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  1. 1. S. Lakshmi, M. Senbagavalli / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 2, March -April 2013, pp.486-491 A New Mathematical Model For Finding MTBF Levels In The Active And Delivery Modes By Using The Hormone Release Of Vasopressin S. Lakshmi * and M. Senbagavalli ** * Associate Professor of Mathematics, Kunthavai Naacchiyar Govt. Arts College for Women (Autonomous), Thanjavur, TamilNadu. ** Research Scholar, Department of Mathematics, Kunthavai Naacchiyar Govt. Arts College for Women (Autonomous), Thanjavur, TamilNadu.ABSTRACT The most common approach for the redistribution of cardiac output and as such playsdevelopment, testing programs include some an important part in the fetal cardiovascularcorrective actions during testing and some response to stress.delayed fixes incorporated at the end of test. This The present study was designed to assesspaper presents an Extended model that addresses the relationship between fetal vasopressin releasethis practical situation given in the application and the progression of parturition as well as thepart of sec 3.In this application the testing mode contribution of specific stimuli to vasopressinis considered as active mode and the delayed secretion during labor. There are practical problemsfixes mode as at the delivery. The result in carrying out experiments of this nature duringconcludes that MTBF is lesser in the action mode spontaneous term labor because of the difficulties inand then increases with the delivery mode. predicting the timing of the parturient process. Therefore, we elected to carry out our1. INTRODUCTION investigations utilizing the more consistent and Vasopressin concentrations are markedly predictable model of premature parturition inducedelevated at birth in the umbilical cord plasma of by infusion of adrenocorticotropin (ACTH) to thehuman infants [5, 10, 15]. Hormone concentrations fetus. Liggins [14] has shown that ACTH-inducedare particularly high in the newborn after vaginal labor is, accompanied by many of the samedelivery or cesarean section preceded by labor. The hormonal changes that accompany spontaneousstimulus to vasopressin release has been attributed the stress and hypoxia that may accompanydelivery, but in the human there are no clear 2. NOTATIONcorrelations [15]. More specifically, Hadeed [10] λ – Scale Parameterhave suggested that cranial compression during β – Shape Parameterpassage through the cervical canal may be the t – Test timeimpetus for fetal vasopressin hyper secretion, T – Total test timewhereas others have linked vasopressin secretion MTBF – Mean time between failureswith the hypothalamic mechanisms initiating or Xi – The i-th successive failure timemaintaining parturition [4]. N – Total number of failures The potential ramifications of enhanced λ A – Type A modes failure intensityvasopressin secretion in the fetus have been more λ B – Type B modes failure intensityextensively explored in the chronically instrumented λ P – Projected failure intensityfetal lamb model. Alexander [1] reported generally λCA – Achieved failure intensityundetectable values for fetal vasopressin (below the λBD – Type BD modes failure intensitylevel of sensitivity of the bioassay) until the last few λEM – Extended model failure intensitydays of gestation; thereafter, hormone MCA – Achievedfailureintensityof MTBFconcentrations rose markedly. In contradistinction, MEM – Extended model MTBFwe found an increased vasopressin secretion onlyafter the onset of spontaneous uterine contractions 3. APPLICATION[18]. A number of stressful conditions including Vasopressin concentrations in fetal plasmafetal hypoxia [3], hemorrhage [2], umbilical cord during various phases of labor are presented inocclusion [9], and hyperosmolar challenge are Figure (i). The mean value before the institution ofaccompanied by increases in fetal plasma infusion was 1.85 ± 0.3 pg/ml (N = 9) and wasvasopressin. The potential for any one of these to similar to that found during the postoperativestimulate the parturient rise in vasopressin is recovery period, 1.3±0.3 pg/ml (N = 25). In fetusesapparent. Iwamoto [13] have speculated that a role infused with ACTH, the mean vasopressinfor vasopressin in the fetus may be in its effect on concentration was 2.0±1.4 pg/ml before labor (N = 486 | P a g e
