J. Biomedical Science and Engineering, 2010, 3, 308-316
doi:10.4236/jbise.2010.33042 Published Online March 2010 (http://www.SciRP.org/journal/jbise/
JBiSE
).
Published Online March 2010 in SciRes. http://www.scirp.org/journal/jbise
An index for evaluating distance of a healthy heart from
Sino-Atrial blocking arrhythmia
Hossein Gholizade-Narm1, Morteza Khademi2, Asad Azemi3, Masoud Karimi-Ghartemani4
1Electronic & Robotic Engineering, Shahrood University Technology, Shahrood, Iran;
2Ferdowsi University of Mashhad, Mashhad, Iran;
3College of Engineering, Penn State University, Philadelphia, USA;
4Sharif University of Technology, Tehran, Iran.
Email: Gholizade@shahroodut.ac.ir
Received 29 November 2009; revised 15 December 2009; accepted 28 December 2009.
ABSTRACT
In this paper, an index for evaluating Distance of a
healthy heart from Sino-Atrial Blocking Arrhythmia
(SABA) is presented. After definition of the main
pacemakers' model of heart, Sino-Atrial (SA) and
Atrio-Ventricular nodes (AV), the boundary of syn-
chronization, which demonstrates the boundary of
blocking arrhythmia, is obtained using perturbation
method. In order to estimate of healthy heart char-
acteristics, a parameter estimator is introduced. The
distance from SABA is calculated using Lagrange
method and Kohn-Tucker conditions. In addition, the
maximum admissible decrease in the coupling inten-
sity and the maximum admissible increase in the dis-
crepancy between the natural frequencies of two
pacemakers are determined in order to maintain the
synchronization between the two pacemakers.
Keywords: Healthy Heart; Blocking Arrhythmia;
Perturbation method; Synchronization Boundary;
Optimization; Bifurcation
1. INTRODUCTION
The heart arrhythmias are the first cause of death ac-
cording to World Health Organization (WHO) reports
[1]. The lack of information about the health degree is
one of the many factors for this. In other words, if peo-
ple know their relative degree of health, the demise will
be decreased. Fortunately, the medical science has the
ability to measure the distance from illness in many is-
sues. For instance, the fat and glucose of blood can be
determined by some easy checks. The treatment could be
initiated if it was near dangerous zone. Many works have
also been done on cardiac arrhythmias' prediction to
reduce the cardiac sudden death. Most of them have
been done by physicians. The statistical methods are
often used in those researches. In statistical methods, the
QRS complex and QT interval and other segments of
Electro Cardio Graph (ECG) are investigated and car-
diac sudden death is predicted [2,3,4,5]. The nonlinear
features such as fractal dimensions of Heart Rate Vari-
ability (HRV) are used by other methods to determine
the risk of sudden death [6]. Finally, the General Regres-
sion Neural Network (GRNN), Learning Vector Quanti-
zation (LVQ) and wavelet transform are used to predict
the life-threatening Ventricular tachycardia (VT) and
Ventricular Fibrillation (VF) [7].
All of the works are based on signal processing and
their result is a number which demonstrates the prob-
ability of being prone to arrhythmia or death. Unfortu-
nately, signal processing based methods are not of
enough accuracy to predict life-threatening cardiac ar-
rhythmias (less than 70%) [6,7]. On the other hand, it
cannot evaluate the relative degree of health if the heart
is assumed healthy.
In this research, our goal is to evaluate the distance
from SABA by observing action potential signals of SA
and AV nodes. The blocking arrhythmias arise when two
pacemakers are not in synchrony. The main reasons for
asynchrony are either decreasing coupling intensity or
increasing the discrepancy of two pacemakers' natural
frequencies or both. For evaluation of distance from
SABA, four stages should be passed. At first, a proper
model should be considered for pacemakers. Then, the
synchronization boundary must be obtained based on the
model parameters. At the next stage, the model of pa-
rameters should be estimated from pacemakers' action
potentials. Finally, by defining a cost function for dis-
tance from SABA and using optimization methods, the
minimum distance is calculated.
Many models have been proposed for pacemakers and
heart cells till now. Among the most well-known of
them are Van der Pol [8,9,10,11], Fitzhugh-Nagumo
[12,13,14], Hodgkin-Huxley [15], Beeler-Reuter [16]
and Lou-Rudy [17] model. In this research, the Van der
H. Gholizade-Narm et al. / J. Biomedical Science and Engineering 3 (2010) 308-316
