A Trivariate Weibull Model For Effects of Drospirenone and Estrogen in Postmenopausal Women with Hypertension

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1 Iteratioal Joural of Applied Egieerig Research ISSN Volume 13, Number 9 (218 pp A Trivariate Weibull Model For Effects of Drospireoe ad Estroge i Postmeopausal Wome with Hypertesio K.Majella Jevi Igatia * ad Dr. S.Lakshmi ** * Research Scholar, Bharathiyar Uiversity, Coimbatore, Idia. **Research Supervisor, Kudavai Naachiyar Govt. Arts ad Sciece College for wome, Tajore-6137, Idia. Abstract The usage of weibull distributio is wide i parametric aalysis of failure time data. The simple form of the weibull survivor fuctio is a attractive property for right-cesored observatio which occurs frequetly i survival studies. The correspodig hazard rate which icreases, costat or decrease accordig to the value of oe parameter. This behavior is commoly accepted as appropriate i may situatio. I the applicatio part, Drospireoe (DRSP, a progesti with atialdosteroe activity, had bee developed for hormoe therapy i combiatio with 17-β-estradiol(E2 i postmeopausal wome. The atihypertesive efficacy ad safety of various doses of DRSP ad E2 ad estradiol aloe i postmeopausal wome with hypertesio usig ambulatory ad cliic blood pressure (BP moitorig is evaluated. I coclusio the mathematical model we costructed support the medical result that DRSP combied with E2 sigificatly reduces BP i postmeopausal wome with hypertesio ad did ot iduce sigificat icreases i serum potassium. These characteristics may lead to a ew beefit for this ovel hormoe therapy i postmeopausal wome with hypertesio INTRODUCTION Weibull models are used to describe various types of observed failures of compoets ad pheomea. They are widely used i reliability ad survival aalysis. I additio to the traditioal two-parameter ad three-parameter Weibull distributios i the reliability or statistics literature, may other Weibull-related distributios are available. The Weibull distributio is widely used for the parametric aalysis of failure time data. For right - cesored observatios, which occur frequetly i survival studies, the simple form of the Weibull survivor fuctio is a attractive property. It also has a hazard rate, icreasig, costat or decreasig accordig to the value of oe parameter, ad this behavior is commoly accepted as appropriate i may situatios, though ot all [2]. I certai circumstaces a argumet ca be made out for believig that the distributio of survival times really should be Weibull [14,15]. For multivariate failure time data the same cosideratios apply ad it would be useful to have a multivariate Weibull distributio with simple, iterpretable ad flexible applicatio. Several mültivariate expoetial distributios have bee proposed by Johso & Kotz ad we ca obtai a correspodig multivariate Weibull distributio by power trasformatio of the idividual variates [6]. However, each of these distributios has at least oe disadvatage for practical applicatio to failure time data. Gumbel proposed two bivariate expoetials which have rather restricted rages of values [5]. Marshall & Olki s multivariate expoetial is a sigular distributio with o - zero probability cocetrated o a certai subspace [11]. The other distributios described by Johso & Kotz either do ot have expoetial margials or have complicated forms for the desity ad survivor fuctios [6]. Clayto ad Cusick s bivariate proportioal hazards models are very geeral but do ot allow egative associatio [3]. The form of multivariate distributio proposed here is with basic properties ad behavior [8]. MATHEMATICAL MODEL Trivarite Weibull Distributio: Lu & Bhattacharyya [1] developed a joit survival fuctio by lettig h 1(x ad h 2(y be two arbitrary failure rate fuctios o [,, ad H 1(x ad H 2(y be their correspodig cumulative failure rate. Give the stress S = s >, the joit survival fuctio coditioed o s, as they