E-Bayesian Point Estimation and Comparative Analysis of the Operational Reliability Related to Electronic Items for Medical Purpose Julia Garipova, Anton Georgiev, and Toncho Papanchev Department of Electronics Technical University of Varna Varna, Bulgaria
[email protected] Abstract—This paper is focused on the medical items reliability analysis and comparison of the evaluated reliability indices per three types of measuring items containing electronic and mechanical modules. The analysis is based on the study of exponential model parameters by means of reliability parametric E-Bayesian estimation. The point estimation values of operational reliability indices valid for the medical items under study are experimentally obtained. This paper is presented an extension of previous research concerning the E-Bayesian evaluation of medical equipment operational reliability. Keywords—E-Bayesian point estimates; operational reliability data analysis; type I censored data;
I. INTRODUCTION Reliability of medical electronic systems is an important factor in determining the correct diagnosis of patients and accurately analyzing the specific causes of the disease [1], [2], [3], [4]. The approaches discussed below are of general scope and can be used for medical items reliability evaluation and for evaluation in many other cases as well [5]. The reliability analyzes performed in this paper are based on the assumption that the thermal regimes of medical electronic systems are provided [6]. The basis of this research is real statistical data regarding operational events that occurred during the system operation. The case study is aimed at formalization of both the prior data and accumulated new sampling data related to different types of blood pressure monitors, for the purpose of reliability analysis.
are still functioning. The incomplete data (i.e. during the test not all the items are failed and so their failure times are unknown) is called censored. In this case, the data is right (type I) censored [7]. Therefore, the failure times ti are known and hence, there are defined the times for the survived items ti=t0. Thus, through the total likelihood L(λ), MLE Λ for λ is calculated as shown below nf
L
nf
t i
e
i 1
e t0
nf
i t i
.
(1)
B. Empirical Bayesian Estimation (E-Bayesian Estimation) Let t1, t2,…, tn is the sampling data discussed above. As Bayesian estimator, the Gamma distribution conjugate prior function of failure rate λ is modeled. Eqn.2 represents the resultant probability density function (pdf) [7], where E is a function of the hyper parameters (α, β), and α is the number of total failures occurred in βth time interval
| E
1 e .
(2)
According [7], the posterior Gamma distribution equation is obtained by applying the Bayes theorem. It is presented in Eqn.3, where α'=α+nf and β'=β+Σni=1 ti are the updated hyper parameters, and DN is the new accumulated empirical data
II. THE RELIABILITY APPROACH The approach developed includes the implementation of two estimation methods used to calculate the reliability indices estimation values – maximum likelihood estimation (MLE) and empirical Bayesian (E-Bayesian) estimation.
nw
| D N , E
1 e .
(3)
III. CASE STUDY A. Maximum Likelihood Estimation (MLE) Consider a set of n items that are under observation for up to time t0. A number nf of these items are failed with corresponding failure times t1, t2,…, tn. The other nw=n–nf items
978-1-5386-3419-6/18/$31.00 ©2018 IEEE
Several models of blood pressure monitors are studied – semi-automatic (SABPM) [8], automatic (APBM), and manual (MBPM). Service records are used, and information on operating intervals and failure times is fetched and classified as initial statistical data and latest (subsequent) statistical data.
Table 1 and Table 2 represent the information collected for ABPMs. The exponential distribution of the life (useful) time can be applied for these items. The collected information is presented as sampling data regarding operational events occurring during the operation. Therefore, for the purposes of E-Bayesian estimation, the data from both tables is presented as a prior and posterior data respectively.
TABLE I.