Copyright © 2010 SciRes
309
JBiSE
Pol model is used as the model of SA and AV nodes for
its simplicity and sufficiency for synchronization issues.
Synchronization and its boundary have dedicated
many works of engineers and physicists to itself from
the past decades. Synchronization can be divided into
complete synchronization, lag synchronization, func-
tional synchronization, phase synchronization and fre-
quency synchronization [18,19]. The output signal of the
oscillators must be exactly equal for complete synchro-
nization. In other words, two signals should be identical
in amplitude and frequency. The lag synchronization is
similar to complete synchronization except that the out-
put of one oscillator should be coincided on the delayed
signal of another one. The complete and lag synchroni-
zation are accessible only for two identical oscillators. In
functional synchronization, the output signal of one os-
cillator is a function of another one. This type of syn-
chronization is defined for those oscillators whose pa-
rameters are related together. Any type of the aforemen-
tioned synchronization types will not be occurred if two
oscillators are non-identical. However, most of the cou-
pled oscillators are non-identical in nature. In this case,
the phase and frequency synchronization is defined in
which, the coincidence of the output signals of two os-
cillators is not necessary. In phase synchronization, the
phase difference of signals should be fixed for all the
time and for frequency synchronization, it is enough that
the phase difference of two oscillators to be bounded.
The heart pacemakers are in phase synchrony in normal
mode. When they tend to asynchrony, they pass fre-
quency synchronization at first and then become asyn-
chrony. Therefore, the boundary of synchronization is
determined by frequency synchronization, but since it is
difficult to find the frequency synchronization boundary
(it can only be obtained by simulation [20]) and on the
other hand, its distance from phase synchronization
boundary if very tiny [20,21], the phase synchronization
has been taken as arrhythmia boundary in this research.
Perturbation methods, specially averaging method and
bi-variant expansion are more applied in synchronization
boundary determination [20,21,22]. The dynamical equa-
tions of two coupled oscillators are mapped to polar
space using perturbation method. In polar space, the
phase difference of two oscillators is a state variable.
Therefore, the stability of new dynamical system is suf-
ficient for phase synchronization because, in this case,
the phase difference tends to a fixed point.
Unfortunately, there is not any background about
measuring distance from arrhythmia. Our goal in this
research is to evaluate this distance using synchroniza-
tion boundary and pacemakers' parameters. In addition,
we will answer to two questions: 1) How much the cou-
pling intensity can be decreased subject to synchroniza-
tion? 2) How much the natural frequency difference of
two pacemakers can be increased subject to synchroni-
zation?
The paper is organized as follow: After introduction in
Section 1, the synchronization boundary is obtained us-
ing perturbation method in Section 2. A state and pa-
rameter estimator is introduced in Section 3 for two cou-
pled pacemakers system. The distance from SABA is
evaluated in Section 4 and the maximum permitted value
of natural frequency difference and minimum permitted
value of coupling intensity are calculated in Section 5.
Simulation results are demonstrated in Section 6 and the
conclusion is discussed in Section 7.
2. THE PACEMAKER MODEL AND
SYNCHRONIZATION BOUNDARY
The asynchrony between two vital pacemakers, SA and
AV nodes, is the basic reason of blocking arrhythmias. In
a healthy heart, the relationship between these two
pacemakers is unidirectional coupling (from SA node to
AV node) according to physicians' opinion; however,
some physiologists suggest that the relationship is bidi-
rectional but the effect of AV node on SA node is very
smaller than vice versa [23]. According to the feedback
of automatic nerves, the second idea is more acceptable.
Here, we take the coupling as bidirectional, which is the
general form. To change bidirectional case to unidirec-
tional case, it is sufficient to equate the correspondence
coefficient to the effect of AV node on SA node to zero.
The model of two coupled pacemakers using Van der Pol
Equations are,
22
111
1
ttttt
t
x
xx xyx