  2. 2. S. Lakshmi, M. Senbagavalli / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 2, March -April 2013, pp.486-49151); in prodromal and early labor, 5.3±3.4 pg/ml (N the early phase of experimental induction of labor= 25; range, 1.0 to 30 pg/ml); in active labor, were not significantly different from those in39.6± 27.5 pg/ml (N = 28; range, 1.0 to 200 pg/ml); prodromal or early labor; however, hormoneand during the expulsive phase,173.3±152.9 pg/ml concentrations increased significantly during active(N = 20;range,5.0 to1090 pg/ml). The mean plasma labor ( P < 0.05), expulsive labor (P < 0.01), and athormone concentration at delivery in nine animals delivery (P < 0.01).There was a significantwas 584.2 + 433 pg/ml (range, 15 to 4000 pg/ml), increment in hormone concentration between activeand at 30 min after birth, it was 359.8±90.0 pg/ml labor and delivery (P < 0.05), but not between(range, 8 to 2400 pg/ml). Using techniques of values obtained during the expulsive phase of laboranalysis of variance with a fixed random design, the and those obtained at delivery.mean values of fetal vasopressin before and during Fig (i): Plasma vasopressin (pg/ml) in fetal and newborn lambs before , during , and afterpremature parturition induced by infusion of ACTH to the fetus. The values before and during labor representthe mean of from two to individual vasopressin determinations. Fetus 167,( ); fetus 835,( ); fetus 116,( );fetus 195,( ); fetus 396,( ); fetus 884,( ); fetus 482,( ); non-ACTH- infused twin of 396 ( ); fetus 884 ( , ).The fetal plasma vasopressin concentrations at The resting plasma concentrations ofdelivery were strongly correlated with the hormone vasopressin in these experiments in both the ewevalues obtained during the active and expulsive and her fetus are in accord with our previousphases of labor with regression coefficients r = studies [18] and those of other investigators [1, 13,0.955 and 0.985 (P < 0.001). A similar correlation 16, 17] using either bioassay or radioimmunoassaywas not obtained between hormone levels before or techniques for vasopressin determination. Theseduring prodromal and early labor with those at investigations report resting plasma concentrationsdelivery. of the hormone consistently less than 3.5 mu/ml or about 10 pg/ml. The vasopressin levels before labor fit nicely within the lower portion of this range with mean fetal levels of 2.0±1.4 pg/ml and4. DISCUSSION maternal levels of 2.1±0.8 pg/ml. The mean 487 | P a g e
  3. 3. S. Lakshmi, M. Senbagavalli / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 2, March -April 2013, pp.486-491hormone values observed in this study are lower are all modes such that if seen during test nothan have been previously reported. The lower limit corrective action will be taken. This accounts forof sensitivity of our assay is 0.25 pg/ml [11] and all modes for which management determines that itallows for a precise estimate of resting vasopressin is not cost- effective to increase the reliability by aconcentration. design change. Type B failure modes are all modes such that if seen during test a corrective action will5.MATHEMATICAL MODEL be taken. This Type A and Type B determination The most widely used traditional helps define the reliability growth managementreliability growth tracking model and reliability strategy. The basic projection model assumes thatgrowth Projection model address reliability growth the Type A failure modes has constant failurebased on failure modes surfaced during the test. intensity λA , the i-th Type B failure mode followsWith the Projection model all corrective actions are the exponential distribution with failure rate