IJMS, Vol. 11, No. 3-4, (July-December 212), pp. 463-469 Serials Publicaions ISSN: 972-754X THE MATHEMATICAL MODEL FOR THE SECRETION OF LUTEINIZING HORMONE BY USING PROPORTIONAL HAZARD MODEL S. Lakshmi & M. Agalya Absrac: The mixure of increasing failure rae disribuion can decrease, aleas some inervals of ime. We consider a survival model ha generalizes proporional hazard models very well known in reliabiliy and survival analysis. Under reasonable assumpions a medical model is fied wih he disribuion. Chronic sress leads o suppression of he hypohalamicpiuiary gonadal axis wih decreased plasma LH concenraions. For finding serum Lueinizing hormone concenraions, he adminisraion of Naloxone produced a ransien bu significan increase in immuno-radiomeric LH levels compared o he NS conrol value and oher cases also verified in he medical (sec-5) and mahemaical par (sec-6) and conclusion is obained in (sec-7). Mahemaical Subjec Classificaion: 6G xx, 62H xx, 62P xx. Keywords: Mixure of disribuions, Proporional hazard model, LH. 1. INTRODUCTION I is hard o find homogeneous populaions in real life, and mixures of disribuions usually presen an effecive ool for modeling heerogeneiy. A naural specific approach o his modeling explois he noion of a nonnegaive, random, unobserved parameer (fraily), Z, inroduced by Vaupel e al., [11] in a demographic, conex. This in fac leads o he consideraion of a random failure rae, (, Z). As he failure rae is a condiional characerisic, he ordinary expecaion wih respec o Z, E [ (, Z)], does no define a mixure failure rae, m (), and proper condiioning should be performed [3]. A convincing experimen showing he deceleraion of he observed failure rae occurs in naure. I is well known ha human moraliy follows he Gomperz lifeime disribuion wih an exponenially increasing moraliy rae. Assume ha heerogeneiy can be described by he proporional gamma fraily model. (, Z) = Z e, Where and are posiive consans defining a baseline moraliy rae. Owing o is compuaional simpliciy, he gamma fraily model is largely he only one so far used in 1 s Inernaional Conference on Mahemaics and Mahemaical Sciences (ICMMS), 7 July 212.
464 S. Lakshmi & M. Agalya applicaions. I can be shown ha in his case he mixure failure rae, m (), is monoone on [, ] and ends o a consan as. I is monoone increasing, however, for he acual parameer values of his model. 2. PROPORTIONAL HAZARDS MODEL The baseline failure rae is usually denoed as () Therefore, model is and he mixure failure rae is given by (, z) = z (), () = ( u) du, (1) m z e z dz z ( ) ( ) e z ( z) dz. (2) Theorem 2.1: Assume ha he mixing PDF (z) in model (1) has suppor on [a, b], a <, b and for z a, can be defined as (z) = (z a) 1 (z a), (3) Where > 1 and 1 (z) is bounded on [, b a], wih 1 (). Then m () ~ a (). (4) I is quie remarkable ha he asympoic resul in his heorem does no depend on a mixing disribuion even when here is a singulariy a z = a. This differs from he case wih a =. Relaion (4) also describes he convergence o he failure rae of he sronges populaion. The explanaion of his difference is quie obvious: owing o he muliplicaive naure of he model, he behaviour of z () in he neighborhood of z = is differen from he behaviour of his produc in he neighborhood of z = a (for he PDF (3)) The mixure failure rae given by (1) can be obained explicily when he Laplace ransform of he mixing PDF. ~ (z), is easily compued, as in special case below. As he cumulaive failure rae is monoone increasing in, he mixure survival funcions is wrien in erms of he Laplace ransforms as z e z dz ( ) ( ( )).