 
  (1)
22
222
1
ttttt
yyyyx

 
 
t
y
(2)
were 1
and 2
are damping coefficients, 1
and
2
are natural frequencies, and 1
and 2
are cou-
pling coefficients which demonstrate coupling intensity.
Assume Eqs. 1 and 2 are the SA and AV nodes' model
respectively. According to relation between two coupled
pacemakers, we take 12
,1
 
 . For using per-
turbation method, the system should be transformed to
perturbation standard form. By scaling the time as
1
tt
which results 1
1,
t
dx 1
1
dt
dx
x
x
dt

dt
 dt
1
2
1
t
x
x
 the Eqs. 1 and 2 change as,




11111
11111
2
12
2
22
1
11
ttttt
ttttt
xxxxyx
yyyyx
 
 
 
 
 
 
1
t
y
(3)
310 H. Gholizade-Narm et al. / J. Biomedical Science and Engineering 3 (2010) 308-316
Copyright © 2010 SciRes JBiSE
where 22 22
12
1, 1

 2
and 2

. Sup-
pose 11
 1 therefore

1
. On the other hand,
k
 .
Remark: the natural frequency of SA node is greater
than AV node in the healthy heart. Since our goal is the
measurement of the distance of healthy heart from
SABA, therefore according to 3.
0
Consider variables x and y linear versus
as
01 0
,1
x
xxyyy
 . By bi-variant expanding of
Eq. 3 versus time in the form of 11
,tt


2
O
and
eliminating higher order terms (
and higher
terms) we have,
2
0
0
20
xx

(4)

2
2
2
00
1
10
21
21
x
x
xxx


 
 
(5)
2
0
0
20
yy

(6)

2
2
2
00
1
10
22
00
20
21
yy
yxy
xy y



 
 






(7)
Considering x0 and y0 in periodic forms,
 
 
0
0
cos sin
cos sin
xA B
yC D




(8)
By substituting x0 and y0 at the right hand of Eqs. 4
and 6 and equating secular terms (coefficients of sin
and cos
) to zero we have,


22
11
22
11
1
24
1
24
dA
A
AA B
d
dB BBAB
d


 
 
(9)




22
22 2
22
22 2
1
24
1
24
dC CCCD DB
d
dD DDCD AC
d

 

 
D
C
(10)
Define A, B, C and D as follow,
 
 
11 11
22 22
cos ,sin
cos ,sin
AR BR
CR DR
 
 


(11)
Therefore, x0 and y0 are,


 

01 1
02 2
,cos
,cos
xR
yR

 


(12)
By substituting A, B, C and D in 9 and 10 and rear-
ranging them,


2
111 1
11
2
2222 2112
2222 112
1
21
4
20
1
21 sin
4
2cos
RR R
R
RRR R
RR







 


 
(13)
where 121
1212
,,,
dRdR dd
RR
ddd
2
d





. Assume
12
, the slow flow dynamical Equations are,
2
111 1
2
2222 21
1
22
2
1
21
4
1
21s
4
2co
RR R
RRRR
R
R






in
s




 


(14)
where
12
d
d

. The synchronization boundary
is achieved by determining the Saddle-Node (S-N) and
Hopf bifurcation condition curves of 14 [20]. The first
step is to find the equilibrium points curve. Equate the
left hand of 14 to 0,
2
11 11
1
10
4
RR R
 2
 

 (15)
2
222 21
2
22 2
21
1
1sin
4
1
14
sin
RRR
RR
R

 0







(16)
H. Gholizade-Narm et al. / J. Biomedical Science and Engineering 3 (2010) 308-316
Copyright © 2010 SciRes
311
JBiSE

1
22
2
22
21
cos 0
cos
R
R
R
R


 




(17)
By summing the square of 16 and 17 and substituting
R1=2 we have,


2
2
2
2222 22
1
1
4
RRR 2
4


 




p
(18)
Replace and rearrange 18 according to
power of p,
2
2
R


2
23 2222
22 222
11 4
16 2
pp p
 
0
(19)
Eq. 19 shows the curve of equilibrium points vs. the
system's parameters. Using Descartes’ rule of signs we
see that (19) has either one or three positive roots for p.
At bifurcation, there will be a double root which corre-
sponds to requiring the derivative of (19) to vanish:


2
22 22
22 22
30
16 pp
 
 
(20)
Eliminating p from (19) and (20) gives the condition
for S-N bifurcation as,



65224 23
222222
44 222
22 22
4523624
22222222
4246083280
604 12
82440 40
 
 
 
 
 
 
3
(21)
Eq. 21 plots as two curves intersecting at cusp points
in the plane. At the cusp point, a further degen-
eracy occurs and there is a triple root in Eq. 19. Deriva-
tion from 20 gives,
2,
8
3
p (22)
By substituting p from (22) to (19) and (20) gives
cusp points.
221
222
0.544 ,0.033,
0.544 ,1.122
 





2
2
(23)
Next we look for Hopf bifurcations in the slow flow
system 14. Let be an equilibrium point.
The behavior of the system linearized in the neighbor-
hood of this point is determined by the eigenvalues of
the Jacobian matrix.
10 20 0
,,RR
 
 
2
10
1
2
20
2021020
210 210
2
00
2
20 20
20
34 00
4
34
1sin cos
24
cos cossin
R
R
JR
RR
RR
R


0
 
 
(24)
Substituting

00
sin,cos, R
10
from (15–17) in
24 and after simplification,

 
1
22
222202220
222
220
20
200
11 13
11
22 44
1
1
24
J
RR RR
R
R


 
 
 
 
 



(25)
For Hopf bifurcation, it is necessary that matrix J con-
tains 2 pure imaginary and 1 real eigenvalue.

