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JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012

2012 JCSE

www.Journalcse.co.uk

26

Application of 2-D Network Simplification toModellingand Simulation of the Cardiac

Electrical ActivityUsing Bidomain ApproachIsaiah A. Adejumobi and Oluwaseun I. Adebisi

Abstract-This work is an application of 2-D network simplification to modelling and simulation of cardiac electrical activity usingthe bidomain approach. The electrical activity of the heart is governed by differential equations consisting of partial differentialequations (PDEs) coupled to a system of ordinary differential equations (ODEs).Theseequations are challenging to solvenumerically and implement owing to their non-linearity and stiffness. Explicit forward Euler method and 2-D network modellingwere respectively used for the time and space discretizations of the derived bidomain model. We implemented and simulatedthe discretized model to obtain the time characteristic of the transmembrane potential, Vm in the normal cardiac tissue. We alsoobserved the effects of changing the values of extracellular and intracellular resistances e, ion Vm; changing which resulted inits time dilation and gradual to near complete collapse. These are signs of cardiac electrical abnormalities. This work has notonly revealed the nature of propagating cardiac electrical signal based on 2-D network simplification of the bidomain model buthas also revealed that increase in values of electrical coupling between the cells in the 2-D network domain can significantly

impact on the propagating cardiac electrical signal.

Index Terms: bidomain approach, cardiac electrical activity, 2-D network simplification, transmembrane potential

1 INTRODUCTIONThe electrical activity is very important for the

heart to perform its functions. It is particularly

responsible for the periodic contraction and

relaxation of the heart which pumps blood

throughout the body [1]. However, abnormal

cardiac electrical activities have been posing great

threats globally, causing many premature deaths.

Mathematical models and computer

simulations are rapidly becoming vital tools for

investigating the heart conditions and the potential

side effects of drugs on cardiac rhythms [2], [3].

They offer a realistic means of understanding the

underlying mechanisms of heart functions and

many other biological systems without carrying

out physical experiments.

The cardiac electrophysiological models are

governed by differential equations consisting of

I.A. Adejumobi is with Electrical and Electronics EngineeringDepartment, Federal University of Agriculture, Abeokuta,Nigeria.O.I. Adebisi is with Electrical and ElectronicsEngineeringDepartment, Federal University of Agriculture, Abeokuta,Nigeria.

coupled systems of partial differential equations

(PDEs) and ordinary differential equations (ODEs)

[1], [4]. These equations are usually non-linear and

stiff, hence, pose computational challenges.

Nevertheless, we adopted the bidomain approach

in the present work due its ability to give a realistic

simulation of cardiac electrical activity. It consists

of a system of two degenerate non-linear partial

differential equations coupled to a system of

ordinary differential equations. Hence, to handle

the computational complexities posed by the

bidomain model, we have considered 2-D network

simplification of the model in this work where the

cardiac tissue is represented by interconnected

network of cells, each individually described by a

given system of cell model.

This work is an application of 2-D network

simplification to modelling and simulation of

cardiac electrical activity using the bidomain

approach. The electrical property of interest is the

cardiac action potential. Our focus is to increase

the value electrical coupling between the cells in

the 2-D network domain and study the resulting

effects on the cardiac action potential.

1.1Cardiac Action Potential

Action potential is an important basic electrical

property of the heart. It is a time characteristic of


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the transmembrane potential which is usually

followed by a recovering of the resting condition. It

shows different shapes and amplitudes according

to the different kind of excitable media to which

the cells belong to, and in the large muscle cells

makes it possible the simultaneous contraction of

the whole cell [5]. An action potential propagatesacross the heart in a heterogeneous way, keeping

the same shape and amplitude all along an entire

neural or muscular fibre.

