J. Biomedical Science and Engineering, 2010, 3, 375-379 JBiSE
doi:10.4236/jbise.2010.34052 Published Online April 2010 (http://www.SciRP.org/journal/jbise/).
Published Online April 2010 in SciRes. http://www.scirp.org/journal/jbise
Heart pacemaker wear life model based on frequent properties
and life distribution*
Qiao-Ling Tong1, Xue-Cheng Zou1, Jin Tang2, Heng-Qing Tong2
1Department of Electron ic Science & Technology, Huazhong University of Science & Technology, Wuhan, China;
2Department of Mathematics, Wuhan University of Tech nolog y, W uha n, China.
Received 4 January 2010; revised 20 January 2010; accepted 25 January 2010.
ABSTRACT
The lifetime of heart pacemaker is important for pa-
tient and researcher. To forecast the lifetime of heart
pacemakers we must determine its distribution regu-
larity. In this paper, a heart pacemaker wear life
model is introduced by using frequency property, and
a life distribution is presented. The parametric prop-
erties of the density are studied. The moment estima-
tor with negative order is used, which is just the
maximum likelihood estimator. A new m ethod of p a-
rameter estimate, F estimate of parameter, is pro-
posed. This method is suitable to both truncated
samples and complete samples, whether or not the
distribution can be transformed into a standard dis-
tribution without any parameters.
Keywords: Heart Pacemaker Life Distribution; Freq u en c y ;
Negative Order Moment Estimator; Maximum Like-
li hood Estimate; F Estimate
1. INTRODUCTION
The heart pacemakers (HPM) have more and more ex-
tensive clinical applications. Almost all users of HPM
worry the lifetime of HPM. If some time his HPM stops
work out of the blue, that will be serious trouble. Post-
poning or advancing to replace or to repair HPM is not
suitable. So forecasting the lifetime of heart pacemaker
is important for patient and researcher.
Extending HPM lifetime is the significant aim to re-
duce the cost of cardiac pacing [1]. The factors which
impact HPM Life content: “innate” factors and “ac-
quired” factors (such as electromagnetic fields, drugs,
etc.) [2], but the former is more important. The innate
factors include many aspects wh ich will b e intro duced as
follows. The battery capacity is decided by battery en-
ergy and battery cubage. For the same functional circuit,
increasing battery capacity will enhance its capacity,
decrease pacing power and prolong the HPM life. Inter-
nal faradic current is a significant, independent factors
related to battery life. Low internal faradic current is an
important role to save battery life [2,3]. Although there
is a little literature to introduce the relativity between
internal faradic current and battery life, some scholars
for this study think that internal faradic current has a
major impact on HPM life and maybe is greater than the
battery capacity and output energy. Markewitz, etc. [3]
found that decreasing the internal faradic current con-
sumption enables to reduce overall current consumption
of HPM and gain a longer life, which approves that the
internal faradic current is a foremost factor of influenc-
ing HPM life. Pacing frequency also has a marked in-
fluence on HPM life, and its level of energy consump-
tion is proportional to the HPM. HPM electrode function
is the weakest link in the system, which is also the most
important factors to determine the security and life of
HPM system [4]. After the HPM pulse generator (Pulse
generator, PG) was designed, the battery capacity and
current consumption has been identified and remain
constant. Improving the wire electrode function is the
only way to prolong HPM lifetime. Electrodes have
many factors which will affect the HPM life. According
to Ohm’s law, the high impedance (> 1000 ohm) can
reduce the loss current, increasing the HPM life.
Lead is also necessary. The main function of electrode
lead is transmitting electricity, but its mechanism capa-
bility is important for assuring the life of pacemaker. The
capability mostly lies on the material of insulating sur-
face. Silica gel may be the best choice, which has been
used over past 30 years. The resistance of lead rest with
its length. The length of electrode with double pole is
two times compared with unipolar one. In the theory,
electrode with double pole uses more energy. Inductive
circuit and pacing with multi-cavity or double cavity
wastes more energy than single cavity and inductive
circuit.
In this paper, a new heart pacemaker life distribution
is presented according to relative influence factors. The
*The research was supported by the National Natural Science Founda-
tion of China (3057 0611, 60773210).