The Mahemaical Model for he Secreion of Lueinizing Hormone... 465 Therefore (2) becomes ( )) m (log ( )) ( ) and he corresponding inverse problem, namely o obain he baseline failure rae, given he mixure failure rae and he mixing disribuion, can also be solved [4]., 3. SPECIAL CASES Le he mixing disribuion be he gamma disribuion wih PDF c 1 z z / b 1 ( z) e b. (5) b( c) Where b, c >. The Laplace ransform of (z) is ~ () = (b + 1) c and, herefore, he mixure failure rae is given by m bc 1 b ( u) du The expeced value of a random variable Z wih PDF (5) is bc, and he variance is b 2 c. Thus for he fixed expecaion E [Z] = 1, he variance is 2 = b and (6) becomes m 2 1 ( u) du Which firs appeared in [11] in he conex of demographics. This form allows us o compare differen mixures for a fixed baseline disribuion. We can see ha, when he variance of he mixing disribuion increases, he mixure failure rae decreases. Obviously, he asympoic behavior of m () can be explicily analyzed. We consider wo specific cases. Firs, if he baseline disribuion is Weibull wih =, hen he mixure failure rae (6) is [6]. (6) ( 1) bc m. (7) 1 1 b Which converges o as and saisfies m () ~ ( + 1) c 1, (here c = + 1). Second, if he baseline disribuion is Gomperz wih () = e hen simple ransformaions resul in
466 S. Lakshmi & M. Agalya m e ce / b 1. (8) If b = / hen m () c, if b > / hen m () increases o /, and if b < / hen m () decreases o /. Le us reurn o a discussion of he convergence of he mixure failure rae o he failure rae of he sronges populaion. I is reasonable o compare he asympoic behaviors in (7) and (8) for he same mixing disribuion (5). For he Weibull CDF, he mixure failure rae converges o. This means ha wihin he framework of he muliplicaive model (1) where he family of failure raes is ordered in z, we sill can speak in erms of convergence o he failure rae of he sronges populaion, defining he case wih z = as having some generalized (or formal) sronges failure rae, (, ) =. 4. APPLICATION Chronic Sress, such as occurs in anorexia nervosa and depression, may be associaed wih he clinical picure of hypogonadoropic hypogonadism wih decreased LH pulsailiy. This is considered a physiological effec of onically elevaed sress hormones causing a secondary suppression of he hypohalamic-piuiary-gonadal (HPG) axis [2]. Numerous sudies in humans [1, 1], and lower animals [7, 9] have addressed he quesion of wheher CRH, he chief iniiaor of he endocrine sress response, may be he primary hormone ha suppresses LH secreion and hus evenually decreases reproducive capaciy. Furhermore, i has been hypohesized ha he deleerious effec of CRH on he HPG axis is a leas parially mediaed hough he endogenous opiae pepide (EOP) recepor sysem [1]. However, hese repors, all involving he use of exogenously adminisered CRH, have produced conflicing informaion abou he exac role and he sie of acion of his hormone in HPG axis suppression during sress. To dae, he acue effecs of exogenous CRH on LH secreion have no been sudied carefully in he human male. We, herefore, chose o invesigae he effecs of a shor erm infusion of ovine CRH (ocrh) on plasma LH concenraions in a group of healhy young men, employing frequen blood sampling and a sensiive LH immunoassay. We also sough o idenify any poenial ineracions beween ocrh and an EOP recepor anagonis on he secreion of LH in he same subjecs. 5. DISCUSSION This sudy demonsraed ha peripheral ocrh adminisraion does no resul in any significan acue aleraion in serum LH concenraions in healhy young men. This findings confirm he previous observaion ha inhibiion of he EOP recepor wih NAL produces a significan rise in LH levels in men [5, 8], an effec no dampened by exogenous ocrh reamen.
The Mahemaical Model for he Secreion of Lueinizing Hormone... 467 The suppressive effecs of chronic sress on he HPG axis are believed o be due primarily o he influence of elevaed levels of endogenous CRH as well as of arginine vasopressin, inerleukin-1, and a number of oher neuroransmiers [2]. Effors o reproduce he phenomenon of sress-relaed gonadoropin suppression wih exogenous CRH have produced varied resuls depending on he dose and roue of adminisraion of CRH as well as on he species, gonadal sae and opiaergic one of he experimenal subjecs. In ras, he adminisraion of CRH by he inracerebrovenricular, bu no by he iv, roue leads o a promp diminuion in peripheral LH levels (7, 9). Figure 1: Mean (± SEM) serum immunoradiomeric LH levels vs clock ime for all subjecs under various experimenal condiions. For each admission,subjecs received a medicaion or NS bolus a 22 h and an infusion from 22-2 h (see Maerials and Mehods). (a) compares he NS conrol vs. he NAL admission; (b) compares he NS vs. he ocrh admission; (c) compares NS vs. he combinaion (ocrh + NAL) admission; (d) compares he ocrh only vs. he ocrh plus NAL admission. Aserisks and p values denoe poins where here is a significan difference in mean LH levels beween he wo experimenal condiions.