Cardiac cells are characterized by a negative

transmembrane potential at rest and show two

kinds of action potentials: the quick (or fast) and

the slow response [5], [6]. The quick response

action potential is typical in the myocardium fibres

(both atrial and ventricular) and in the Purkinje

fibres, which are fibres specialized in the

conduction. The quick response action potential is

usually identified by five different phases namely:

phase 0 (depolarization phase), phase 1 (partial

repolarization phase), phase 2 (plateau phase),

phase 3 (repolarization) and phase 4 (resting

membrane potential phase). Depolarization phase

occurs due to the opening of the fast sodium ion

(Na+) channels, causing a rapid increase in the

membrane conductance to Na+ and therefore a

rapid influx of Na+ into the cell. The partial

repolarization phase occurs a result of the

inactivation of the fast Na+ channels. The transient

outward of potassium ion (K+) causes the small

downward deflection of the action potential. The

balance between the slow inward calcium ion

(Ca++) currents and the outward K+ currents causes

the plateau phase. Usually, the ventricular

contraction persists throughout the action

potential, so the long plateau produces a long

action potential to ensure a forceful contraction of

substantial duration. Rapid repolarization is

caused by the outward K+ current. Na+

channelrecovery starts during the relative

refractory period. These phases are shown in fig. 1.

Fig. 1: Quick (fast) response action potential [6]

The slow response action potential is typical of

the Sinoatrial Node (SA), the natural pacemaker of

the heart, and the Atrioventricular Node (AV), thetissue meant to transfer pulse from atria to

ventricles. The slow response action potential is

identified by a less negative resting membrane

potential phase, a smaller slope and amplitude

depolarization phase, an absence of the partial

repolarization phase and by a relative refractory

period that continues during resting membrane

potential phase. The slow response action potential

is shown in fig. 2.

Fig. 2: Slow response action potential [6]

The electrical activity of the heart as a whole is

therefore characterized by a complex multiscale

structure, ranging from the microscopic activity of

ion channels in the cellular membrane to the

macroscopic properties of the anisotropic

propagation of the excitation and recovery fronts

in the whole heart and the most complete model

that gives the description of such a complex

process is the anisotropic bidomain model.

4

0

3


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2 MATHEMATICAL MODEL FORCARDIAC ELECTRICAL ACTIVITYThe electrical wave propagation in the thoracic

volume is governed by three fundamental

electrical laws [1], [ 7], [ 8]:

The electrical charge conservation law The electrical conduction law (Ohms law) The consequence to the electromagnetic

induction law

The law of conservation of charge states that an

outward flow of positive charges must be balanced

by a decrease of positive chargeswithin the close

surface [9], [10]. Hence, this requires that:

= . = (1Where

I is the current in Ampere (A)

j is the current density in Ampere

per square meter (A/m2)

Q is the charge in Coulomb (C)

S is the surface area in square meter (m2)

t is the time in seconds (s)

Applicationof divergence theoremwhich the

surface integral to the volume integral to (1) gives

(2) [9], [10]:

. = .v (2Representing the enclosed charge Q by the

volume integral of the charge density, (1) and (2)

can be modified as:

.v = ( ) v (3Whereis the volume charge density in coulombper cubic meter (C/m3).

Keeping the surface constant, the derivatives in(1), (2) and (3) becomes partial derivative and may

appear within the integral as:

.v = ( ) v (4Since (4) is true for any volume no matter how

small [9], [10]then;

. = (5Equation (5) is generally called the continuity

equation [9], [10].

For a good conductor, the volume charge

density is zero, = 0, since the amount of positiveand negative charges are equal [11]. Hence, ifthoracic volume is assumed to be volume of

conductor, equation (5) can be modified as:

. = 0 (6Equation (6) is called the electrical charge

conservation law.

Relatingthe current density j with the electric

field E in volt per metre (V/m), the electric field E

with the electric potential in volt (V) and currentdensity j with the electric potential, the followingfundamental laws emerge [9], [10]: the electric

conduction law (Ohms law), electromagnetic

induction law and modified Ohms law

represented by:

j = E (7

E = (8j =

(9

Where is the conductivity in siemens per metre

(S/m).

