Mathematical Model for the Diurnal Rhythm of Leptin in Healthy Young Women

Transcription

1 Intrnational Journal of Scintific and Rsarch Publications, Volu 7, Issu, January 7 5 ISSN 5-5 Mathatical Modl for th Diurnal Rhyth of Lptin in Halthy Young Won Gtha.T * and Sangtha.B ** * Asst profssor of athatics K.N Govt Arts collg for won,thanjavur. Tailnadu, Southindia. ** Lcturr of athatics, BDU collg Orathanadu,Thanjavur,Tailnadu,Southindia Abstract- Nw paratrs can b introducd to xpand failis of distributions for addd flxibility or to construct covariat odls. Introduction of scal paratr usually lads to th acclratd lif odl, and taking powrs of th survival function introducs a paratr that lads to th proportional hazard odl of Morgnstrn gubl bivariat distribution was applid to find th diurnal rhyth of lptin dpnds on nrgy, or carbohydrat, availability, not intak, and xrcis has no supprssiv ffct on th diurnal rhyth of lptin byond th ipact of its nrgy cost on nrgy availability. Indx Trs- Lutinizing horon; pulsatility paratr, survival functions T I. INTRODUCTION h indpndnt ffcts of nrgy availability and xrcis strss on th 4-h an and aplitud of th diurnal rhyth of lptin.bfor thir participation,voluntrs rcivd a dtaild oral and writtn dscription of th scrning procss and xprintal protocol,which was approvd by th Institutional of univrsity.all voluntrs signd an inford consnt docunt. Th daily xrcis tratnt consistd of a sris of suprvisd in bouts of walking on a tradill at 7% of ach individual s arobic capacity, with -in rst intrvals btwn bouts until ach subjct had xpndd kcal. Aftr th baslin days,ditary nrgy intak was controlld by fding subjcts asurd aounts of th liquid ditary product, Ensur. Subjcts wr prittd to drink watr ad libitu, but nothing ls. Coplianc with th nrgy availability tratnts was chckd ach orning by onitoring urinary actoactat lvls with Multistix dip sticks. For ach subjct, 4-h an lptin concntration was calculatd, and lptin ti sris wr xprssd both as absolut concntrations and as rlativ concntrations noralizd to th 4-h an for ach subjct. Fig.Th 4-h group an lptin rhyths at th nd of 4 days of abalancd (j) and low (n). Lft: sdntary tratnt group (S, n 5 7). Right: xrcising tratnt group (X, n 5 9). In S, but not inx, low nrgy availability bluntd th aplitud of th diurnal lptin rhyth as a prcntag of th 4-h an.valus ar ans 6 SE. Mal tis wr at 8, 9, (solid arrows),and (B only, opn arrows).

2 Intrnational Journal of Scintific and Rsarch Publications, Volu 7, Issu, January 7 6 ISSN 5-5 Th rsulting two ti sris for ach subjct wr analyzd by cosinor rhythotry with th assuption of a priod of 4 h. All lptin paratrs wr calculatd fro th raw lptin data and not fro th fittd cosinor curv. Th acrophas and nadir wr dfind as th axiu and iniu lptin concntrations, rspctivly. Aplitud was dfind as on half of th diffrnc btwn th acrophas and th nadir. Figur shows th lptin profils asurd during th 4-h frqunt blood sapling priod. Th profils ar prsntd sparatly as concntrations and as prcntag changs fro ach individual s 4-h an. Profils for balancd and low nrgy availability tratnts ar plottd togthr for coparison. Tabl prsnts suary statistics for th profils shown in Fig. Coparing xrcising won (X) to sdntary won (S) at th sa nrgy availabilitis rvald that xrcis strss had no supprssiv ffct on ithr th 4-h an (P.) or th aplitud (P.) of th diurnal lptin rhyth. As ntiond abov, howvr, th unxpctdly xcssiv riss in lptin during th aftrnoon of th nd day of sapling wr sallr in th xrcising than in th sdntary won. Coparing won rciving low nrgy availability tratnts with thslvs whn thy rcivd balancd nrgy availability tratnts rvald that low nrgy availability strongly supprssd both th 4-h an and aplitud of th diurnal rhyth of lptin. Low nrgy availability bluntd th aplitud of th lptin rhyth by % in all svn sdntary won, and cosinor rhyth analysis was unabl to dtct a significant rhyth in two of th won. By contrast, th rhyth was aintaind in all nin of th xrcising won during th low nrgy availability tratnt, and th aplitud was bluntd by % in only two. Thus thr was an intraction btwn nrgy availability and xrcis strss on both th 4-h an and aplitud of th diurnal lptin rhyth. This intraction was du to th gratr supprssion of both th 4-h an and aplitud of lptin in th sdntary won by low nrgy availability. Th diurnal rhyth of lptin displayd by th won in this xprint aftr th balancd nrgy availability tratnts in figur was siilar to th rhyths obsrvd in othr groups who consud thir dits as oral als.as in so othr xprints howvr, lptin lvls wr uch highr at th nd of th 4-h frqunt blood sapling priod than at th bginning bcaus of an xcssiv ris in lptin during th aftrnoon of th nd day of our 4-h frqunt blood sapling priod in figur probably as an artifact of our fding schdul. Throughout th tratnt priod, w controlld nrgy availability on a 4-h clock fro 8 to 8, but th frqunt blood sapling priod was schduld fro 7 to 7. During th frqunt blood sapling, w adinistrd th balanc of th first day s food btwn 7 and idnight, in kping with th fding schdul during th tratnt priod, but w adinistrd th ntir nxt day s food btwn 9 and, causing an unusually strong driv for lptin scrtion in th aftrnoon. Exrcis strss had no supprssiv ffct on ithr th 4-h an or aplitud of th diurnal rhyth of lptin. II. MATHAMATICAL MODEL Nw paratrs can b introducd to xpand failis of distributions for addd flxibility or to construct covariat odls. Introduction to scal paratr usually lads to th acclratd lif odl, and taking powrs of th survival function introducs a paratr that lads to th proportional hazard odl. Marshall and Olkin (997) introducd a thod of obtaining an xtndd faily of distributions including on or paratr. For a rando variabl with a distribution function and survival function,w can obtain a nw faily of distribution functions calld univariat Marshall-Olkin faily having cuulativ distribution G( givn by G ( ; < x < ; < α < α + ( α) Thn th corrsponding survival function is α ; < x < ; < α < G ( ( α)