A. MLE of ABPMs Let the statistical data from Table 1 is presented as sampling data regarding operational events that occurredduring the operation of 27 ABPMs (n=27). As can be seen, 7 of all medical items are demonstrated a failure with corresponding failure times t1, t2,…, tnf (nf=13). Concerning the failed items, the failure times are evaluated from statistical data. Therefore, assuming that that the observed time period is
INITIAL STATISTICAL DATA REGARDING AUTOMATIC BLOOD PRESSURE MONITORS (ABPMS)
No
Product
Model
Serial number
Adoption date
Transmission date
Guarantee
Status
Comment
1
Microlife
BP 3AG1
241308***
24.06.2015
24.06.2015
12.06.2014
repair
***
replacement of arm cuff, testing – without deviation testing – without deviation testing – without deviation testing – without deviation testing – without deviation testing – without deviation testing – without deviation replacement of DC power jack, testing – without deviation testing – without deviation operating instructions, testing – without deviation testing – without deviation unfounded claims, testing – without deviation testing – without deviation repairing the AC adapter, testing – without deviation testing – without deviation adjusting of PRV, testing – without deviation testing – without deviation adjusting of PRV, testing – without deviation operating instructions, testing – without deviation testing – without deviation testing – without deviation testing – without deviation testing – without deviation cleaning of ADV, testing – without deviation transistor replacement, testing – without deviation testing – without deviation testing – without deviation
2 3 4 5 6 7
Microlife Microlife Microlife Microlife Microlife Microlife
BP 3AG1 BP 3AG1 BP 3AG1 BP 3AG1 BP 3AG1 BP 3AG1
001306 141208*** 141102*** 141208*** 261405*** 511204***
03.07.2015 06.07.2015 28.07.2015 28.07.2015 31.08.2015 30.11.2015
03.07.2015 06.07.2015 28.07.2015 28.07.2015 31.08.2015 30.11.2015
04.11.2014 18.12.2012 12.10.2012 04.08.2014 04.08.2015 17.04.2013
test test test test test test
8
Microlife
BP 3AG1
491301***
30.11.2015
30.11.2015
10.10.2015
repair
***
9
Microlife
BP 3AG1
241308
18.12.2015
18.12.2015
10.03.2014
test
10
Microlife
BP 3AG1
141501***
23.12.2015
23.12.2015
09.12.2015
test
11
Microlife
BP 3AG1
141500***
22.01.2016
22.01.2016
13.01.2016
test
12
Microlife
BP 3AG1
011407***
22.01.2016
22.01.2016
08.10.2014
test
13
Microlife
BP 3AG1
261403***
14.03.2016
14.03.2016
01.02.2016
test
14
Microlife
BP 3AG1
031301***
30.03.2016
30.03.2016
15.12.2015
repair
15
Microlife
BP 3AG1
261403***
26.04.2016
26.04.2016
13.01.2016
test
16
Microlife
BP 3AG1
011205***
27.05.2016
27.05.2016
05.05.2016
calibration
17
Microlife
BP 3AG1
281511***
20.06.2016
20.06.2016
16.04.2016
test
18
Microlife
BP 3AG1
031301***
20.06.2016
20.06.2016
30.12.2013
calibration
19
Microlife
BP 3AG1
261406***
25.08.2016
25.08.2016
04.09.2015
test
20 21 22 23
Microlife Microlife Microlife Microlife
BP 3AG1 BP 3AG1 BP 3AG1 BP 3AG1
261403*** 141208*** 361506*** 131402***
21.09.2016 21.09.2016 21.09.2016 15.12.2016
21.09.2016 21.09.2016 21.09.2016 15.12.2016
08.06.2015 04.08.2014 15.09.2016 28.06.2015
test test test test
24
Microlife
BP 3AG1
241307***
28.02.2017
28.02.2017
10.02.2014
prevention
***
25
Microlife
BP 3AG1
047900
28.02.2017
28.02.2017
15.07.2016
repair
26 27
Microlife Microlife
BP 3AG1 BP 3AG1
341407*** 481403***
19.05.2017 22.05.2017
19.05.2017 22.05.2017
10.04.2016 23.01.2017
test test
TABLE II.