The adopted bidomain modelassumes the

cardiac tissue as a homogenized two-phase Ohmic

conducting medium with one phase representing

the intracellular space and the other, extracellular

space. The phases are linked by a network of

resistors and capacitors representing the ion

channels and the capacitive current driven across

the cell membrane due to a difference in potential

respectively as shown in fig. 3.


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Fig. 3: Schematic model of the bidomain space [12]

Considering a post homogenization process,

the intracellular and extracellular domains can be

assumed to be superimposed to occupy the whole

heart volume H [1], [13], [14], [15] and also appliesto the cell membrane. Hence, the average

intracellular and extracellular current densities,

and, conductivity tensors and and electricpotentials and are defined in H.Application of (6) to the heart volume gives:

. = . = (10Where is the surface to volume ratio of the cellmembrane per meter (m-1) is the cell membrane current in ampere (A)

From (10), (11) is obtained:

. + = 0 (11Putting(9) in (10) yields:

. = . (12The transmembrane potential,, defined as

difference in potential between intracellular and

extracellular spaces is represented by:

(13Where is the intracellular electric potential in volt (V) is the extracellular electric potential in volt (V)

Substitution of (13) in (12), gives:

. ( + ) = . (14

Extending the cell model formulated by

Hodgkin and Huxley as reported in Matthias [12]

with its electric circuit equivalence diagram as

shown in fig. 4 to our reference model gives:

= + , (15Where is the membrane capacitance in per area unit.Imis the membrane current in ampere (A)

Iappis the excitation current in ampere (A)

Iion is the ionic current in ampere (A)

Fig.4: Cell model equivalent circuit diagram;ionic currentsare parallel-connected tomembrane capacitor[12]

The use of (13) and (15) in (10) yields:

. + . =( + , ) in (16

The ionic variable w satisfies a system of ODEof the type given by (17):

= ,in (17Where g is a vector-valued function.

The bidomain model described by (14), (16) and

(17) depicts a non-linear elliptic equation for the

extracellular potential coupled with theparabolic differential equation for the

transmembrane potential Vm as well as an ordinary

differential equation representing the ionic current

w.


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Equations (14) and (16) describe the

propagation of the electrical signal through the

cardiac tissue while (17) describes the

electrochemical reaction in the cell.

The bidomain model described by (14), (16) and

(17) has to be coupled to an ionic model andcomplemented with appropriate initial and

boundary conditions for complete description of

electrical wave propagation in the cardiovascular

system.

2.1 Ionic Model

Some of the ionic models that have been to obtain

the expressions forIionand g include [16]:FitzHugh-

Nagumo model, Aliev-Panfilov Model, Roger

McCulloch Model and MitchellSchaeffer model.

However, we considered FitzHugh-Nagumo

(FHN) model because it qualitatively gives therepresentation of the most basic features of the

action potential coupled with its

straightforwardness as well as wider theoretical

and computational applications. Thevariantof the

FHN model adoptedis represented by (18) and (19)

[17].

= 1 1 3 3 (18 = 2 + (19Where 1, 2,, are positive constant parametersrespectively called excitation rate constant,recovery rate constant, recovery decay constant

and excitation decay constant. They are typical

assumed to be positive constant. 1 controls thesharpness of the action potential which determines

its mobility while2 controls the action potentialduration.

2.2 Initial and Boundary Conditions

The bidomain equations described by (14), (16) and

(17) are subjected to the initial conditions given by

(20):

,, 0 = ,, 0 = (20

The boundary conditionimposed on this (14),

(16) and (17) is that of a sealed boundary where no

current flows across the boundary between the

intracellular and extracellular domains, that is:

. = . (21Where n is the normal vector to the domain

boundary.

