Mathematical Model for the Effect of Ghrelinon basal,GNRH Induced FSH and LH Secretion in Normal Wom

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American Journal of Engineering Research (AJER) 2016 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-5, Issue-12, pp-124-129 www.ajer.org Research Paper Open Access

Mathematical Model for the Effect of Ghrelinon basal,GNRH Induced FSH and LH Secretion in Normal Women by using four Variate Weibull Distribution S. Lakshmi1 and P. Velvizhi2 1

Principal (Rtd) (K.N. Govt Arts College for Women, TNJ), Thanjavur-613001, Tamilnadu Associate Prof. of Mathematics, M.I.E.T Engineering College, Gundur, Trichy-620007, Tamilnadu

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ABSTRACT: In this paper, we introduce probability density function of four variate Weibull distributions. A multivariate survival function of Weibull Distribution is used for four variables. From the survival function, the probability density function and cumulative probability function are derived. Ghrelin may affect reproductive function in animals and humans.In the application part the experimental conditions of an acute injection of ghrelin (1ď ­g/kg) to normal women, basal and GnRH-induced LH and FSH secretion were not affected and suggested that ghrelin does not play a major physiological role in gonadotrophin secretion in women.In the mathematical part, we have found that the Survival function of the curveis suddenly decreased in Mid-luteal phase compare with other phases. Pdf of the curve is suppressed in late follicular phase and it will be increased at the time of7min.Pdf for early follicular phase of control cycleis increased from 4 min .Also Pdf curve for early follicular phase with ghrelin administration and mid-luteal phase with ghrelin and GnRH are also increased at the time of 5 and 3 minutes respectively. Keywords: Ghrelin, FSH, LH, GnRH, Estradiol, Progesterone, four variate Weibull Distribution, Survival function, PDF.

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MATHEMATICAL MODEL

1.1. Introduction Lu and Bhattacharyya (1990, [10]) developed a joint survival function by letting h 1(x) and h2(y) be two arbitrary failure rate functions on [0,ď‚Ľ ), and H1(x) and H2(y) their corresponding cumulative failure rate. Given the stress S= s 0, the joint survival function conditioned on s,as they defined is đ??š (x, y đ?‘ = expâ Ąâˆ’ đ??ť1 đ?‘Ľ + đ??ť2 đ?‘Ś đ?›ž đ?‘ , Where measures the conditional association of X and Y. Further, based on the joint survival function, they proved a theorem that a bivariate survival function đ??š (x, y đ?‘ can be derived with the Marginalđ??šđ?‘Ľ and đ??šđ?‘Ś given the assumption that the Laplace transform of the stress S exist on [0,ď‚Ľ ) and is strictly decreasing. From the theorem, they derived a bivariate Weibull Distribution đ??š (x, y) = đ?‘’đ?‘Ľđ?‘? −

đ?‘Ľ

ď §1 ď Ą

ď Ź1

+

đ?‘Ś

ď Ź2

ď §2 ď Ą ď Ą

,

Where 0ď€źď Ąď‚Ł 1, 0ď€źď Ź1 , ď Ź2 , and 0ď€źď §1 ,ď §2 . This bivariate Weibull Distribution is exactly the same as developed by Hougaard(1986, [3]). By the same steps, the theorem can be expanded to more than two random variables and therefore, a multivariate survival function of Weibull distribution is constructed as ď §1

S(x1, x2 ‌‌‌, xn) =đ?‘’đ?‘Ľđ?‘? −

đ?‘Ľ1 ď Ą

ď Ź1

ď §2

+

đ?‘Ľ2 ď Ą

ď Ź2

+ ⋯‌..+

đ?‘Ľđ?‘›

ď Źđ?‘›

ď §đ?‘› ď Ą ď Ą

,

(1)

Where ď Ą measures the association among the variables, 0ď€źď Ąď‚Ł 1, 0 ď€źď Ź1 , ď Ź2 , ‌ . . ď Źđ?‘› , and 0ď€źď §1 ,ď §2 ‌ ‌ ‌ ď §đ?‘› . 1.2 Probability density function of the Multivariate Weibull Distribution The multivariate probability density function f( x1, x2, ‌‌., xn ) of a multivariate distribution function can be obtained by differentiating the multivariate survival function with respect to each variable. Li (1997, [9]) has shown that

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