22
1220222332332
0
10
2
IJ
RJJJJ
 








(26)
It can be seen from 26 that the coefficient of
must
be 0 for Hopf bifurcation.
2
220220
10
2RpR

2
2
  (27)
Substituting p from 27 in Eq. 19 gives the Hopf bi-
furcation condition curve.
22
222
1
20
4

2
  (28)
Eqs. 21 and 28 identify the phase synchronization
boundary. A typical sample for synchronization bound-
ary is brought in Ex. 1.
Synchronization boundary Equations show that the
boundary depends on coupling coefficients, damping
coefficients and discrepancy of two pacemakers’ natural
frequency. Therefore, they should be estimated in prac-
tice for distance evaluation from SABA.
3. THE STATE-PARAMETER ESTIMATOR
Measurement of all states and parameters of the cardiac
pacemakers is impossible in practice but the measure-
ment of only their action potentials is practical using two
implantable leads. In this section, a state-parameter es-
timator (adaptive observer) is introduced which is able to
312 H. Gholizade-Narm et al. / J. Biomedical Science and Engineering 3 (2010) 308-316
Copyright © 2010 SciRes
The adaptive rules are,
estimate necessary parameters from pacemakers' action
potentials for distance evaluation. This estimator uses
the high-gain method. The high-gain method is very
effective in state estimation [24]. When the parameters
are unknown in addition to states, the estimator is modi-
fied which is called adaptive observer. An adaptive ob-
server is introduced in [25] which can estimate parame-
ters and states simultaneously. We use an adaptive ob-
server which is introduced in [25] in this research. The
system form is as follow,
JBiSE
 
 

 

 

 

 
1
1
2
1
1
2
1
2
,
0
,
0
,
,
0
0
x
tut
A
xt xt
x
tut
A
xt ut
xt ut
C
yt xt
C





















(29)
 



 


 
 
1
ˆˆ
,
ˆˆ ,,
ˆ
ˆ
ˆˆ ,.
ˆ
ˆ
ˆˆ ,.
ˆ
TT
TT
tAKCt xtut
x
tAxtxt utxt utt
KttCytCxt
tI tCytCxt
x
xx ifxXXotherwise
x
if otherwise



 
 
 

 


 




,
(31)
X
are the upper bound,
 
0, 0x
1
2
0
0
K
K
K
where
where , , and
n
x 2
y q
,,
ii i
A
are the state vector,
output vector and the unknown parameters vector re-
spectively. The matricesC
, i = 1, 2 are, and K1,2 are selected such that the ii

 

,1 ,
2
1
1
01 0
,
1
00
00
,
00
,,
100 ,
ii
i
i
i
inin q
ini
i
A
xu
x
ux
Cnn





















i
A
KC matrix be
stable.


1
12
1,2
1
2
1,,,,0 ,
0
0
i
n
idiag

 





and
1
2
0
0
n
n
I
I
I
u
(30)
For using the introduced adaptive observer, the
system 1 and 2 must be mapped to state space rep-
resentation.
 

 
,,
x
tAxtxt utxt ut
yt Cxt

 
(32)
where,
H. Gholizade-Narm et al. / J. Biomedical Science and Engineering 3 (2010) 308-316
Copyright © 2010 SciRes
313
00
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  




 


 
1234
22
11 1222
2
12113
2
34313
010 0
1000 0000
,,
0010 0001
0000
,, 0
000 000
10
,
000 000
0001
T
T
CAxtxtxtxt xt
xu
xtxt xt xtxt
xu
x
txtxtxtxt
 







 















 
 
 
 

 
 
 
 
 
 

 
(33)
The performance of introduced estimator is demon-
strated in Ex. 2.
4. DISTANCE EVALUATION FROM SABA
In this section, our goal is to evaluate the distance from
SABA using synchronization boundary and pacemakers'
parameters. As it was mentioned in Section 2, the
boundary is determined by S-N and Hopf bifurcation
conditions. If 2
1.122 0
, the boundary is de-
termined by S-N bifurcation condition curve. Otherwise,
the synchronization boundary is identified by Hopf bi-
furcation condition curve. For distance evaluation, at
first, the pacemakers' parameters should be mapped from

12
,, ,
 
space to

,,

space.