defied, is F (x, y s = exp{ [H 1 (x + H 2 (y] γ s}, where γ measures the coditioal associatio of X ad Y. Further, based o the joit survival fuctio, they proved a theorem that a bivariate survival fuctio F (x, y/s ca be derived with the margials F x ad F y give the assumptio that the Laplace trasform of the stress S exists o [, ad is strictly decreasig. From the theorem, they derived a bivariate Weibull Distributio F (x, y s = exp { [( x x + ( ]} Where < 1, <, <, < γ 1, γ 2 < Followig the same steps, the theorem ca be expaded to more tha to radom variables, ad therefore, a multivariate survival fuctio of Weibull distributio is costructed as S (x 1, x 2,.. x = exp { [( x 1 + ( x ( x γ ] λ } 6628

2 Iteratioal Joural of Applied Egieerig Research ISSN Volume 13, Number 9 (218 pp Where measures the associatio amog the variables, < 1, ad <,,. λ <, ad < γ 1, γ 2,.. γ < Probability Desity Fuctio of the Multivariate Weibull Distributio The bivariate probability desity fuctio of a multivariate distributio fuctio ca be obtaied by differetiatig the multivariate survival fuctio with respect to each variable. Li [9], ad Yi ad Weg [24] had show that f(x 1, x 2,.., x = ( 1 S(x 1,x 2,..,x x 1 x 2.. x Usig Li s derivatio ad oe of the special cases of the multivariate Faa di Bruo formula by Costatie ad Savits [4], the probability desity fuctio is f(x 1, x 2,.., x = ( 1 exp { [(x 1 + ( x ( x γ ]} λ [( γ 1 ( γ 2.. ( γ ] [( x ( x ( x γ 1 ] λ λ p( {( 1 k ip s (, i ( j [( x 1 + ( x ( x γ k i ] } λ Where k i is the umber of summads of the i th partitio of such that ki =, 1 2 ki >, 1 k i : j is equal to ( 1.( j + 1, the fallig factorial of ; P( is the total umber of set partitios of the set S = {1, } correspodig to the i th partitio of. PDF of trivariate Weibull is f(x 1, x 2, x 3, = ( S(x 1,x 2,x 3 x 1 x 2 x 3 x ( x 1 γ 1 γ 2 γ 3 = exp( (( x 1 2 x 1,x 2,x 3 + ( x 2 + ( x 3 x{2 + 3 ( 1 + (( x 1 ( x 2 + ( x 3 γ3 γ3 + (( x 1 (( x 1 + ( x 2 + ( x 2 + ( x 2 + ( x 2 + ( x 3 γ3 + ( x 3 + ( x 3 + ( x 3 γ3 γ3 3+ γ (1 3 (( x 1 } + Let deote the umber of observatios, the, the log likelihood fuctio becomes log (f(x 1, x 2, x 3 γ 1 γ 2 γ 3 log ( 2 (( x 1i x 1, x 2, x 3 + ( x 2i + ( x γ3 3i + log + log (( x 1i + ( x 2i + ( x γ3 3i (( x 1i + ( x 2i + ( x γ3 3+ 3i (2 + 3 ( 1 + ( x 1i + ( x 2i + ( x γ3 3i + 2 (1 3 (( x 1i + ( x 2i + ( x γ3 3i APPLICATION Itroductio + (( x 1i + ( x 2i + ( x γ3 2 3i Drospireoe (DRSP is a ovel progesti with atialdosteroe effects that, i combiatio with 17-βestradiol (E2, has bee developed for use i postmeopausal wome as a hormoe therapy[ 7,12,13] The combiatio DRSP/E2 was approved for estroge deficiecy symptoms i postmeopausal wome. Durig its developmet for relief of meopausal symptoms, DRSP/E2 was show to have sigificat atihypertesive effects i studies of postmeopausal, hypertesive wome aloe or i combiatio with ealapril[16,17,21]. Furthermore, DRSP/E2 was foud to have a sigificat atihypertesive effect i patiets with ad without type 2 diabetes mellitus ad cocomitat use of agiotesi-covertig ezyme ihibitors ad agiotesi receptor atagoists[17]. Whe compared with other hormoe therapies ad oral cotraceptives, DRSP results i a sigificatly greater icrease i plasma aldosteroe[7,12,13] i respose to the atialdosteroe effect of the compoud. The primary objective of this study was to determie whether this ew hormoe therapy has cliically sigificat effects o 24-hour ambulatory ad cliic blood pressure (BP i a large populatio of hypertesive postmeopausal wome usig varyig doses of 1 mg DSRP with E2 compared with placebo.the reductios i cliic BP were sigificatly greater o 2- ad 3-mg DRSP/E2 compared with placebo at the ed of the study, whereas chages from baselie for 1-mg DRSP/E2 ad E2 aloe were similar to placebo. Reductios i BP occurred at 4 weeks of DRSP/E2 therapy ad stabilized by 6 weeks of therapy (Figure 1. At the ed of the study, the mea reductios i cliic BP i the 3- ad 2-mg DSRP/E2 groups averaged 13.8/ 8.5 mm Hg ad 12.1/ 9.2 mm Hg respectively, whereas the reductios for placebo were 8.7/ 5. mm. The chages from baselie i heart rate were similar for DRSP/E2 ad placebo. 6629