LATEST STATISTICAL DATA ACCUMULATED REGARDING AUTOMATIC BLOOD PRESSURE MONITORS (ABPMS)
BP 3AG1
Serial number 481603***
Adoption date 08.06.2017
Transmission date 08.06.2017
BP 3AG1
191405***
27.06.2017
***
No
Product
Model
1
Microlife
2
Microlife
Guarantee
Status
Comment
25.05.2017
test
27.06.2017
12.01.2015
repair
testing – without deviation replacement of arm cuff, testing – without deviation testing – without deviation testing – without deviation testing – without deviation testing – without deviation testing – without deviation replacement of arm cuff, testing – without deviation replacement of AC/DC adapter, testing – without deviation testing – without deviation testing – without deviation repair of pump motor, testing – without deviation testing – without deviation testing – without deviation testing – without deviation
3 4 5 6 7
Microlife Microlife Microlife Microlife Microlife
BP 3AG1 BP 3AG1 BP 3AG1 BP 3AG1 BP 3AG1
191406 261403*** 479006*** 101400*** 481603***
27.06.2017 18.07.2017 15.08.2017 12.09.2017 21.09.2017
27.06.2017 18.07.2017 15.08.2017 12.09.2017 21.09.2017
12.03.2016 04.08.2015 15.07.2016 14.08.2014 25.08.2017
test test test test test
8
Microlife
BP 3AG1
491301***
28.09.2017
28.09.2017
12.11.2014
repair
***
9
Microlife
BP 3AG1
191406
23.10.2017
23.10.2017
24.05.2015
repair
10 11
Microlife Microlife
BP 3AG1 BP 3AG1
261405*** 191406***
02.11.2017 22.11.2017
02.11.2017
13.11.2015 06.06.2016
test test
12
Microlife
BP 3AG1
321602***
24.11.2017
24.11.2017
21.03.2017
repair
13 14 15
Microlife Microlife Microlife
BP 3AG1 BP 3AG1 BP 3AG1
521600*** 481601*** 321603***
12.12.2017 15.12.2017 02.02.2018
12.12.2017 15.12.2017 02.02.2018
25.10.2017 15.08.2017 30.08.2017
test test test
equal to the failure time (t0=ti) and by means of (1), the likelihood function can be found (Fig.1). λ, [hours-1]
0,00025
λabpm ABPM
0,0002
λmbpm MBPM
0,00015 0,0001
0,00005 0 0
5000
10000
15000
20000
25000
30000
t, [hours] Fig. 1. Failure Rate Estimation of ABPMs and MBPMs.
The MLE point estimation regarding ABPMs failure rate is evaluated by means of (1) as follows
nf
i t i
8,9705.10 6 item/hour
B. E-Bayesian Estimation of ABPMs Consider sampling data accumulated regarding ABPMs life testing (Table 1). The conjuage prior on λ is modeled through (2). In relation to prior data from Table 1, the hyper parameter α is equaled to the total number of failures occurred from all the previous data, and the hyper parameter β is equaled to the total of all the failure hours in previous test, i.e. α=7 and β=58464. The expression of Gamma posterior pdf is given by (3) with updated hyper parameters α' and β'. From Table 2, the actual hyper parameter values are evaluated as α'=11 and β'=132360. The both of obtained conjugate prior and posterior pdf of the ABPMs under test are presented on Fig.2. 0.3
π(λ|E), π(λ|D N,E)
prior 0.25
posterior
0.2 0.15 0.1 0.05 0 0
0.000005
0.00001
0.000015
0.00002
0.000025
0.00003
0.000035
λ , [hours-1] Fig. 2. Prior and Posterior Gamma Distribution Function of ABPMs.
Consider initial sampling data accumulated regarding MBPMs (Table 3) and latest statistical data accumulated (Table 4). Applying the described approach, the reliability analysis of these type medical items is performed.