22
212
1
1,,
2
 
  (34)
The second step is to find a point on synchronization
boundary in which has the minimum distance from
. Assume the point
,,

22
,,


is the desired
point. By inverse transformation of this point using (35),
the minimum distance from SABA can be obtained.
22
2min 122
1
,1


(35)


22
2
22min 2222
Min. Dist. from Blocking Arrh.
 


(36)
The crucial point is how to find the desired point. The
Eq. 21 contains two curves that one of them pass through
origin point (0, 0, 0) and forms the first part of synchroniza-
tion boundary. For simplicity, we approximate this curve by
a line which passes from origin and cusp point.
(37)
20.48485
Therefore, the synchronization boundary is defined by
Eqs. 28 and 37. Suppose

22
,,
PP
P

 is the
pacemakers' parameters which is transformed to
,,

space. Define the distance of a point on
boundary
,,Q

from P as,

22
2
22 22
PP
Dis
 
  
2
P
(38)
By minimizing 38 subject to 37 and again subject to
28 individually and taking into account their minimum
value, the desired point
is achieved.
Therefore, two minimization problems should be solved.
22
,,



2
122 2
2
. .,,0.484850
1.122 0
Minimize Dis
ST f
 
 

(39)
Using Lagrange coefficient and Kohn-Tucker method
for considering inequality constraints we have,

2
112212
, ,1.122LDis f
2
 
 (40)


22 1
,,
1
1
22
0
0
0
1.122 0
L
L





 
(41)
314 H. Gholizade-Narm et al. / J. Biomedical Science and Engineering 3 (2010) 308-316
Copyright © 2010 SciRes
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The parameters
22 12
,,,,,

22

are gained by
solving 41. If conditions 42 and 43 are satisfied, the dis-
tance is evaluated from obtained , otherwise,
the extremum of those parameters which do not satisfy
the conditions is substituted to evaluate distance.
,,
2
0
1.122 0

 (42)
and
1
2
0
0
(43)
In second time, the desired point is sought on the (37)
surface.

2
22
22222 2
2
Minimize Dis
1
.., ,20
4
1.122
ST f
 


2
(44)
Again, using Lagrange coefficient and Kohn-Tucker
method, values are obtained and distance is
evaluated. By comparing the two evaluated distance and
considering the minimum value, the desired point is
identified. By inverse transformation of this point to
22
,,

12
,
,,

space according to Eq. 35, the minimum
distance from SABA is evaluated using (36). The distance
evaluation is illustrated in Ex. 3 for a typical sample.
5. IDENTIFICATION OF MAXIMUM
ADMISSIBLE VARIATION OF
PACEMAKERS' PARAMETERS
In this section, our goal is to evaluate the maximum
variation of a parameter subject to remaining synchroni-
zation whiles other parameters are fixed. For instance,
how much the coupling coefficient can be decreased
whiles two pacemakers remain in phase synchrony or
how much the discrepancy of two natural frequencies
can be increased and the synchronization remain un-
changed.
5.1. Maximum Admissible Decrease of Coupling
Coefficient
For identification of maximum admissible variation of
coupling coefficient, at first, we transform the pacemak-
ers' parameters to same usable form in synchronization
boundary equations using Eq. 34. Now, we consider
coupling coefficient as unknown parameter and obtain it
from Eq. 37. If the condition (42) is satisfied, the ob-
tained value is the desired coefficient otherwise we will
compute it from Eq. 28. In any way, assume the desired
coefficient is2
. Therefore the maximum admissible
decrease of coupling coefficient is,
2
max22
 
 (45)
In other words, the minimum of coupling coefficient
subject to remaining synchronization is,
2
2min
(46)
5.2. Maximum Admissible Increase Discrepancy
between Two Pacemakers’ Natural
Frequencies
For identification of maximum admissible increase dis-
crepancy between two Pacemakers' natural frequencies,
like Section 5-1, at first, we transform the pacemakers'
parameters from
12
,, ,
 
space to
,,

space using Eq. 34. Now, we assume that is the un-
known parameter and obtain it from Eq. 37. If the con-
dition (42) is satisfied, the obtained value is desired
otherwise we compute it from Eq. 28. In any way, as-
sume the desired value is the. Therefore the maxi-
mum admissible increase of discrepancy between two
Pacemakers' natural frequencies is,
max12 min
22
2min 1
1