3 Iteratioal Joural of Applied Egieerig Research ISSN Volume 13, Number 9 (218 pp Figure 1. Effects of DRSP/E2 vs placebo o cliic BP after 8 weeks of double-blid therapy. A is chages i systolic BP ad B is chages i the diastolic BP. Chages from baselie were sigificat for the 3- ad 2-mg DRSP/E2 groups oly (systolic BP, P=.4 ad.195 for 3- ad 2-mg DRSP/E2 groups, respectively; diastolic BP, P<.1 for both 3- ad 2-mg groups. The mea chages from baselie i the 24-hour ambulatory BPs are show i Figure 2. Sigificat reductios from baselie i the coprimary ed poit of mea 24-hour systolic BP were observed i the 2- ad 3-mg DRSP/E2 treatmet group compared with placebo. I cotrast, there were o differeces i ambulatory BP for 1-mg DRSP/E2 ad E2 aloe versus placebo. As show i the 24-hour curves of the systolic BP i Figure 2, the reductios i systolic BP were greater as the dose of DRSP icreased ad persisted for all 24 hours of the dosig period. Less proouced dose-related reductios i 24-hour diastolic BP were observed o DRSP/E2 compared with E2 aloe or with placebo. 663

4 Iteratioal Joural of Applied Egieerig Research ISSN Volume 13, Number 9 (218 pp Figure 2. Effects of DRSP/E2 vs placebo o the co-primary ed poit of ambulatory systolic BP over 24 h ours at baselie ad after 8 weeks of double-blid therapy that icluded placebo, E2 aloe, 1-mg DRSP/E2, 2-mg DRSP/E2, ad 3-mg DRSP/E2. Baselie period is the,, ad treatmet period is the,. Five patiets i each of the 3 DRSP/E2 groups ad 5 patiets i the placebo group developed a serum potassium of 5.5 meq/l. The mea maximal chage from baselie i the DRSP group was ot sigificatly differet for the 5 treatmet groups ad raged betwee.29 meq/l ad.37 meq/l (P=.48. I additio to o differeces i the proportio of patiets developig predefied hyperkalemia (> 5.5 meq/l, there were also o differeces for 1 icrease of serum potassium betwee >.5 meq ad <1. meq as oted here: 3 mg DRSP/E2: =44 (29.1%; 2 mg DRSP/E2: =45 (3.2%; 1 mg DRSP/E2: =25 (16.6%; 1 mg E2: =41 (27.3%; ad placebo: =43 (29.3%. Reductios i total ad low-desity lipoprotei cholesterol were sigificatly greater with all of these active treatmet groups compared with placebo, with the largest reductio o 3-mg DRSP/E2. There were o chages i serum triglycerides i ay of the treatmet groups. There were mixed fidigs for the high-desity lipoprotei cholesterol. Dose-related icreases i the serum aldosteroe cocetratios were observed o 2- ad 3-mg DRSP/E2. First of all, all subgroups showed sigificatly differetial coditioig i me or i wome, as compared to their respective cotrol groups. The survival fuctios for correspodig values of E2, DRSP1, DRSP2, DRSP3 combied with E2 fuctios for placebo ad cortisol have bee foud. 6631

5 Iteratioal Joural of Applied Egieerig Research ISSN Volume 13, Number 9 (218 pp Cliic ad Ambulatory BP As show i Figure 1, DRSP/E2 lowered both the cliic ad 24-hour ambulatory BPs sigificatly compared with placebo, whereas E2 aloe had o effect o BP levels. There was evidece of a dose-related reductio i both types of BP measuremets, although the most distict relatioship was for the 24-hour systolic BP. The atihypertesive effect of DRSP/E2 becomes sigificat with 2-mg DRSP but the higher dose (3-mg DRSP yielded more cosistet ad larger decreases i both cliic ad 24-hour BP (Figure 2. The level of ambulatory BP reductios observed i this study is comparable to may other atihypertesive agets, icludig the selective aldosteroe blocker eplereoe[2]. For example, the placebo-subtracted mea reductio from baselie i 24-hour BP o eplereoe was &7/4 mm Hg for the 5-mg dose, a value similar to what was observed with the 3-mg dose of DRSP both i the preset study (Figure 2 ad i our prelimiary