2.3 Discretization

The bidomain equations described by (14), (16) and

(17) are non-linear. For easy handling and

manipulation, these equations need to be

linearized (discretized). Also, the bidomain

equations are both time and space dependent,

therefore they must be separately linearized.

Various explicit and implicit time discretization

techniques exist for linearizing differential

equations, though, implicit method offers greater

stability than explicit method but the latter is avery simple and straightforward method and

problem of instability can be reduced by making

the time step size very small. Hence, we employed

explicit forward Euler time discretization scheme

to linearize (16) and (17), which contain time

derivatives.Equations (14) and (16) are space-

discretized using 2-D discrete (network)

modelling. The final discretized equations are

given by (22), (23) and (24).

+1 = +

1

1 3

3

+ + (22With and assumed unity; Gi, the intracellularadmittance matrix equivalent to . , and t,timestep size.

+1 = + 2 + (23 = + 1 .(24Where Ge, the extracellular admittance matrix

equivalent to . .2.3.1. Construction of Admittance MatricesGiand Ge

Owing to the adoption of 2-D discrete (network)

modelling, we were able replace Del operators on

the intracellular and extracellular conductivity

tensors i and e with intracellular and

extracellular admittances Gi and Ge. Gi and Ge were

constructed by considering node arrays Nx-by-Ny


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defined in the 2-D network domain to be linked by

network of resistors arranged along x- and y-

direction with ex, ey,ix,and iy representing the

extracellular and intracellular resistance values

along these directions.These resistors arrays were

then transformed into matrices in the

implementation code. This procedure is illustratedhere using 2 by 3 nodes in figure 5 as an example

and the result was generalized to the Nx by Ny

nodes considered in this work.

Each resistor was represented by five indices;

the xi and yi indices of one the nodes to which the

resistor was connected, the xj and yj indices of the

other nodes to which the resistor was connected

and resistor value . These five-index arrays are

presented in table 1. The two separate indices

representing each node (cell) to which the resistor

was connected were then converted into a single

index (encircled numbers in fig.5) which now

represented the first node (x'i) to which the resistor

was connected and the second node (y'j) to which

the resistor was connected. It is these two single-

indexed arrays that were stored in the matrices of

Gi and Ge to represent the positions pij where the

resistors are to be placed in the admittance

matrices. This is presented in table 2.

Fig. 5: Network of Resistors Connecting the Nodes

TABLE 1FIVE-INDEX ARRAYS

xi yi xj yj

1 1 1 2 x

1 2 1 3 x

2 1 2 2 x

2 2 2 3 x

1 1 2 1 y

1 2 2 2 y

1 3 2 3 y

TABLE 2TWO SINGLE-INDEXED ARRAYS

x'i y'j

1 2

2 3

4 5

5 6

1 4

2 5

3 6

Scanning through the network of resistors in

fig.5 from left to right and right to left, it was

observed that the resistor values were equal in

both directions, that is, 12is equal to 21 and so on.

Based on this, size of the admittance matrices Gisnd Ge is 6 by 6 matrices that is (2 x 3 by 2 x 3). The

final matrix elements positions pij and their values

are respectively shown in the matrices below.

, =

11 12 1321 22 2331 32 3314 15 1624 25 2634 35 3641 42 4351 52 5361 62 6344 45 4654 55 5664 65 66

, =

0 0 0 0 0

0 00

0

0 0 0 00 00 0

0 0 0 0 0

Where gx is 1 and gy is 1 The admittance matrices finally obtained

represented homogeneous but anisotropic system

since the resistors appeared the same everywhere

but current flows in the two different directions x

and y due to difference in the resistances in the x

and y directions were different.

3 APPLICATION OF COMPUTERThe discretized equations were implemented in

Java programming language (Java 6.0 version).

Java is an object-oriented programming language.