λi ,delayed until the end of test. This Paper Presents an and the initial failure intensity for Type B failureExtended Model that addresses this practical modes is λB .situation and allows for pre emptive corrective An effectiveness factor (EF) dj is theactions. The tradition reliability growth models fraction decrease in λ j after a corrective action has[6,7,8], provide assessments when the failure been made for the j-th Type B mode. The failuremodes corrected are surface during the testing . The rate for the i-th Type B failure mode after afocus of this paper is a combination of these corrective action is (1 ‒ dj ) λ j . In Practice, forapproaches for the practical situations where some application of the Projection Model , the EFs arecorrective actions are incorporated during the test assigned based on engineering assessments, testand some corrective actions are delayed until the results, etc.end of the test. The system failure intensity is constant, Suppose a development testing program say, λS , during the testing and then jumps to abegins at time 0 and is conducted until time T and lower value due to the incorporation of correctivestopped. Let N be the total number of failures actions. The intensity at the end of the test T,recorded and let 0 < X1 < X2 <…..XN < T denote the before delayed corrective actions are introducedN successive failure times on a cumulative time into the system, is the achieved intensity. Thescale. We assume that the Non-Homogeneous reciprocal of the intensity is the achieved MeanPoisson Process (NHPP) assumption applies to this time between failure (MTBF) MS .set of data. Under the basic model the maximum We estimate the achieved failure intensitylikelihood estimates (MLEs) for λ and β( numerator λS byof MLE for β adjusted from N to N-1 to obtain λ̂S =λ̂A+λ̂B, λ̂A= NA /T, λ̂B = NB /T ----- (4)unbiased estimate) are The estimated achieved Mean time 𝑁 𝑁−1 between failure MS at time T, before the jump is λ̂ = T β̂ , β̂ = 𝑁 𝑇 ------------------ (1) 𝑖=1 𝑋𝑖 ̂ MS . We estimate the jump next.The achieved or demonstrated failure intensity and The estimated projected failure intensity [8],Mean time between failures are estimated by isλ̂CA = λ̂ β̂T β̂‒1, -------------- (2) 𝑁𝑗 ̄ λ̂P = λ̂A + 𝑗 𝑀 (1 − 𝑑𝑗 ) 𝑇 + d ĥ (T) --------- (5) =1 𝑀̂M CA = [ λ̂CA ]-1 ----------------- (3) Where ̄ d= 𝑗 =1 𝑑 𝑗 is the average EF, 𝑀 and It is important to note that this model does ĥ (T) = λ̂ β̂T β̂‒1 , --------- (6)not assume that all failures in the data set receive acorrective action. Based on the strategy some The Projection Model λ̂ and β̂ for use only the Mfailures may receive a corrective action and some first occurrence failure times of the seen andmay not. unique Type B failure modes.6. PROJECTION MODEL 7. Extended Reliability Growth Model Suppose a system is tested for time T. The assessment and management metricDuring the testing problem failure modes are structure for corrective actions during and after aidentified, but all corrected actions are delayed and test. We define two types of B modes. Type BCincorporated at the end of the test phase. These failure modes are corrected during test. Type BDdelayed corrective actions are usually incorporated failure modes are delayed to the end of the a group and the result is generally a distinct Type A failure modes, as before, are those failurejump in the system reliability. The projection modes that will not receive a corrective action.model [8], estimates this jump in reliability due to These define the management strategy and can bethe delayed fixes. This is called a “Projection.” changed. This BC and BD failure mode designated The Projection model places all failure is an important aspect of the Extended Model.into two groups, A and B. Type A failure modes 488 | P a g e