study with DRSP/E2 [ 21]. I additio, Presto et al[ 16 ] reported that 3-mg DRSP/E2 lowered 24- hour ambulatory BP by 9/5 mm Hg whe the drug was added to ealapril after just 2 weeks of therapy. As i our preset study, these reductios i BP are associated with icreases i aldosteroe by &3 g/dl (4% above baselie, attestig to the pharmacological effect of DRSP i blockig the mieral corticoid receptor. Reductios from baselie i the cliic BP versus reductios i the 24-hour ambulatory BP for DRSP/E2 are variable. Differetial effects for cliic ad ambulatory BP measuremets are commoly observed i atihypertesive therapy trials, but typically the mea reductio i ambulatory pressures i cliical trials is &4% less tha the average reductio i the cliic BP[ 21]. I the preset trial, there is a fairly large BP reductio i the placebo group for both the cliic systolic ad diastolic pressure ( 8.6/ 5. mm Hg, which are attributable i part to both observer bias ad i part to regressio to the mea [22]. Because each treatmet group had betwee 119 ad 129 patiets who uderwet ambulatory BP recordigs, the fidigs for those data are exceedigly robust ad, ulike the cliic BP measuremets, show that 2 of 3 doses of DRSP/E2 have atihypertesive activity (Figure 2 ad that a doserelated respose occurs amog the 3 doses tested i our study. Fially, it appears that the atihypertesive effect of DRSP/E2 is attributable primarily to the effects of DRSP, because E2 aloe did ot iduce ay reductio i either cliic or ambulatory BP. Alteratively, there may be syergism betwee DRSP ad E2. Coclusios provided i biological system The study demostrated that DRSP/E2, a ew hormoe therapy with aldosteroe blockig activity, was effective i reducig ambulatory systolic ad diastolic BP with doserelated reductios betwee 1 ad 3 mg of DRSP. The 2-mg DRSP is the lowest effective dose that provided sigificat ad adequate BP reductio i postmeopausal hypertesive wome. The hormoe therapy was well tolerated for the 2- moth period with modest subjective or objective adverse evets. Because reductio of systolic BP of the magitude observed i our study has sigificat implicatios i postmeopausal wome with hypertesio,[ 18,19,23] especially for reducig stroke ad heart failure, DRSP/E2 may have a advatage for the treatmet of meopausal symptoms over covetioal progestis. Future studies desiged to evaluate the loger-term (ie, 2 years cardiovascular effects associated with these sigificat atihypertesive effects of DRSP/E2 might be warrated at this time. RESULTS Estroge s Effect o Diastolic Blood pressure combied with DRSP SYSTOLIC-E2 SYSTOLIC-DRSP1/E2 6632

6 Iteratioal Joural of Applied Egieerig Research ISSN Volume 13, Number 9 (218 pp SYSTOLIC-DRSP2/E2 SYSTOLIC-DRSP3/E SYS-E2 SYS-DRSP1/E2 SYS-DRSP2/E2 SYS-DRSP3/E2 Estroge s Effect o Diastolic Blood pressure combied with DRSP DIAS-E2 SYSTOLIC-DRSP1/E2 6633

7 Iteratioal Joural of Applied Egieerig Research ISSN Volume 13, Number 9 (218 pp SYSTOLIC-DRSP2/E2 + DIAS-DRSP1/E DIAS-E2 DIAS-DRSP1/E2 CONCLUSION I sectio 4, we have developed mathematical curves which show the decliig effect of DRSP with E2 over the week time period. I particular, all the dosages of DRSP with E2 are compared with placebo which are sigificat i toto. It reaches the X-axis declied towards the time period which shows that blood pressure reduces util the 5 th week ad stabilizes over the 6 th ad 7 th week period. I coclusio the mathematical model we costructed supports the medical result that DRSP combied with E2 sigificatly reduces BP i postmeopausal wome with hypertesio ad did ot iduce sigificat icreases i serum potassium. These characteristics may lead to a ew beefit for this ovel hormoe therapy i postmeopausal wome with hypertesio REFERENCE [1] Archer DF, Thoreycroft IH, Foegh M, Haes V, Glat MD, Bitterma R, Kempso RL. (25, Log-term safety of drospireoe-estradiol for hormoe therapy: a radomized, double-blid, multiceter