We adopted Java basically because of its enriched

mathematical library and well designed Graphical

User Interface (GUI) for displaying graphical

representation of results. Another benefit of Java is

1,

1,2 1,3

2,1 2,2 2,3

x

x

y y

1 2 3

4 5 6


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its portability across various operating systems. A

flow chart for the implementation algorithms is

shown in fig. 6 below. Simulation experiments

were carried out on a 4GB RAM, Intel (R) Core

(TM) i7 CPU M620 @ 2.67GHz and 32-bit operating

system computer. Running time of one simulation

experiment ranged between 3 and 7minutes forunchanged and changed intracellular and

extracellular resistances.

Fig. 6: Flow chart for the bidomain code

4 SIMULATION RESULTS ANDANALYSISIn this section of the work we present the results of

our simulations. We performed simulation

experiment using the developed 2-D Java

programme based on the linearized bidomain

equations given by (22), (23) and (24) and the

parameters in table 3.

Start

Input parameters

Construct admittance

matrices Gi and Ge

Sum matrices Gi and Ge

Invert the sum ofmatrices Gi and Ge

Construct the right hand

side of equation (24)

Solve for e at timestep n(24)

Solve Vm at timestep

n+1(22)

Solve w at timestep

n+1 (23)

Plot results V

against Tim

Stop

Keep matrix

Next

timeste

Impose initial conditionson Vm and w

Define arrays


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TABLE 3VALUES OF BASIC PARAMETERS [17]

Parameter Symbol Value

Excitation rate constant 1 0.2

Recovery rate constant 2 0.2

Excitation decay constant 0.7

Recovery decay constant 0.8

Time step size t 0.01Extracellular resistance in x-

direction

1.0Extracellular resistance in y-

direction

3.0Intracellular resistance in x-

direction

1.0Intracellular resistance in y-

direction

3.0Resting transmembrane

potential

-1.2Initial value of ionic variable wo -0.62

The selected cells 8, 10, 15, 17 of the 50-by-50

nodes (cells) specified in 2-D network domain

produced the propagated electrical waves in the

normal cardiac tissue as shown in fig. 7a to d

respectively with depolarization, partial

repolarization, plateau, repolarization and resting

membrane potential phases identified using the

values 0, 1, 2, 3, and 4 in fig. 7b for clarity. This

electrical signal produced is called the action

potential (time characteristic of transmembrane

potential), with the highest period observed in thiswork around 600ms. Fig. 7a to d are typically of

the same wave pattern, consistent with the

theoretical standard and the experimental findings

from other researchers [1], [18, [19].

(a)

(b)

(c)

(d)Fig. 7: Electrical wave propagation in the normal cardiactissue: (a) at cell 8, (b) at cell 10, (c) at cell 15, (d) at cell 17

In order tostudy the effects ofincreasing the

value of electrical coupling between the cells in the

0

12

3

4


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2-D network domain on the propagated cardiac

signal, the extracellular and the intracellular

resistancesex, ey,ix, and iy connecting different

cells along x and y directions in the 2-D network

domain were subjected to five levels of increment.

The values of ex, ey,ix, and iygiven as 1, 3, 1 and

3 respectively were increased by a factors of 1.5, 2,2.5, 3 and 10.During the simulations, time dilation

effect in which the cardiac signals witnessed

extended excitation cycles was observed when the

extracellular and intracellular resistances ex, ey,

ix, and iy along the x and y directions in the 2-D

network domain were increased by factors of 1.5, 2,

2.5, 3 and10 respectively. Also, it was observed that

the slopes of cardiac signals witnessed gradual

collapse which became more obvious when ex, ey,

ix, and iy were increased by factors of 3 and

10.The implication of this slope collapse is that the

cardiac signals were not able to return to the

resting state especially under the incremental

factor of 10 for the period around 2 seconds which

when compared to the excitation values in fig. 7 is

an over-excitation period and is an indication of

abnormal electrical wave propagation in the

cardiac tissue. The obtained propagated cardiac

signals for each incremental value are presented in

fig. 8, 9, 10, 11, 12.