  4. 4. S. Lakshmi, M. Senbagavalli / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 2, March -April 2013, pp.486-491Those failure modes that received a corrective 9. VASOPRESSIN CONCENTRATIONSaction during the test (BC modes) and also those Fetal vasopressin concentrations rosefailure modes that will receive a corrective action at significantly during the active and expulsive phasesthe end of the test (BD modes) . During test the of ACTH-induced labor (Fig.i) in a similar fashionType A and Type BD failure modes do not to that we have previously described duringcontribute to reliability growth. The corrective spontaneous labor. These data further reinforce theactions for the BC failure modes affect the increase postulate that fetal vasopressin secretion increasesin the system reliability during the test and this is in response to labor and not before labor. They dothe same for both the management strategy. After not support the concept that fetal vasopressin isthe incorporation of corrective actions for the Type integrally related to the onset of parturition, asBD failure modes, the reliability increases. others have suggested [4].Estimating the increased reliability with the Vasopressin hyper secretion into the fetalobjective of this paper. lamb circulation is not just a momentary Here, for the Extended Model we assume phenomenon occurring at delivery, but alsothat the achieved Mean Time between Failure proceeds during active labor. The significant(MTBF), before delayed fixes, based the data correlations between elevated vasopressinshould be exactly the same as the achieved MTBF concentrations in lamb plasma at delivery with thefor the data. If K is the total number of distinct BD hormone concentrations obtained in the lattermodes then, in the intensity to be estimated is stages of labor may be important in interpreting 𝐾λp = λs ‒ λB + 𝑖=1(1 − di ) λi + dh(T) ----- (7) data obtained from human umbilical cord blood. High concentrations of the hormone are found inTo allow for BC modes in the extended model we human fetal plasma at vaginal delivery [5, 10, 15].replace λs by λCA in (1). Let λBD be the constant These observations of vasopressin hyper secretionintensity for the Type BD modes, and let h(t BD) at the moment of birth may similarly imply a morebe the first occurrence function for the Type BD prolonged exposure of the human fetus tomodes. vasopressin at concentrations which may have The Extended model Projected failure extensive effects on the fetal cardiovascularintensity is system. 𝐾λEM = λCA ‒ λBD + 𝑖=1(1 − di ) λi + dh(T BD) (8) One of the factors responsible for vasopressin release in the latter stage of delivery ̂The Extended model Projected MTBF is MEM = 1/ may be mechanical compression of the fetus duringλEM. uterine contractions and passage through theThis is the MTBF after the incorporation of the cervical canal. Hadeed [10] have attributed thedelayed BD modes. hyper secretion of vasopressin during normal human labor to cerebral compression. We have8. Estimation for the Extended Reliability observed a parallel phenomenon in the rat whereGrowth Model body compression for 60 sec before decapitation The estimate of the projected failure intensity for results in an 100-fold increase in vasopressin levelsthe Extended Model is [12]. It is speculated that compression causes 𝑁𝑗 ̄ impaired oxygenation and/or hemodynamicλ̂EM = λ̂CA ‒ λ̂BD + 𝑗 𝑀 (1 − 𝑑𝑗 ) =1 + d ĥ (T/ BD) 𝑇 alterations which trigger vasopressin release.---------------------(9) 489 | P a g e
  5. 5. S. Lakshmi, M. Senbagavalli / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 2, March -April 2013, pp.486-49110.RESULT 12 M̂ CA = 3.1546 10 8 M̂ EM = 7.837 Vasopressin 6 4 2 0 1 2 3 4 5 6 7 Time in hours 1cm = 7hours Here we have found that, according to fetus,116, 167, 195, 835, 396, 884, 482 cases the corresponding ̂ ̂MEM Values and MCA Values increases in the above curves. We can find the bounds of the above curves to checktheir Monotonicities.11. Conclusion 3. Alexander, D. P., Forsling, M. L., Martin, The Extended Model Projected Mean M. J., Nixon, D. A,, Ratcliff, J. G., ̂Time Between Failure MEM = 7.8370. The achieved Redstone, D., and Tunbridge, D. : TheMean Time Between Failure(MTBF) before the 8 effect