anaction potential from celli can propagate to cellj. The TAbegins in theidle location and transitions to theannihilatelocation if both cellsi and j go into the upstroke phaseq2.A transition back to theidle location is made when bothcells exit their upstroke phaseq2. If only cell i goes into theupstroke phaseq2, then a transition from theidle to directionilocation is made. Next, if a partial propagation from celljoccurred recently, then a transition to theannihilate locationis made. Otherwise, a transition to thewaiti location is made.The TA transitions to theannihilate location if cell j entersthe upstroke phaseq2 during cell i’s conduction timeδij .Otherwise, the TA remains in thewaiti location until theconduction time has elapsed, at which point the signalstarti isemitted (denoted by “!”) and a transition to therelayi location

is made. Next, a transition to theidle location is made after atime of δj has elapsed to ignore cellj’s upstroke phase. Theright half of the TA uses similar logic to determine if an actionpotential from cellj can propagate to celli.

The TA in Figure 15b relays celli’s action potential atcell j when the signalstarti is received (denoted by “?”). Therelay stops when cellj has successfully depolarised or whencell i’s action potential reaches the resting phaseq0. Similarlogic is used by the TA in Figure 15c to relay cellj’s actionpotential at celli when the signalstarti is emitted. A cell canbe connected to multiple neighbours and, therefore, receivemultiple action potentials simultaneously. We determine thevoltage induced at cellk by itsn neighbours with the followingreaction-diffusion [10] function:

hk(~v) =

n∑

i=1

Γikσik

AmCm

(vouti − vk) (3)

whereΓik is the cross-sectional area (units ofmm2) fromcell i to k, σik is the electrical conductivity (units ofmS/mm)from cell i to k, Am is cellk’s surface area to volume (units ofmm−2), Cm is cell k’s specific membrane capacitance (unitsof µF/mm2), and vouti is cell i’s membrane potential afterpropagating along the path.

B. Discussion

We have developed an improved path model specificallyfor continuous-time action potentials. We can successfullymodel propagations that are annihilated by a full or partialpropagation from the opposite direction. This allows us tomodel conduction block that only occur in one direction, e.g.,due to the source-sink relationship [11], which the UPennpath model [7] can not. Bartocci et al. [31] modelled cardiactissue with a high-resolution grid of Stony Brook cells [13]. In

10

Cell i

Path

Cell j

(a) Full and partial propagations.

Cell k

Path ik

Cell i

Path jk

Cell j

(b) Multiple neighbours needed to stimulate a cell.

Fig. 16. Examples of propagation behaviour that can be modelled by theimproved path model, but not by the Oxford [6] or UPenn [7] path models.

that model, the cells are connected electrically by a reaction-diffusion equation because the conduction times between thecells are not significant.

Figure 16a demonstrates two cellsi andj with a conductionblock from cellj to i. The conduction time between the cellsis 30ms. At 10ms, cell i generates an action potential thatsuccessfully stimulates cellj. At 260ms, cell j generatesan action potential that does not stimulate celli because ofconduction block. Shortly after at360ms, cell i generates anaction potential that does not reach cellj because the cellsalong the path have not returned back to the resting phaseq0.At 710ms, cell i generates an action potential that successfullystimulates cellj.

Figure 16b demonstrates a cellk that will only stimulateif the action potentials of its neighbours, cellsi and j, arrivetogether. At times10ms and260ms, cellsi andj, respectively,generate action potentials that take30ms to propagate tocell k. However, the voltage induced on cellk is neitherstrong enough nor long enough to stimulate cellk. At time610ms, both neighbours generate action potentials that arrivesimultaneously at cellk. This time, the voltage induced oncell k is enough to stimulate it.