3 Intrnational Journal of Scintific and Rsarch Publications, Volu 7, Issu, January 7 7 ISSN 5-5 This nw faily involvs an additional paratr α. In bivariat cas if (X,Y)b a rando vctor with joint survival function F (,thn G( ; < x < ; < y < ; < α <. α + ( α) Constitut Marshall Olkin bivariat faily of distributions. Th nw paratr α rsults in addd flxibility of distributions and influnc th rliability proprtis. Considr th brvariat xponntial distribution of Gub(96) with joint cuulativ distribution function. x (, ) ( ) x x + F x x + x, x, Th survival function of th Morgnstrn Gubl bivariat distribution can b obtaind as x + x x, x ) ( ) G x α ( x ) ; α, x x ( α) X Thr ar univariat Marshall-Olkin xponntial distributions Thor 7.. Lt N b a gotric rando variabl with p(nn)pq n-, N,.,<p<,q-p Considr a squnc {(X ii,y ii ), ii } of ii. ii.d rando variabls with coon survival function F (, N and U N in X i (X ii,y ii ),ar indpndnt for all ii. Lt i N and V N Yi in i N. Thn th rando vctor (U N,V N ) is distributd as Marshall Olkin Gubl Exponntial(MOGE (,p))if and only if (X ii,y ii ),has th Gubl bivariat distribution with paratr. Considr th survival function F. Th nw faily of survival functions is constructd using Marshall-Olkin thod by taking α G( x; α) ; < α < ( α) Whn α, w gt G F Marshall Olkin Gubl bivariat xponntial distribution can b writtn as α x ) G( x ; x ) ; α, x, x ( α) x, x ) α ( x + x ) ( α) Whn Fro ths w gt ( x + x ) α, it asily follows that G F Proof. Lt S (, b th survival function of (UN,V N ). By dfinition p ( p) S ( P(UN x, V N n ( F ( x, y )) i p n pq ( p) MOBGE(,p) n Convrsly lt(u N, V N ) has MOGE(,p)distribution. Thn solving th quation was gt. And G x α ( x ) ; α, x x ( α) X p ( p) p ( p)

4 Intrnational Journal of Scintific and Rsarch Publications, Volu 7, Issu, January 7 8 ISSN 5-5 Thn w gt, ( x + y ) Which is th survival function of th Morgnstrn Gubl xponntial distribution with paratr. Lptin lvl in sdntary tratnt group X X G (X, X ) x x G (X,X) Man for lptin in sdntary tratnt group X X G (X, X ) x x G (X,X x x G (X,X) Lptin lvl in xrcising tratnt group.5 X X G (X, X ) Man for lptin in xrcising tratnt group X X G (X, X )

5 Intrnational Journal of Scintific and Rsarch Publications, Volu 7, Issu, January 7 9 ISSN III. CONCLUSION X X G (X,X) Th survival function of th M.G bivariat distribution was applid in diurnal rhyth of lptin dpnds on nrgy, or carbohydrat, availability, not intak, and xrcis has no supprssiv ffct on th diurnal rhyth of lptin byond th ipact of its nrgy cost on nrgy availability and also found cuulativ distribution function of lptin horon. REFERENCES [] Gubl, E.J. (96) Bivariat xponntial distributions, Journal of th Arican Statistical Association, 55, [] Jana, P.K. (994) Estiation of P(Y < X) in th bivariat xponntial cas du to Marshall-Olkin, Journal of th Indian Statistical Association,, 5-7. [] Langndonk, J. G., H. Pijl, A. C. Toornvlit, J. Burggraaf, M. Frolich, R. C. Schoakr, J. Doornbos, A. F. Cohn,and A. E. Mindrs. Circadian rhyth of plasa lptin lvls in uppr and lowr body obs won: influnc of body fat distribution and wight loss. J. Clin. Endocrinol. Mtab. 8: 76 7, 998. [4] icinio, J., C. Mantzoros, A. B. Ngrao, G. Cizza, M. L. Wong, P. B. Bongiorno, G. P. Chrousos, B. Karp, C. Alln, J. S. Flir, and P. W. Gold. Huan lptin lvls ar pulsatil and invrsly rlatd to pituitary-adrnal function. Natur Md.: , 997. [5] Marshall, A.W., Olkin, I. (997) A nw thod for adding a paratr to a faily of distributions with application to th xponntial and wibull failis, Biotrica,84(),64-65 [6] Schollr, D. A., L. K. Clla, M. K. Sinha, and J. F. Caro. Entrainnt of th diurnal rhyth of plasa lptin to al tiing. J. Clin. Invst. : , 997. AUTHORS First Author Gtha.T, Asst profssor of athatics K.N Govt Arts collg for won,thanjavur. Tailnadu, Southindia Scond Author Sangtha.B, Lcturr of athatics, BDU collg Orathanadu,Thanjavur,Tailnadu,Southindia