of maternal hypoxia on fetal ̂delayed fixed is estimated by MCA = 3.1546. We pituitary hormone release in sheep. Biol.therefore have based on the Extended Model Neonate, 21: 219 (1972).estimates that the MTBF grew to 3.1546 as a result 4. Challis, J. R. G., and Thorburn, G. D.: Theof corrective actions for BC failure modes during fetal pituitary. In: R. W. Beard, P.W.the test , and then jumped to 7.8370 as a result of Natanielsz: Fetal Physiology andthe delayed corrected actions after the test for the Medicine. pp. 233-253 (W. B. SaundersBD failure modes. In the medical mode we Co., Philadelphia, 1976).estimate the MTBF grew to 3.1546 as a result of 5. Chard. T., Boyd, N. R. H., Edwards, C. R.corrective actions for before active mode during the W., and Hudson, C. N.: The release oflabor and then jumped to 7.8370 as a result of the oxytocin and vasopressin by the humandelayed corrected actions after the labor for the fetus during labor. Nature (Lond.), 234:after delivery mode. 352 (1971). 6. Crow .L.H, Achieving High Reliability,REFERENCES RAC Journal, Vol, 4, 2000, Reliability 1. Alexander, D. P., Bashore, R. A,, Britton, Analysis Center, Rome ,NY. H. G., and Forsling, M. L. Maternal and 7. Crow .L.H, Reliability Analysis for fetal arginine vasopressin in the Complex, Repairable Systems, in chronically catheterized sheep. Biol. Reliability and Biometry, ed. By F. Neonate, 25 : 242 (1974). Prochan and R.J Sarling, pp. 379-410, 2. Alexander, D. P., Britton,H . G., Forsling, 1974, Philadelphia, SIAM. M. L., Nixon, D. A,, and Ratcliff. J G.: 8. Crow, L.H., Reliability Growth Projection Pituitary and plasma concentrations of from Delayed Fixes, Proceedings 1983, adrenocorticotrophin, growth Annual Reliability and Maintainability hormone, vasopressin and oxytocin in Symposium, pp.84-89. fetal and maternal sheep during the latter 9. Daniel, S. S., Husain, M. K., Milliez, J., half of gestation and response to Stark, R. I., Yeh, M. N., and James, L. S.: hemorrhage. Biol. Neonate, 24: 206 Renal response of the fetal lamb to (1974). complete occlusion of the umbilical 490 | P a g e
  6. 6. S. Lakshmi, M. Senbagavalli / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 2, March -April 2013, pp.486-49110. cord. Am. J. Obstet. Gynecol., 131: 514 fetal lambs. J. Endocrinol., 42: 323 (1978). (1968).11. Hadeed. A. J., Leake, R. D., and 16. Polin, R. A., Husain, M. K., James, L. S. Weitzman, R. E.: Possible mechanisms of and Frantz, A. G. : High vasopressin high blood levels of vasopressin during concentrations in human umbilical cord perinatal period. J. Pediatr. 94: 805-08, blood. J. Perinat. Med., 5 : 114 (1979). (1977).12. Husain, M . K., Fernando, N., Shapiro, 17. Rurak, D. W.: Plasma vasopressin levels M., Kagan, A., and Glick. S. M.: during hypoxemia and cardiovascular Radioimmunoassay of argining effects of exogenous vasopressin in foetal vasopressin in human plasma. J. and adult sheep. J. Physiol. (Lond.). Clin. Endocrinol. Metab., 37: 616 (1973). 277:341 (1978).13. Husain, M. K., Manger, W. M., Rock, T. 18. Skowsky, W. R., Bashore, R. A., Smith, F. W.. Weiss, R. J., and Frantz, A. G.: G., Jr., and Fisher. D. A.: Vasopressin Vasopressin release due to manual metabolism in the fetus and newborn. In: restraint in the rat: role of body R. S. Comiline, K. W. Cross, G. S. Dawes, compression and comparison with other P. W. Natanielsz: Fetal and Neonatal stressful stimuli. Endocrinology, 104: 641 Homeostasis. Proc. Sir Joseph Barcroft (1979). Centenary Symp. pp. 439-477 (Cambridge14. Iwamoto. H. S., Rudolph, A. M.. Keil, L. University Press. Cambridge, England. C..and Heymann, M. A.: Hemodynamic 1973). responses of the sheep fetus to vasopressin 19. Stark. R. I., Daniel, S. S.. Husain. K. M.. infusion. Clrc. Res., 44: 430 (1979). James, L. S.. and Vande Wiele , R. L.:15. Liggins, G. C.: Premature parturition after Arginine vasopressin during gestation infusion of corticotropin or cortisolinto and parturition in sheep fetus. Biol. Neonate, 35: 235 (1979). 491 | P a g e