Figure 17 demonstrates atrioventricular node re-entranttachycardia (AVNRT [12]), caused by dual pathways aroundthe atrioventricular (AV) node shown in Figure 17a. An electri-cal stimulus propagates faster down the left pathway than the

right pathway, called thefast andslowpathways, respectively.In normal cardiac behaviour, shown in Figure 17a, an electricalstimulus from the atriums splits and propagates down the fastand slow pathways. The stimulus in the fast pathway willreach the bottom of the AV node before the stimulus in theslow pathway. The stimulus in the fast pathway continues topropagate down into the ventricles and up the slow pathwaywhere it will annihilate the existing stimulus. Thus, only thestimulus that propagates down the fast pathway will enter theventricles. Figure 17c demonstrates our ability to replicate thisnormal behaviour by modelling the dual pathways around theAV node with only four cells and paths.

An early electrical stimulus from the atriums can endup propagating continuously around the AV node, shownin Figure 17b. This is because the early stimulus cannotpropagate down the fast pathway, which has cells that arestill repolarising. The cells in the fast pathway take longerto repolarise than those in the slow pathway and, hence, havelonger action potential durations. Due to this, the stimuluscan propagate down the slow pathway because its cells haverepolarised. By the time the stimulus reaches the bottom ofthe AV node, the cells in the fast pathway have reached theirresting phase. Thus, the stimulus continues by propagatingup the fast pathway, during which time the cells in the slowpathway reach their resting phase. The stimulus continuesby propagating back down the slow pathway and the cyclecontinues. Every time the stimulus passes the bottom and topof the AV node, a stimulus is propagated into the ventriclesand atriums, respectively. Figure 17d demonstrates our abilityto replicate this abnormal behaviour by modelling the dualpathways around the AV node with only four cells and paths.

VI. U OA HEART MODEL

Following the approach of Oxford’s heart model [6], wemodel the cardiac conduction system with a two-dimensionalnetwork of 33 cardiac cells, shown in Figure 1. Our heartmodel uses the improved cardiac cell and path models (Sec-tions IV-B, IV-C, and V-A). The following cell and pathmodel parameters were modified to capture the range ofaction potential durations, conduction velocities, and electricalconductivities present in the conduction system:

• The value ofα3

y in Figure 10 was increased to increasethe cell’s rate of repolarisation and, therefore, decreaseits action potential duration.

• The values ofδij andδji in Figure 15 were increased toincrease the conduction time and, therefore, decrease theconduction velocity.

• The value ofσik in equation (3) was increased to increasethe conductivity.

Cardiac electrophysiology data in the literature [22], [32]was used to estimate the parameters. The data of interestinclude the morphology of myocyte action potentials and theconduction velocities of electrical stimuli along the pathways.Table IV provides a qualitative comparison of Oxford’s heartmodel [6] with our improved heart model (UoA). Thanks toimprovements in our cell and path models, we are able tomodel more heart conditions and with realistic results. This is

11

(a) Schematic of the fast andslow pathways around the AVnode.

(b) Re-entry due to an early stim-ulus from the atriums.

(c) Normal behaviour simulated. Cell 1 stimulated at10 and260ms. (d) Abormal behaviour simulated. Cell 1 stimulated at10and160ms.

Fig. 17. Atrioventricular node re-entrant tachycardia.

demonstrated in the next section where we provide simulationresults of the common arrhythmias described in Section II-D.

VII. S IMULATION RESULTS

We implemented our heart model in MathWorksR©

Simulink R© and StateflowR© R2015b Pro. A fixed simulationtime step of0.0005ms and the Dormand-Prince ODE45 solverwas used. Figure 19 shows the propagation of electrical stimulithrough the cardiac conduction system (Figure 1) during anormal cardiac cycle. The membrane potential of each nodeis denoted by a shade of grey, according to Figure 18. In thefollowing, we provide simulation results of the arrhythmiasdescribed in Section II-D.

A. Heart Block

This is modelled by reducing the voltage that the neigh-bouring atrial cells induce on the AV node and by increasingthe conduction time through the AV node. Figure 20 showsa simulation of heart block where an electrical stimulus doesnot propagate through the AV node.