trial, Meopause. Vol.12, pp [2] Beett S. (1983, Log-Logistic regressio models for survival data, Appl. Statist, Vol. 32, pp [3] Clayto D., Cusick J. (1985, Multivariate geeralizatio of the proportioal hazard models, J.R, Statist. Soc. A, Vol. 148, pp [4] Costatie GM., Savits TH. (1996, A multivariate faa di Bruo formula with applicatios, America mathematical society, Vol. 358, pp [5] Gumbel EJ. (196, Biovariate expoetial distributios, joural of the america statistical associatio, Vol. 55,pp [6] Johso NL., Kotz S. (1972, Distributios i statistics : cotiuous multivariate distributios, NewYork, Wiley. [7] Krattemacher R. (2, Drospireoe: Pharmacology ad pharmacokietics of a uique progestoge.cotraceptio, Vol. 62, pp [8] Lakshmi S., Durgadevi N.(214, A trivariate weibull model for cotextual cotrol over expressios of feat i huma, the joural of the 6634

8 Iteratioal Joural of Applied Egieerig Research ISSN Volume 13, Number 9 (218 pp idia academy of mathematics, Vol. 36,No. 2, ISSN : , pp [9] Li CL. (1997, A model for iformative cesorig Ph.D. Dissertatio, The uiversity Alabama Birmigham. [1] Lu J-C., Bhattacharyya G.(199, Some New Costructios Of Bivariate Weibull Models, Auals Of The Istitute Of Statistical Mathematics, Vol. 42, pp [11] Marshall AW., Olki I. (1967, A multivariate expoetial distributio, joural of the America statistical associatio, Vol. 62, pp [12] Oelkers W., Foidart JM., Dombrovicz N., Welter A., Heithecker R. (1995, Effects of a ew oral cotraceptive cotaiig a ati mieralocorticoid progestoge, drospireoe, o the rei-aldosteroe system, body weight, blood pressure, glucose tolerace, ad lipid metabolism, J Cli Edocriol Metab, Vol. 8, pp [13] Oelkers W. ( 1996, Effects of estroges ad progestoges o the rei-aldosteroe system ad blood pressure.steroids, Vol. 61, pp [14] Peto R., Lee P. (1973 Weibull distributios for cotiuous carciogeesis experimets, biometrics, Vol. 29, pp [15] Pika MC. (1966, A method of aalysis of a certai class of experimets i carciogeesis, biometrica, Vol. 22, pp [16] Presto RA., Aloso A., Pazitta D., Zhag P., Karara AH. (22, Additive effect of drospireoe/17-beta estradiol i hypertesive postmeopausal wome receivig ealapril, Am J Hypertes, Vol. 15 pp [17] Presto RA., White WB., Pitt B., Bakris G., Norris PM., Haes V. (25, Effects of drospireoe/17-β estradiol o blood pressure ad potassium balace i hypertesive postmeopausal wome, Am J Hypertes, Vol. 18, pp [18] SHEP. (1991, Cooperative Research Group. Prevetio of stroke by atihypertesive drug treatmet i older persos with isolated systolic hypertesio: fial results of the Systolic Hypertesio i the Elderly Program (SHEP, JAMA, Vol. 265, pp [19] Staesse JA., Fagard R., Thijs L., Celis H., Arabidze GG., Birkehager WH., Bulpitt CJ., de Leeuw PW., Dollery CT., Fletcher AE., Forette F., Leoetti G., Nachev C., O Brie ET., Rosefeld J., Rodicio JL., Tuomilehto J., Zachetti A. ( 1997, Radomised double-blid compariso of placebo ad active treatmet for older patiets with isolated systolic hypertesio, Lacet, Vol. 35, pp [2] White WB., Carr AA., Krause S., Jorda R., Roiker B., Oigma W.( 23, Assessmet of the ovel selective aldosteroe blocker eplereoe usig ambulatory ad cliical blood pressure i patiets with systemic hypertesio, Am J Cardiol, Vol. 92, pp [21] White WB., Pitt B., Presto RA., Haes V. (25, Atihypertesive effects of drospireoe with 17βestradiol, a ovel hormoe treatmet i postmeopausal wome with stage 1 hypertesio, Circulatio, Vol. 112, pp [22] White WB. (2, Advaces i ambulatory blood pressure moitorig for the evaluatio of atihypertesive therapy i research ad practice, White WB, ed. Blood Pressure Moitorig i Cardiovascular Medicie ad Therapeutics, Totowa, NJ, Humaa Press, pp [23] White WB. (2, Beefits of atihypertesive therapy i older patiets with hypertesio, Arch Iter Med, Vol. 16, pp [24] Yi Z., Weg C. (26, A Applicatio of the Power Approximatio i Multiple Life Isurace, Vol. 33, pp