Q. L. Tong et al. / J. Biomedical Science and Engineering 3 (2010) 375-379 376
heart pacemaker may be worn or damaged after working
for a long time, therefore, its vibration and frequency
may be changed. The frequency of a heart pacemaker
can be recorded by a frequency record meter. The heart
pacemaker life and its reliability can be investigated by
the analysis of the heart pacemaker frequency.
Let 1, ,
i
A
in
 be the amplitude values of the fre-
quency of a normal heart pacemaker, 1, ,
i
A
in be
that of the damaged heart pacemaker. We define the fre-
quency param eter k:
1
1
1
ni
ii
A
kn
A




(1)
Experiments have shown that we can deduce the level
of the wear or the damage of a heart pacemaker accord-
ing to the analysis of the frequency parameter k. When
max
kk, the heart pacemaker can not work normally.
The value of the frequency parameter increases as the
work time of a heart pacemaker increases. In general, we
suppose krt, then, we will get k
tr , where r
denotes the change rate of k. It may be affected by
many independent little factors such as capability of
battery, the function of electrode, lead, material and so
on. We suppose that r is a random variable with nor-
mal distribution with definition 0r:
 
2
2
exp, 0
2
2
r
rr
C
fr r



 


 
(2)
From

1
r
frdr
, we know

1
Cr  , where

is the distribution function of standard normal.
Let t be the heart pacemaker wear life, max Tk r

,
then t is also a random variable. The density function
of heart pacemaker wear life distribution is
 

2
22
2exp, 0
2
2
krt
C
fr r
t
rt






 
(3)
When max
kk, its distribution function is


 

1ktr
Ft r
 
 (4)
Now, we particularly investigate statistical property of
heart pacemaker wear life distribution. First, we see the
graph of

f
t. Put 0t , then

0ft  , because
 
2
2exp 2
kt
t





 , put t ,

0ft  . Differentiate

f
t, then
 
22
2kkt r
ft fttt








 (5)
The solution of equation

0ft
 is


22
28
4
k
trr

 
 (6)
Second, we see the change of failure rate .









 
22
2
1
exp 2
1 2
ft
tFt
kt r
kr
rktr rt


 


 
 
(7)
Let

2
2
2
1
exp 1
2
krkr
gt tr
tt


 


 


 


(8)
then




tcgt , where c is a constant. Because


0
lim
tgt

 (9)
and





2
22
2
2
2
1
lim explim
2
1
explimexp 22 22
tt
t
rktr
r
gt t
rktkr
t

 

 







 
  

 

 
  


We have


0
limlim 0
t
ttt



. The graph of


t

is also located on the first quadrate. Its left limit is origin
and its right asymptotic line is t axis.
2. MOMENT ESTIMATE WITH
NEGATIVE ORDER FOR COMPLETE
SAMPLE
When we investigate the parameter estimate of the den-
sity function


f
t of heart pacemaker wear life distri-
bution, we notice that it has two independent parameters
only. Let ak

, br

, then the density function
of heart pacemaker wear life distribution is
 
2
2
1
exp, 0
2
2
aa
ftb t
t
bt



 



 

(10)
In Eq.3,


f
t has three parameters ,,kr

. They
are independent in the physical process, but they are
dependent in the density function. We can estimate ,ab

Q. L. Tong et al. / J. Biomedical Science and Engineering 3 (201 0) 375-379 377
from Eq.10. We need to estimate k in addition, then
we can estimate ,r.
First, we consider moment estimator. For common
moment estimator (), 1,2,
j
Et j, we can not calcu-
late its integration. If we make use of moment estimator
with negative order (), 1,2,
j
Et j  , the integration
can be calculated easily.

 

2
1
3
0
2
0
2
1
exp 2
2
111
exp 2
2
11
exp 2
2
aa
ETb dt
t
bt
yby dy
ab
bb
aab
a
yb
t







 


 











 

 





 (11)

 

2
2
4
0
22
3
22
2
1
exp 2
2
111
exp 2
2
11
exp 2
2
b
aa
ETb dt
t
bt
yby dy
ab
bb b
aab







 



 











 

 

 (12)
Let 1,,
n
tt be i.i.d. sample data of the heart pace-
maker wear life. We make use of the sample moment to
estimate the population moment. The difference is the
sign of the order of moment. From the system of equa-
tions