(47)
The damping coefficient is one of the determinative
parameters for synchronization boundary but we ignore
to find its maximum admissible variation for its little
importance.
6. SIMULATION
The distance evaluation from SABA is completed in 3
steps: at first, the synchronization boundary is deter-
mined, and then the pacemakers' parameters estimated.
Finally, the minimum distance from boundary is com-
puted. In this section, the synchronization boundary is
obtained for a typical sample and plotted (Ex. 1). After-
ward, the performance of introduced adaptive observer is
illustrated in Ex. 2. Then the distance from SABA is
shown in Ex. 3. As a final point, the maximum admissi-
ble variation of parameters is calculated in Ex. 4.
Ex. 1. Assume 21
, the synchronization boundary
is determined by (21) and (28). Figure 1 shows the syn-
chronization boundary and highlights the cusp points.
Ex. 2. Assume pacemakers' parameters as 12,
22
11222
64,4,1,25,2


2
. The para-
meters are chosen such that the synchronization fre-
quency is 70 per minute and the natural frequency of
pacemakers equal to 76 and 47 per minute for SA and
AV nodes respectively. Figure 2 illustrates the state-
H. Gholizade-Narm et al. / J. Biomedical Science and Engineering 3 (2010) 308-316
Copyright © 2010 SciRes
315
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-1 -0.8 -0.6 -0.4 -0.20
0
0.1
0.2
0.3
0.4
0.5
0.6
X: 0.033
Y: 0.5352
SN and Hopf Bifurcation Condition Curves
Synchronization Boundary
2
X: -1.122
Y: 0.5417
SN Bif.
SN Bif.
Hopf Bif.
(a)
-2 -1.5-1-0.5 00.5 1
0
0. 5
1
1. 5
2
2. 5
SN and Hopf Bifurcation Condition Curves
Synchronization Boundary
2
SN Bif.
SN Bif.
Hopf Bif.
(b)
Figure 1. Phase synchronization Boundary. a. small discrepancy
of natural frequency b. big discrepancy of natural frequency.
parameter estimator performance.
Ex. 3. Suppose pacemakers’ parameters as Ex. 2
for using synchronization boundary to evaluate dis-
tance from SABA, we should transform parameters
at first.

222
0.125 ,0.05,
,,2.75,1, 4.875
PPP
P



Since , therefore the desired
point is on Hopf bifurcation condition curve. By solving (44)
we have,. We
see that condition (44) is satisfied. The distance from
SABA is,
4.875 1.122
P

22
2.1409, 1.021


4, 5.1252

0 1 23 45 678 910
-100
0
100
1
0 1 23 45 678 910
-100
0
100
1
2
0 1 23 45 678 910
-100
0
100
1
Tims (S)
012345678910
-200
0
200
2
012345678910
-50
0
50
2
2
012345678910
-50
0
50
2
Tims (S)
Figure 2. System's parameter estimation.


2
2min
22
22
2
22min22 22
648 5.125222.9284,
8 2.140917.1272,1.0214
Min. Dist. from Blocking Arrh.
23.7894

 


 
 
 
Ex. 4. The maximum admissible variation for Ex. 3 pa-
rameters are,
2min
222
max2
2
max
max222 min
1
2.0374
16.2992 5.7008
6.61 1.7318,
11.1456 1.6615



 



 

7. CONCLUSIONS
The evaluation of distance from SA blocking arrhythmia
for a healthy heart was investigated in this paper. To do
316 H. Gholizade-Narm et al. / J. Biomedical Science and Engineering 3 (2010) 308-316
Copyright © 2010 SciRes
this, the synchronization boundary was obtained using
perturbation method. The synchronization boundary de-
fines the illness boundary approximately. To compute
the distance from this boundary, an estimator was intro-
duced which can estimate SA and AV nodes’ parameters
from their action potential signals. Using optimization
methods, the minimum distance from SABA was evalu-
ated. This value can be considered as an alarm when it is
very small. The coupling intensity and discrepancy of
natural frequency of two pacemakers play a crucial role
in synchronization of two basic pacemakers. Hence, a
method was proposed to evaluate the maximum admis-
sible variation of these parameters. The maximum varia-
tion value demonstrates the confidence margin.
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[12] Fitzhugh, R. (1961) Impulses and physiological in theo-
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