468 S. Lakshmi & M. Agalya 6. MATHEMATICAL RESULTS (a) (b) (c) Figure 2 (d) (i) LH secreion due o saline and Naloxone [Fig. 2(a)] (ii) LH secreion due o saline and ocrh [Fig. 2(b)] (iii) LH secreion due o saline and ocrh + NAL [Fig. 2(c)] (iv) LH secreion due o Naloxone and ocrh + NAL [Fig. 2(d)] These figures are well explained of LH secreions in differen siuaions and corresponding medical conclusions are compared. 7. CONCLUSION By using mixures of increasing failure rae disribuions and proporional hazard model our following mahemaical resuls and conclusion have been obained. LH secreion due
The Mahemaical Model for he Secreion of Lueinizing Hormone... 469 o saline and Naloxone, LH secreion due o saline and ocrh, LH secreion due o saline and ocrh + NAL, LH secreion due o Naloxone and ocrh + NAL are well explained in Figure 2 and hey are monoonically decreasing when ime increasing. REFERENCES [1] Barbarino A., De Marinis L., and Folli G., e al., (1989), Coricoropin-Releasing Hormone Inhibiion of Gonadoropin Secreion During he Mensrual Cycle, Meabolism, 38: 54 56. [2] Chrousos G. P., and Gold P. W., (1992), The Conceps of Sress and Sress Sysem Dis Orders: Overview of Physical and Behavioural Homeosasis, JAMA, 267: 1244 1252. [3] Finkelsein M. S., (24), Minimal Repair in Heerogenous Populaions, J. Appl. Prob., 41, 281 286. [4] Finkelsein M. S., and Esaulova V., (21b), On an Inverse Problem in Mixure Hazard Raes Modeling, Appl. Sock Models Business Indusry, 17: 221 229. [5] Foresa C., Marra S., Federspil G., and Scandellari C., (1983), Effecs of Naloxone on Gonadoropin Secreion in Casraed Young Males Before and Afer Tesoserone Treamen, Horn Meab Res., 15: 567 568. [6] Gupa P. L., and Gupa R. C., (1996), Ageing Characerisic of Weibull Mixures, Prob. Eng. Inf. Sci., 1: 591 6. [7] Ono N., Lumpkin M. D., Samson W. K., McDonald J. K., McCann S. M., (1984), Inrahypohalamic Acion of Coricorophin-Releasing Facor (CRF) o Inhibi Growh Harmone and LH Release in he Ra, Life Sci., 35: 1117 1123. [8] Peragila F., Bernasconi S, and Lughei L., e al., (1986), Naloxone Induced Lueinizing Hormone Secreion in Normal, Precocious and Delayed Pubery, J. Clin Endo Crinol Meab., 63: 1112 1116. [9] River C., and Vale W., (1984), Influence of Coricoopin Releasing Facor on Reproducive Funcions in he Ra, Endocrinology, 114: 914 921. [1] Thomas M. A., Rebar R. W., Labarbera A. R., Penningon E. J., and Liu J. H., (1991), Dose Response Effecs of Exogenous Pulsaile Human Coricoropin-Releasing Hormone on Adreno Coricoropin Corisol and Gonadoropin Concenraion in Agonodal Women, J. Clin Endocrinol Meab., 72: 1249 1254. [11] Vaupel J. W., Manon K. G., and Sallard E., (1979), The Impac of Heerogeneiy in Individual Fraily on he Dynamics of Moraliy, Demography, 16: 439 454. S. Lakshmi M. Agalya Associae Professor of Mahemaics, Assisan Professor in Mahemaics, Kunhavai Naachiyar Gov. Ars College (W), Annai Valankanni Ars and Science College, Thanjavur, India. Thanjavur, India.