2
11
22
122
1
11 11
exp 2
2
11 1 11
exp 2
2
n
ii
n
ii
b
sb
nta ab
b
s
b
na tab



 

 

 

 


(13)
we have

2
1
2
1
11
11 1
exp 2
2
bas
sa
ab b
sab


 






 



(14)
Moreover we can obtain the solution a

and b

from
Eq.14. They are the moment estimate of a and b.
Second, we consider the maximum likelihood estima-
tor of pa r a meters. The likelihood function is
 
2
2
1
11
,exp
2
2
n
ni
i
i
aa
Lab b
t
bt


 

  
 
 
   (15)
Taking derivation of


,Lab, we have
 

12
21
1
22
211
1
11 1
exp 2
22
11 1
exp 2
2
0
nn
nii
i
i
nnn
nii
iii
i
i
La a
nb
at
bb
t
aaa
bb
ttt
bt






 

  
 
 
 

  


   
 
 
 
 





(16)
and




12
21
1
22
211
1
11
exp 2
22
11
exp 2
2
0
nn
nii
i
i
nnn
nii
ii
i
i
ab
La a
nb
bt
bb
t
aaa
bb
tt
bt










  
 
 
 

  

    
 
 
 
 





(17)
Simplifying Eq.16 and Eq.17, we have

 
2
221 2
1
1
2
1
1
11
1
1
2
b
as absbas
sa
be
sa bab
bs ab





 
 





 





(18)
Noting that Eq.18 is just Eq.19, we know that the
moment estimator with negative order of heart pace-
maker wear life distribution is just its maximum likeli-
hood estimator. It is more reasonable to take the moment
estimator with negative order for heart pacemaker wear
life distribution.
3. PARAMETER ESTIMATE FOR
TRUNCATED TESTING
Life testing is often truncated by the given size of sam-
ples, because the time and the cost of testing are lim-
ited. Suppose n products are taken as life testing.
We have observed m products which have been
failures. 12m tt t

 are m preceding order sta-
tistics of products life. For exponential distribution,
Weibull distribution, normal and lognormal distribution,
we can give parameter estimator in truncated testing.
These distributions can be transformed into standard
distributions without any parameter. According to the
order statistic of the standard distribution, we can com-
pute its expectation, variance and covariance. Then, we
can obtain the Best Linear Unbiased Estimation by the
least square method. But we can not do it in this way for
heart pacemaker wear life distribution. The reason is it
can not be transformed into any standard distribution
without any parameter. The maximum likelihood esti-

Q. L. Tong et al. / J. Biomedical Science and Engineering 3 (2010) 375-379 378
mate of heart pacemaker wear life distribution from the
order statistics of the truncated samples is not certain to
converge, because its density function does not satisfy
some of the convergency conditions.
In this paper, a new statistical method,
F
estimate of
parameter, is proposed. It is suitable to any distribution
whether or not it can be transformed into a standard dis-
tribution. What is more, it has no definition of sample
size. We can consider the problem in this way. For the
lives of n products in life testing, we have observed
m preceding lives of products. We have not observed
nm
 late lives of products but they are existent. For
the empirical distribution function of life t, we have
observed



11,, nnm
F
tnFtmn. We have not
observed the late values, but they also exist. From
Glivenko-Cantelli Theorem, we know that empirical
distribution function of random variable with probability
one converges to its distribution function uniformly.
That is,
 
lim sup01
n
nt
PFtFt
  







(19)
Therefore we can obtain m approximate equations
 
,1,,
ni i
F
tFti m, i.e.,



2
1
2
i
m
Ft n
Ft n
m
Ft n













(20)
The parameters to be estimated are contained in the
system of Eq.20. The problem is changed into solving a
nonlinear regression model. Solving the model by the
least square method, we can obtain the parameters to be
estimated. We call it
F
estimate of parameter. Obvi-
ously, this method is reliable in theory, simple in com-
putation and extensive in application.
From [5], we know that heart pacemaker wear life dis-
tribution is
 
1ab
t
Ft b

 


 (21)
The regression equations are

1
,1,,
i
ab
tiim
bn

 



 (22)
Making use of the least square method, we need to
seek the minimum by changing parameters ,ab:
 
2
1
1
,min
mi
i
ab
ti
Qab bn



 










 (23)
Or we solve the following system of equations:



2
1
1
2
2
111
exp 0
2
1
11
exp1 exp
22
mi
iii
mi
i
ii
ab
tia
b
bnt t
ab
ti
bn
aa
bb bb
tt





 


 






 












 















 




 

 




 






0







Of course, this method is also suitable to complete
samples, that is mn

.
4. COMPUTATION EXAMPLE AND
COMPARISON OF PRECISION
In order to investigate the properties of heart pacemaker
wear life distribution carefully, and test the new statisti-
cal method,
F
estimate of parameter, we compute
examples on a computer. The parameters ,ab are given
and random numbers are generated and are sequenced
in terms of Eq.10. Using these data, first we compute
the moment estimators with negative order (and also
the maximum likelihood estimators) ˆ
ˆ,
mm
ab based on
Eq.14 or Eq.18. Second, we compute the
F
estimators
of parameters ˆ
ˆ,
F
F
ab using Eq.23. Third, we take
mpreceding data and compute the
F
estimators of
parameters 11
ˆ
ˆ,
F
F
ab according to Eq.23 also. Com-
pared them repeatedly by changing sample size m, pa-
rameters ,ab and truncated number m.
From the results of comparison, we know that it is
robust to take absolute value in Eq.23.
 
1
1
,min
mi
i
ab
ti
Qab bn


 




 (24)
We take the Powell algorithm improved by Sargent to

Q. L. Tong et al. / J. Biomedical Science and Engineering 3 (201 0) 375-379 379
compute Eq.24 and the bisection method to compute
Eq.14.
When data are small samples, we take sample size
20n, truncated number 10m, initial parameters
0.9, 3.0ab. The computed results are in Table 1 . The
random data are generated 5 times and computations are
repeated 5k times. When the random data are generated,
their seeds are differe nt. First of all, these seeds are generated
as random data independently. If it is necessary to calculate a
great number of random data, we can change the modules of
pseudorandom data yet. Using this method, the repeated
number k can be very large. We can calc ulate the em pirical
distribution function of the statistics ,,
mmF
aba and
F
b.
Therefore, we can draw up the distribution function tables of
the statistics for given precision according to Glivenko-
Cantelli Theorem. In any case, we are satisfied with t he com-
puted results in Table 1, beca use the sam ple size is so small.
When data are common samples, we take sample size
100n, truncated number 50m

, initial parameters also
use 9.0a, 3.0b. Computed results are in Ta b l e 2.
Of course, they are more accurate and also make us sat-
isfied.
5. CONCLUSIONS
More and more functions are gathered in heart pacemaker.
It is used more frequently along with the development
ofbiomedical engineering. Extending and forecasting the
lifetime of heart pacemaker is very important to user. In
this paper, we introduced a heart pacemaker wear life
model by making use of the frequency property, and
deduced its life distribution. The parametric properties of
the distribution density are interesting because its moment
Table 1. Computed results with n = 20, m = 10, a = 0.9, b = 3.0.
k m
a m b
F
a
F
b 1
F
a 1
F
b
1 9.746 3.403 8.7893.144 6.541 2.223
2 8.295 2.711 9.8533.228 6.067 1.804
3 8.298 2.523 7.6622.394 7.740 2.331
4 10.675 3.712 13.0324.629 6.980 2.223
5 9.362 3.489 9.9303.836 6.112 2.223
Table 2. Computed results with n = 100, m = 50, a = 9.0, b = 3.0.
k m
a m b
F
a
F
b 1
F
a 1
F
b
1 10.4913.54410.504 3.568 11.2103.823
2 9.7873.2819.628 3.223 9.5473.187
3 9.3373.0868.809 2.890 8.1272.641
4 9.7503.3409.502 3.261 7.7102.485
5 9.4173.3188.957 3.152 9.0463.178
estimator with negative order is just its maximum like-
lihood estimator. We also propose an
F
estimate of
parameter that is suitable to both truncated samples and
complete samples. Through computation example and
comparison of precision, we can get satisfying results.
That means whether or not the distribution can be trans-
formed into a standard distribu tion without any parameters,
we can estimate the parameters in truncated data sample.
6. ACKNOWLEDGEMENTS
We would like to thank the experts who take their time to check, ap-
prove this paper and give us valuable suggestions.
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