B. Bundle Branch Block

This is modelled similarly to heart block. For one of thebundle branches, the voltage induced by the AV node isreduced and the conduction time along the branch is increased.Figure 21 shows a simulation of bundle branch block wherepropagation down the right branch is slower than the leftbranch.

C. Long Q-T Syndrome

This is modelled by decreasing the value ofα3

y in theventricular cells to decrease their rate of repolarisation, therebycreating a longer action potential duration. Figure 22 shows asimulation of the syndrome.

D. VA Conduction

This is modelled by stimulating a ventricular node beforethe SA node generates a stimulus. Figure 23 shows a simu-lation of VA conduction by stimulating the RV1 node in theright ventricle.

12

Fig. 18. Grayscale colourmap used to represent the voltage of each node.

(a) 35ms. Electrical stimulus propa-gates from the SA node.

(b) 88ms. Electrical stimulus arrivesat the AV node.

(c) 193ms. Electrical stimulus propa-gates into the ventricles.

(d) 245ms. Electrical stimulus propa-gates up the ventricles.

Fig. 19. Simulation of a normal cardiac cycle. The SA node depolarises at10ms.

(a) 35ms. Electrical stimulus propa-gates from the SA node.

(b) 88ms. Electrical stimulus isblocked at the AV node.

(c) 193ms. Electrical stimulus fails toenter the ventricles.

(d) 245ms. Ventricles fail to depo-larise.

Fig. 20. Simulation of heart block. The SA node depolarises at 10ms.

(a) 35ms. Electrical stimulus propa-gates from the SA node.

(b) 88ms. Electrical stimulus arrivesat the AV node.

(c) 193ms. Electrical stimulus propa-gates slower into the right ventricle.

(d) 245ms. Right ventricle depolarisesafter the left ventricle has depolarised.

Fig. 21. Simulation of right bundle branch block. The SA nodedepolarises at10ms.

13

(a) 35ms. Electrical stimulus propa-gates from the SA node.

(b) 193ms. Electrical stimulus propa-gates into the ventricles.

(c) 640ms. Repolarisation of normalventricles would already have com-pleted.

(d) 740ms. Repolarisation of the ven-tricles finally completes.

Fig. 22. Simulation of long Q-T syndrome. The SA node depolarises at10ms.

(a) 35ms. Electrical stimulus propa-gates from the RV1 node.

(b) 150ms. Electrical stimulus arrivesat the AV node.

(c) 220ms. Electrical stimulus propa-gates into the atriums.

(d) 311ms. Electrical stimulus propa-gates down the atriums.

Fig. 23. Simulation of VA conduction. The RV1 node depolarises at10ms.

(a) 35ms. Electrical stimulus propa-gates from the SA node.

(b) 125ms. Electrical stimulus propa-gates into the left ventricle via the LA1→ LV1 accessory pathway.

(c) 176ms. Left ventricle depolarisesbefore the right ventricle.

(d) 245ms. Ventricles have depo-larised.

Fig. 24. Simulation of Wolff-Parkinson-White syndrome. The SA node depolarises at10ms.

14

(a) 88ms. Electrical stimulus arrivesat the AV node via fast path.

(b) 222ms. Early stimulation of theSA node.

(c) 243ms. Early stimulus propagatesthrough the atriums.

(d) 320ms. Early stimulus arrives atthe AV node via the slow path.

(e) 400ms. Early stimulus re-entersthe atriums via the fast path.

(f) 430ms. Early stimulus re-entersthe slow path a second time.

(g) 515ms. Refractory of FP1 nodeprevents a second re-entry into theatriums.

(h) 570ms. Atriums and ventricles re-polarise.

Fig. 25. Simulation of AV node re-entrant tachycardia. The SA node depolarises at10 and220ms.

TABLE IVDETAILED QUALITATIVE COMPARISON OF HEART MODELS FROMUOA

AND OXFORD.