(a)

(b)

(c)

(d)Fig. 8: Electrical wave propagation due to increment in ex,ey, ix, and iy by factor of 1.5: (a) at cell 8, (b) at cell 10, (c)at cell 15, (d) at cell 17


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(a)

(b)

(c)

(d)Fig. 9: Electrical wave propagation due to increment in ex,ey, ix, and iy by factor of 2: (a) at cell 8, (b) at cel 10, (c) atcell 15, (d) at cell 17

(a)

(b)


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(c)

(d)

Fig. 10: Electrical wave propagation due to increment in ex,ey, ix, and iy by factor of 2.5: (a) at cell 8, (b) atcell 10, (c) atcell 15, (d) at cell 17

(a)

(b)

(c)

(d)Fig. 11: Electrical wave propagation due to increment in ex,ey, ix, and iy by factor of 3: (a) at cell 8, (b) at cell10, (c) atcell 15, (d) at cell 17


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(a)

(b)

(c)

(d)Fig. 12: Electrical wave propagation due to increment in ex,ey, ix, and iy by factor of 10: (a) at cell 8, (b) at cell10, (c) atcell 15, (d) at cell 17

5 CONCLUSIONIn this work, we applied 2-D network

simplification to the modelling and simulation of

cardiac electrical activity using bidomain

approach. Apart from the fact that our adoption of

2-D network simplification of the bidomain model

enabled to avoid excessive computations, we have

also been able to provide some insights into the

electrical behaviour of human heart, revealing the

nature of the electrical wave propagation pattern

in the normal cardiac tissue.This work further

revealed that increase in the intracellular andextracellular resistances coupling different cells in

the 2-D network domain beyond certain limit can

cause time dilation effect and collapse of the

cardiac electrical waves with the overall effect of a

delayed repolarization.If this persists for a long

time, it may result in sudden cardiac death.

Research is still on-gong on the application of

continuum modelling (specifically finite element

method) to analysis of cardiac electrical activity

using bidomain approach. This model takes care of

the pitfall our adopted network (discrete)

modelling which uses cell counts below the actualcell counts of the heart.

6 REFERENCES[1] M.S. Shuaiby, A.H. Mohsen and E. Moumen,

Modelling and Simulation of The Action

Potential in Human Cardiac Tissue Using

Finite Element Method, J. of Commun. &Comput. Eng., vol. 2, no. 3, pp. 21-27, 2012.


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[2] J.A. DiMasi, R.W. Hansen and H.G. Grabowski,

The Price of Innovation: New Estimates of

Drug Development Costs, J. Health Econ., vol.22, no. 2,pp. 151185, 2003.

[3] R.J. Spiteri and R.C. Dean,On The

Performance of an Implicit-Explicit Runge

Kutta Method in Models of Cardiac ElectricalActivity, IEEE Trans. Biomed. Eng., vol. 55, no.5,pp. 14881495, 2008.

[4] Y. Belhamadia, Recent Numerical Methods in

Electrocardiology, in: D. Campolo (Ed.), New

Development in Biomedical

Engineering,http://www.intechopen.com/book,

2010.

[5] L. Gerardo-Giorda, Modelling and Numerical

Simulation of Action Potential Patterns in

Human Atrial Tissues. Technical Report TR-

2008-004, Mathematics and Computer science,

Emory University, Atlanta, USA, 2008.

[6] R.E. Klabunde, Cardiovasculas PhysiologyConcepts.New York, NY: Lippincott Williamand Wilkin, 2011.

[7] A.D. Alin, M.M. Alexandru, M. Mihaela and

M.I. Corina, Numerical simulation in

electrocardiography, Rev. Roum. Sci. Techn.lectrotechn. et nerg., vol. 56, pp. 209218, 2011.

[8] C.S. Henriquez, Simulating The Electrical

Behaviour of Cardiac Tissue Using The

Bidomain Model,Crit. Rev. BiomedicalEngineering,vol. 21, pp. 1-77, 1993.