C. E-Bayesian Estimation of MBPMs Concerning this case, the conjuage prior on λ is also modeled by (2). In relation to prior data from Table 3, the hyper parameter values are α=17 and β=85032. From Table 4, the actual hyper parameters values are evaluated as α'=23 and β'=138216. The both of obtained conjugate prior and posterior pdf of the MBPMs under test are presented on Fig.3. 0.0016
prior
π(λ|E), π(λ|DN ,E)
0.0014
posterior
0.0012
estimates obtained through Bayesian application are expressed the items failure-resistance. This approach is appropriated to defining deadlines for prevention procedures as well as design electronic items with improved failure-resistance based on past experience. As opposed to other reliability methods such as classical structure reliability method, Markov chains, pattern recognition, etc., the E-Bayesian techniques are provided more flexible reliability analysis and ongoing update of reliability parameters values when the new data is accumulated. REFERENCES
0.001 0.0008
[1]
0.0006 0.0004 0.0002
0 0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
λ , [hours-1]
Fig. 3. Prior and Posterior Gamma Distribution Function of MBPMs.
Fig.4 represents a comparison of the results obtained by applying MLE approach for SABPMs [8], ABPMs, and MBPMs. It is obvius that the MLE point estimation of MBPMs having the highest value due to the frequently observed failures demonstrated in the manometers and cameras of the medical items under test.
λ , [h-1]
0.00004
SABPMs 3.1244E-05
0.00003 0.00002
[2]
[3]
[4]
[5]
ABPMs MBPMs
1.53191E-05 8.97049E-06
0.00001
[6]
0
SABPMs
ABPMs
items under test
MBPMs
Fig. 4. Comparison of MLE failure rate values of the medical items under study.
IV. CONCLUSION This paper discusses two methods (MLE and E-Bayesian techniques) for operational reliability assessing of electronic items during their operational lifetime. The reliability
[7]
[8]
Ts. Georgiev, M. Ivanova, A. Kopchev, Ts, Velikova, A. Miloshov, E. Kurteva. Rheumatol Int (2018) 38: 821, “Cartilage oligomeric protein, matrix metalloproteinase-3, and Coll2-1 as serum biomarkers in knee osteoarthritis: a cross-sectional study,” Springer Berlin Heidelberg. Springer-Verlag GmbH Germany, part of Springer Nature 2017, Rheumatology International, pp 1-10, May 2018, Volume 38, Issue 5, pp 821–830, First Online: 21 November 2017. Ts. Georgiev, R. Dacheva, M. Ivanova, R. Stoilov and R. Rashkov, “The place of autologous platelet-rich plasma in the treatment of knee osteoarthritis: A systemic review,” Revmatologiia (Bulgaria) 24(4), 2016, pp. 3-13. Ts. Georgiev, R. Stoilov, M. Penkov, M. Ivanova and A. Trifonov, “Radiographic assessment of knee osteoarthritis”, Revmatologiia (Bulgaria) 24(2), 2016, pp. 16-24, Medical Information Center. Ts. Georgiev, R. Stoilov, and M. Ivanova, ”Biomarkers in osteoarthritis,” Revmatologiia (Bulgaria) 21(4), 2013, pp. 21-25, ISSN: 13100505, Medical Information Center. N. Nikolaev, S. Yordanov, R. Vasilev, “An optimization algorithm for simulating smart-grid means for distribution grid balacing,” Proc. Of the Second International Scientific Conference “Intelligent Information Technologies for Industry” (IITI’17). IITI 2017. Advances in Intelligent Systems and Computing, vol.679, Springer Cham pp 359-369, 2018. K. Yordanov, T. Mechkarova, A. Stoyanova and P. Zlateva, “Determination of the temperature of cathode unit of indirect plasma burner through a computer simulation model,” Proceedings of the Second International Scientific Conference “Intelligent Information Technologies for Industry” (IITI’17). IITI 2017. Advances in Intelligent Systems and Computing, vol. 680, Springer, Cham, pp. 403-409, 2018 A. O’Conner, M. Modarres, and A. Mosleh, “Probability Distribution Used in Reliability Engieering,” published by Center for Risk and Reliability. University of Martlang, College Park, 2011. J. Garipova, P. Georgieva, and A. Georgiev, “Empirical Bayesian estimates of operational reliability related to electronic items for medical Purpose,” Physics – Eternally Young Science, Kamchia, Bulgaria, 2018.