UoA Oxford [6]

HeartAnatomy(Figure 1)

2D conduction system

Morphologyof actionpotentials(Figure 6)

Realistic because a morerecent Stony Brook HAmodel [13] is used (withthe improvements fromSections IV-B and IV-C)

Unrealistic because asimplified Stony BrookHA model [19] is used

ElectricalRestitution(Figure 12)

Shows good dynamicbehaviour

Shows limited dynamicbehaviour

Path Model(Figure 16)

Electrical propagation is time delayed, bi-directional,and anisotropic.

Correctly disallowscolliding electrical stimuli

from passing througheach other

Incorrectly allowscolliding electrical stimuli

to pass through eachother

DiseaseConditions(Figures 20to 25)

Bradycardia andtachycardia by modifying

the SA rate, AV nodere-entrant tachycardia, VAconduction, heart blocks,and long Q-T syndrome

Bradycardia andtachycardia by modifying

the SA rate, and heartblocks

E. Wolff-Parkinson-White Syndrome

This is modelled by creating an additional pathway betweenthe left atrium and left ventricle. Figure 24 shows a simulationof the syndrome, which causes the left ventricle to depolariseearlier than usual.

F. AV Node Re-entrant Tachycardia

This is modelled by the fast and slow pathways around theSA node, demonstrated previously in Figure 17. Figure 25simulates re-entrant tachycardia by stimulating the SA node asecond time with an early stimulus.

G. Time to Simulate the Cell and Heart Models

The execution time needed to simulate the cell and heartmodels of Oxford and UoA are compared in this section. For afair comparison, we wanted to avoid the execution overhead ofthe SimulinkR© environment. Thus, for each model, SimulinkCoderTM was used to generate a single-threaded executable Cprogram that is optimised for execution speed. Each C programwas then executed on a single core of a3.40GHz Intel Corei7-4770 processor with16GB of RAM and running MicrosoftWindows 7 Enterprise. Each reported execution time is theaverage of ten runs of the same program.

The Oxford cell model takes12.376s to simulate60s ofcell activity with a 1s base cycle length, compared with9.586s for the UoA improved cell model. The Oxford cell

15

model takes longer to simulate because it uses a StateflowR©

block to manually check the transiton guards and producea corresponding Boolean event whenever a guard is true.The Boolean event triggers another StateflowR© block to taketransitions between the hybrid automaton states. For our UoAimproved cell model, the hybrid automaton is modelled bya single StateflowR© block. The hybrid automaton transitionguards are modelled directly as StateflowR© transition con-ditions. StateflowR© (automatically) handles the checking oftransitions, which leads to a more efficient simulation, eventhough the Oxford cell model only uses thevx variable of theStony Brook 2005 cell model. Moreover, the UoA improvedcell model uses SimulinkR© components that are more efficientto simulate wherever possible.

The Oxford and UoA heart models use the same arrange-ment of 33 nodes, as shown in Figure 1. The Oxford heartmodel has a total of34 paths between the nodes and takes8.516s to simulate1s of a normal cardiac cycle. The UoAimproved heart model has a total of68 forward and backwardpaths between the nodes and takes30.771s to simulate1sof a normal cardiac cycle and31.295s to simulate1s of AVnode re-entrant tachycardia. The UoA improved heart modeltakes longer to simulate because it implements a much morecomplex path model that considers the backward propagationand annihilation of electrical impulses throughout the cardiacconduction system. Thus, it is imperative to further improvethe efficiency of the path model without sacrificing its pre-dictive power. The execution time of the UoA improved heartmodel could be reduced by parallelising the simulation of thenodes over multi-cores [5].

VIII. C ONCLUSION

We carefully reviewed the heart models designed for theclosed-loop testing of pacemakers and identified key areasof improvement. To this end, we advanced the state-of-the-art in three ways: (1) stabilising and enforcing a minimumaction potential duration on the Stony Brook cardiac cellmodel, (2) developing a path model that handles the partialand full propagation of continuous-time action potentials, and(3) demonstrating the predictive power of our heart modelthrough the extensive simulation of arrhythmias. We are ableto model many more arrhythmias than are possible with theexisting heart models.

