A Trivariate Weibull Model For Oestradiol Plus Progestrone Treatment During The Preovulatory Period In Normal Premenopasual Women

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1 American Journal of Engineering Research (AJER) 7 American Journal of Engineering Research (AJER) e-issn: -847 p-issn : -96 Volume-6, Issue-, pp-6- Research Paper Open Access A Trivariate Weibull Moel For Oestraiol Plus Progestrone Treatment During The Preovulatory Perio In Normal Premenopasual Women S. Lakshmi * an K. Sheeja** *Principal(Rt.), Kunthavai Naacchiyaar Govt. Arts College for Women,(Autonomous),Thanjavur-67 lakshmi98@yahoo.co.in **Assistant Professor of Mathematics,SIES (Nerul) College of Arts Science an Commerce, Nerul, Navi Mumbai- 4, Maharastra. sheejak@sies.eu.in ABSTRACT: In the previous stuies a hypothesis was evelope from the results that the increase in leptin concentrations uring the secon half of the menstrual cycle may be relate to changes in the steroial milieu uring the preovulatory perio an the luteal phase. The present stuy was unertaken to test this hypothesis further by examining the effect of treatment with oestraiol an progesterone on leptine concentrations in normal pre menopausal women. The trivariate Weibull moel is use for fining survival functions an probability ensity functions for corresponing values of oestraiol, LH an FSH for both untreate an treate with oestraiol an oestraiol plus progesterone respectively. Keywor: Trivariate Weibull, LH, FSH, oestraiol an progesterone AMS Classifications: 6HXX, 6EXX I. INTRODUCTION Normal women have significantly higher serum leptin concentrations than normal men []. Also in the luteal phase of the normal menstrual cycle, serum leptin concentration are significantly higher than in the follicular phase [7,9], while significant positive correlations have been foun between leptin an oestroiol or progesterone concentrations[]. Although these stuies suggest that ovarian sterois may play a role in leptin secretion in humans no irect evience has been provie as yet. A significant ecrease in serum leptin concentration was foun following bilateral ovariectomy in normal women [], an although treatment with oestraiol was without any effect, the aition of progesterone prevente this ecrease, suggesting that progesterone plays an important role in the control of leptin secretion []. A hypothesis was evelope from these results that the increase in leptin concentration uring the secon half of the menstrual cycle may be relate to changes in the steroial milieu uring the periovulatory perio an the luteal phase. The present stuy was unertaken to test this hypothesis further by examining the effect of treatment with oestroiol an progesterone on leptin concentration in normal premenopausal women. II. MATHEMATICAL MODEL Trivariate Weibull Distribution: Lu an Bhattacharya [5] evelope a joint survival function by letting h (x) an h (y) be two arbitrary failure rate functions on [, ), an H (x) an H (y) be their corresponing cumulative failure rate. Given the stress S=s >, the joint survival function conitione on s, as they efine is, F (x,y/s)=exp{- [H (x)+ H (y)] γ s}, where γ measures the conitional association of X an Y. Further, base on the joint survival function, they prove a theorem that a bivariate survival function F (x,y/s) can be erive with the marginals F x an F y given the assumption that the Laplace transform of the stress S exists on [, ) an is strictly ecreasing. From the theorem, they erive a bivariate Weibull Distribution w w w. a j e r. o r g Page 6