[9] J.A. Edminister, Electromagnetics. New Delhi,India: Tata McGraw-Hill Publishing Company

Limited, 2006.

[10] W.H. Hayt and J.A. Buck, Engineeringelectromagnetics.New Delhi, India:McGraw-Hill Education Private Limited, 2006.

[11] S.R. Reddy, Electromagnetic theory. Chennai,India: V. Ramesh Publisher, 2002.

[12] G. Matthias, 3D bidomain equation for muscle

fibers, masters thesis, Fredrich-Alexander-

Universitt Erlangen-Nrnberg, Germany,

2011.

[13] E.J. Vigmond, R.W. dosSantos, A.J. Prassl, M.

Deo and G. Plank, Solvers for The Cardiac

Bidomain Equations, Prog. Biophys. Mol.Biol., vol. 96, pp. 3-18, 2008.

[14] M. Boulakia, M.A. Fernandez, J.-F. Gerbeau,

and N. Zemzemi, Mathematical modelling of

electrocardiograms: A numerical study, AnnBiomed Eng., vol. 38, pp. 1071-1097, 2009.

[15] M. Boulakia, M.A. Fernandez, J.-F. Gerbeau,

and N. Zemzemi, towards the numerical

simulation of electrocardiograms, in: F.B.

Sachse and G. Seemann (Eds.), Functional

Imaging and Modeling of the Heart, Springer-

Verlag, pp. 240-249, 2007.

[16] Boulakia, M., Fernandez, M.A., Gerbeau, J.-F.andN.Zemzemi, A coupled system of

PDEs and ODEs arising in electrocardiograms

modelling. Appl. Math. Res. Exp. (abn 002):28, 2008.

[17] O.F. Niels, Bidomain model of cardiac

excitation, http://pages.physics.cornell.edu,

2003.

[18] F.H. Nigel, Efficient simulation of action

potential propagation in a bidomain, doctoral

dissertation, Duke University, Durhan,

Northern Carolina, USA, 1992.

[19] B.M. Rocha, F.O. Campros, G. Planck, R.W.

dos-Santos, M. Liebmann, and G.C. Haase,

Simulation of electrical activity in the heart

with graphical processing units, SFB Report

No. 2009-016, Austria, 2009.

Isaiah A. Adejumobi obtained his B.Eng., M.Eng.

and Ph.D degrees in Electrical Engineering from

University of Ilorin, Ilorin, Nigeria in 1987, 1992

and 2003 respectively. He started his academic

career form University of Ilorin in August 1990

where he worked for about fifteen and half years

before moving to his present place of work;

Federal University of Agriculture Abeokuta,

Nigeria. He is a Corporate Member of Nigeria

Society of Engineering and a registered

Engineering for Council for the Regulation of

Engineering in Nigeria. Academically Dr

Adejumobi has led some joint researches in

electrical and related disciplines. He has over thirty

journal publications both locally and

internationally. He is currently working on a jointresearch on Cardiac Electrical Activities, a research

that was motivated due to the increasing

cardiovascular problems in Nigeria. He is

currently an Associate Professor of Electrical

Engineering.

Oluwaseun I. Adebisi obtained his B.Eng. degree

in Electrical and Electronics Engineering from
http://www.intechopen.com/bookhttp://www.intechopen.com/bookhttp://www.intechopen.com/bookhttp://pages.physics.cornell.edu/http://pages.physics.cornell.edu/http://pages.physics.cornell.edu/http://www.intechopen.com/book


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Federal University of Agriculture, Abeokuta,

Nigeria. He is currently a Junior Research Fellow

in the Department of Electrical and Electronics

Engineering in the University. His research interest

is on Modelling of Cardiac Electrical Activities.

application of 2-d network simplification to modellingand simulation of the cardiac electrical...

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