For future work, we look to implement our heart model withmore computationally efficient methods to achieve the goal ofreal-time heart simulation, i.e., heart emulation. We alsolookto investigate the automatic generation and parameterisation ofa network of cardiac cells and paths for patient-specific heartmodelling.

ACKNOWLEDGMENT

This research was funded by the University of AucklandFRDF Postdoctoral grant No.3707500. The authors wouldlike to express their gratitude to Prof. Marta Kwiatkowsk andher research group from the University of Oxford for providingan implementation of their heart model.

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Eugene Yip received his B.E. (Hons) and Ph.Ddegrees in electrical and computer systems engineer-ing from the University of Auckland, New Zealand.At the start of 2015, he joined the Heart-on-FPGAresearch group in the Department of Electrical andComputer Engineering, University of Auckland, asa research assistant working on the real-time mod-elling of cardiac electrical activity. In the middleof 2015, he joined the SWT research group at theUniversity of Bamberg as a research assistant work-ing on synchronous languages. His current research

interests include synchronous mixed-criticality systems, parallel programming,static timing analysis, formal methods for organ modelling, and biomedicaldevices.

Sidharta Andalam received his Ph.D degree fromthe University of Auckland, New Zealand, in 2013,where his thesis focused on developing a predictableplatform for safety-critical systems. He is currentlya research fellow in embedded systems at the Uni-versity of Auckland, New Zealand. His principleresearch interest is in the design, implementation,and analysis of safety-critical applications. He hasworked at TUM CREATE, Singapore, exploringsafety-critical applications in the automotive domain.

Partha S. Roop received his Ph.D degree in com-puter science (software engineering) from the Uni-versity of New South Wales, Sydney, Australia, in2001. He is currently an Associate Professor and isthe Director of the Computer Systems EngineeringProgram with the Department of Electrical and Com-puter Engineering, the University of Auckland, NewZealand. Partha is an associated team member ofthe SPADES team INRIA, Rhone-Alpes, France, andheld a visiting position in CAU Kiel, Germany, andIowa State University, USA. His research interests

include the design and verification of embedded systems. In particular, he isdeveloping techniques for the design of embedded applications in automotive,robotics, and intelligent transportation systems that meet functional-safetystandards.

Avinash Malik is a lecturer at the University ofAuckland, New Zealand. His main research interestlies in programming languages for multicore anddistributed systems and their formal semantics andcompilation. He has worked at organisations suchas INRIA in France, Trinity College Dublin, IBMresearch Ireland, and IBM Watson on the designand the compilation of programming languages. Heholds B.E. and Ph.D degrees from the University ofAuckland.

Mark Trew attained a Bachelor of Engineering inEngineering Science in 1992 and a Ph.D Engineer-ing Science in 1999, both from the University ofAuckland, New Zealand. He is currently a SeniorResearch Fellow at the Auckland BioengineeringInstitute. Mark constructs computer models andanalysis tools for interpreting and understanding de-tailed images of cardiac tissue and cardiac electricalactivity.

Weiwei Ai received the B.S. degree in 2003 fromQingdao University, China, and the M.E. degree in2006 from Beijing University of Technology, China.From 2006 to 2013, she worked as a reliability andfailure analysis engineer in CEC Huada ElectronicDesign Co.,Ltd. She is currently working towardsthe Ph.D. degree at the the University of Auckland,New Zealand. Her research interests are in verifica-tion and validation with a focus on medical devices.

Nitish Patel received the B.E. degree from Manga-lore University, Karnataka, India, and the Ph.D. de-gree from the University of Auckland, New Zealand.He is currently a Senior Lecturer at the University ofAuckland. His research interests include embeddedsystems, robotics, artificial neural networks, controlsystems, and hardware implementation of real-timesystems for control.