2 American Journal of Engineering Research (AJER) 7 γ + y F (x,y) = exp x where <, < λ,, <, < This bivariate Weibull Distribution is exactly the same as evelope by Hougaar [8]. Following the same steps the theorem can be expane to more than ranom variables an, therefore, a multivariate survival function of Weibull Distribution is constructe as S(x,x, x n )=exp x + x + x n Where measures the association among the variables, <, an <,,. < an <,,.. γ n < [,] Probability Density Function of the Multivariate Weibull Distribution: The multivariate probability ensity function f(x,x, x n ) of a multivariate istribution can be obtaine by ifferentiating the multivariate survival function with respect to each variable. Li [4], an Yi an Weng [] ha shown that f(x,x, x n )= (-) n n S x, x.x n x x x n using Li s erivation an one of the special cases of the multivariate Faa i runo formula by Constantine an Savits [], the probability ensity function is f(x,x, x n ) = n exp x + x + x n. p(n) i= γ n x k ip s n, i + x n i= n j + x n x + x + x n k i n where k i is the number of summans of the i th partition of n such that n +n +.n ki = n, n n n ki >, k i n; n j is equal to ( ).. ( n j ), the falling of ; p(n) is the total number of set partitions of the set S n ={,,n} corresponing to the i th partition of n. The PDF of trivariate Weibull is f(x,x, x n )= (-) S x, x, x x x x = x x x exp x. x x x. + + x xλγ+xλγ+xλγ + x. x + x + x + x + x + x + + x + x + x + III. RESULTS AND DISCUSSION Serum concentrations of oestraiol, progesterone LH an FSH uring the aministration of oestraiol oestraiol cycles an the control cycles in the women are shown in Figure [ to ]. Concentrations of these hormones on cycle ay were similar between the oestraiol an the control cycles. A marke increase in oestraiol concentrations was seen in all women from the onset of treatment on cycle ay to ay 5. Subsequently, oestraiol concentrations ecline rapily an returne to pretreatment level on cycle ay 7, an showe no significant changes thereafter. In the control cycle, serum oestraiol concentrations increase slightly from ays to 7 an progressively from ays 7 to. From ays to 6 oestraiol concentrations was slightly higher, an on ays 9 an w w w. a j e r. o r g Page 7

3 American Journal of Engineering Research (AJER) 7 were slightly lower, in the oestraiol cycle than in control cycle. Serum progesterone concentrations were low throughout the experimental perio in both the oestraiol an control cycle. The changes in serum oestraiol, progesterone,lh an FSH concentrations in the six cycles treate with oestraiol plus progesterone an in the corresponing control cycles are shown in Figure [4 to6]. Pretreatment concentrations of these hormones on cycle ay i not iffer significantly between the two groups of cycle. During treatment with oestraiol an progesterone serum oestraiol concentration showe a pattern of increase similar to that in oestraiol cycle, with no significant ifference at the corresponing points. Oestraiol concentrations were significantly higher on ays, 4, an 5 an significantly lower on ays 9 an in the oestraiol plus progesterone cycles than in the control cycles. Serum progesterone concentrations in the oestraiol + progesterone cycles showe an abrupt increase from ays to 4, remaining high on ays 5 an 6 an ecreasing thereafter. This stuy is first to show an increase in preovulatory concentration in normal women uring treatment with exogenous oestraiol an progesterone. It is of interest that when oestraiol alone was given to the women there was no change in preovulatory concentration, but the increase was observe when progesterone was ae to the oestraiol regimen. This suggests that in the presence of progesterone an normal oestraiol the preovulatory concentration increases, but however whether the progesterone has the same effect in the presence of low oestraiol nees to be investigate. The aim of this stuy was to examine whether a sequence of events in terms of changes in oestraiol an progesterone concentrations resembling those in the normal menstrual cycle, can affect preovulatory concentration. During treatment with oestraiol an progesterone, serum concentration of oestraiol showe a pattern of increase similar to that seen uring the preovulatory perio of the normal menstrual cycle, an although progesterone concentration showe an abrupt increase, the concentration was similar to those seen in luteal phase of the cycle. IV. CONCLUSION The meical conclusion shows that the preovulatory concentrations of oestraiol inuce in the early follicular phase of the cycle by the aministration of oestraiol. However inuction of luteal phase concentrations of progesterone, in aition to the increase concentration of oestraiol can stimulate ovulatory concentration. Mathematical Figure 7 an 8 shows the survival function of combine effect of LH, FSH an Oestraiol with aministration of oestraiol an oestroiol plus progesterone for the untreate an treate cases. Figure 9 an show that the probability ensity function of combine effect of LH, FSH an Oestraiol with aministration of oestraiol an oestroiol plus progesterone for the untreate an treate cases. REFERENCES [] Constantine, G.M., Savits, T.H., (996). A multivariate Faa Di Bruno Formula with Applications. American Mathematical Society, Vol 58, pp 5-5. [] Crower, M., (989). A Multivariate Distribuition with Weibull Connections. Journal of the Royal Statistical Society, Series B, Vol 5, pp -7. [] Frees, E. W., Valez, E. A., (998). Unerstaning Relationships using Copulas. North American Actuarial Journal Vol, pp -5. [4] Li, C.L., (997). A Moel for Informative Censoring. Ph.D. Dissertation: The University Alabama at Birmingham. [5] L u, J-C., Bhattacharya, G., (99). Some New Constructionsof Bivariate Weibull Moels. Annals of the Institute of Statistical Mathematics, Vol 4, pp [6] Luwig. M., Klein, H.H., Dierich, K. an Ortmann, O. (). Serum leptin concentrations throughout the menstrual cycle. Arch. Gynecol. Obstet., pp [7] Harie, L.,Trayhum, P., Abramovich, D. an Fowler, P. (997) Circulating leptin in women: a longituinal stuy in the menstrual cucle an uring pregency. Clin. Enocrinol., Vol 47, pp -6 [8] Hougaar, P., (986) A Class of Multiariate Failure Time Distributions Biometrica Vol 7, pp [9] Mannucci, E., Ognibene, A., Becorpi, A. et al. (998) Relationship between leptin an oestrogen in healthy women. Eur. J. Enocrinol., Vol 9, pp 8- [] Messinis, I.E., Milingos, S.D., Alexanris, E. et al. (999) Leptin concentration in normal women following bilateral ovariectomy. Hum. Repro., Vol 4, pp 9-98 [] Messinis, I.E., Kariotis, I., Milingos, S, et al. ()Treatment of normal women with oestroiol plus progesterone prevents the ecrease of leptin concentrations inuce by ovariectomy. Hum. Repro., Vol 5, pp 8-87 [] Rosenbaum, M., Nilcoson, M., Hirsch, J et al. (996) Effects of gener, boy composition an menopause on plasma concentration of leptin. J. Clin. Enocrinol. Metab., Vol 8, pp w w w. a j e r. o r g Page 8

