application of 2-d network simplification to modellingand simulation of the cardiac electrical...
TRANSCRIPT
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
1/14
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
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
2/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
27
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
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
3/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
28
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.
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
4/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
29
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.
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
5/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
30
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
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
6/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
31
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
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
7/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
32
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
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
8/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
33
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
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
9/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
34
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
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
10/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
35
(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)
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
11/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
36
(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
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
12/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
37
(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.
-
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
13/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
www.Journalcse.co.uk
38
[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 -
7/30/2019 Application of 2-D Network Simplification to Modellingand Simulation of the Cardiac Electrical Activity Using Bidom
14/14
JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 15, ISSUE 2, OCTOBER 2012
2012 JCSE
39
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.