4 American Journal of Engineering Research (AJER) 7 [] Yi, Z.,Weng, C., (6). An Application of the Power Approximation in Multiple Life Insurance, Mathematics an Economics, Vol 8, pp 98-. Figure: Figure:4 oestraiol Oestraiol Figure: LH.5 Figure:5 LH Figure: Figure:6 FSH. FSH untreate Figure[,,]: Serum oestraiol, progesterone,lh an FSH concentration uring the follicular phase untreate spontaneous cycles an cycles treate with oestraiol through skin patches in normally cycling women. Figure[4,5,6]: Serum oestraiol, progesterone,lh an FSH concentration uring the follicular phase untreate spontaneous cycles an cycles treate with oestraiol through skin patches plus progesterone intravaginally in normally ovulating women. w w w. a j e r. o r g Page 9

5 American Journal of Engineering Research (AJER) 7 MATHEMATICAL FIGURES Survival functions Figure:7 Figure: Oestroiol, LH an FSH without progestrone S(x,x, x n ) oestraiol, LH,FSH with Progestrone S(x,x, x n ) Figure:7 : Denotes Survival functions of oestroiol, LH an FSH when treate with only oestroiol Figure:8 : Denotes Survival functions of oestroiol, LH an FSH when treate with oestroiol plus progesterone. Pf Figures: Figure Oestraiol, LH &FSH without progestrone f(x,x,x ) Figure: 9: Denotes trivariate Probability ensity function of oestroiol, LH & FSH when treate with only oestroiol Figure: Figure: : Denotes trivariate Probability ensity function of oestroiol, LH an FSH when treate with oestroiol plus progesterone w w w. a j e r. o r g Page

Mathematical modeling of follicular development in bovine estrous cycles

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