Fitting Compartmental Models to Multiple Dose Pharmacokinetic Data using SAS PROC NLMIXED

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1 Ppr SP-2009 Fttng Comprtmntl Modls to Multpl Dos Phrmcokntc Dt usng SAS PROC NLMIXED Jng Su, Png Sun, Xodong L nd Aln H. Hrtford Mrck & Co., Inc, North Wls, PA ABSTRACT Phrmcokntc (PK) modlng uss systms of ordnry dffrntl qutons drvd from bologcl consdrtons long wth sttstcl modls to modl th tm cours of drug n th body. Th sttstcl modl rqurs lgorthms for fttng nonlnr mxd ffcts modls. Whl th NLMIXED procdur n SAS s vlbl, t dos not llow for ndvdul subjct dt to ffct th structurl form of th modl. In th cs of multpl dos study whr subjcts xprnc dffrnt dosng tms, suprposton prncpl cn b usd to rcursvly ccount for ch ddtonl dos. A SAS tmplt progrm ws wrttn to mnpult dt nd thn construct mn functons for fttng phrmcokntc dt from multpl dos studs. On-comprtmnt modls for orl dos dmnstrton r consdrd to llustrt th mthodology nd chllngs for fttng multpl dos dt usng PROC NLMIXED. Th tmplt progrm contns smpl SAS cod for fttng two typs of modls: gnrl modl nd smplfd modl. Th gnrl structurl modl cn hndl mny stutons such s whn subjct hs rrgulr dosng ntrvls, chngs dosg durng thrpy or hs non-gnorbl dffrncs btwn ctul dosng tm nd schduld dosng tm, tc. Th smplfd modl s for th most common scnro n whch ll subjcts r constrnd to hv th sm multpl dosng schduls wth rgulr dosng ntrvls nd constnt doss. Undr ch modl, w lso dscuss how to hndl mssd dos problms. KEYWORDS: Phrmcokntc (PK); Comprtmntl Modls; Multpl dos; Suprposton; SAS; PROC NLMIXED.. INTRODUCTION Phrmcokntc (PK) modlng s n r of study whr systms of ordnry dffrntl qutons (ODEs) r usd to modl th tm cours of drug n th body. Sttstcl modls r usd n conjuncton wth th ODEs to ccount for both ntr- nd ntr-subjct vrblty; PK prmtrs of th modl r stmtd for ndvduls undr sttstcl dstrbuton ssumptons nd thn smultons r prformd to dtrmn th ffct of dffrnt clncl trl or dosng dsgns. Spcfc to PK modlng s tht th modls do not rs mprclly by choosng th smplst lgbrc xprssons tht ft th dt. Instd, bologcl consdrtons for how th body s known to bsorb, dstrbut, mtbolz, nd lmnt drug s wll s how th mchnsm of cton of th drug works r usd to drv mnngful modls to d n drug dvlopmnt nd dosng dcsons. Bcus of ths bologcl consdrtons, th functonl form of th modl, th mn functon for nonlnr sttstcl modl, cn b vry complx. As th mn functon s componnt of th soluton to n ODE, t mght not vn hv closd form xprsson. Undr multpl dos scnros th suprposton prncpl for ccumultng doss s ndd to rcursvly construct th mn functon. To llustrt ths, ssum tht drug s gvn vry 24 hours (.., t hours 0, 24, 48 tc.), wth th sm dos mount t ch dosng tm. Tbl nd Fgur dmonstrt th bsc d. Lt C(t) dnot th drug concntrton ftr sngl dos. If 0 t < 24 hrs, thn only th frst dos s n ffct nd th prdctd concntrton s C(t). If 24 t < 48 hrs, thn th frst two doss r n ffct. Th contrbuton of th scond dos sc ( t 24) ; hnc th prdctd concntrton s C ( t) + C( t 24). Th prdctd concntrtons t othr tm ponts cn b smlrly drvd.

2 Tbl. Rcursv Modl Formul for Prdctng Concntrton Tm Prdctd concntrton 0 t < 24 hrs C(t) 24 t < 48 hrs C(t) + C(t-24) 48 t < 72 hrs C(t) +C(t-24) +C(t-48) Concntrton (nm) Concntrton Profl for Multpl Doss Concntrton Profl for Sngl Dos Tm (hour) Fgur Concntrton Profls for Sngl nd Multpl Doss Furthr complctons r tht th form of th mn functon vrs wth dffrnt dosng schduls. In rl-lf stutons such s n clncl trl, th dos my b djustd bsd on vlus of covrts,.g., wght. Anothr complcton s whn on subjct msss dos s comprd to ll othr subjcts rsultng n ths subjct rqurng mn functon wth on fwr pk thn th othr subjcts. Dffrnt ptnts wll lwys hv t lst slghtly dffrnt dosng tms nd dffrncs on th ordr of only fw mnuts my hv n mpct on prmtr stmton. Bcus of th dstrbuton ssumptons on th prmtrs, nonlnr mxd modl mthodology s mployd for prmtr stmton. Howvr, PROC NLMIXED n SAS s not sy to us whn dosng tms from subjct to subjct ffct th structurl modl. To ddrss th lmtton of PROC NLMIXED, SAS tmplt progrm ws wrttn to mnpult dt nd thn construct mn functons for fttng dt from multpl dos studs. On-comprtmnt modls for orl dos dmnstrton r consdrd to llustrt th mthodology nd chllngs for fttng multpl dos dt usng PROC NLMIXED. Th tmplt progrm contns SAS smpl cods for fttng two typs of modls: gnrl modl nd smplfd modl. Th gnrl modl pproprtly djusts th mn functon for ch subjct bsd on thr own rlzd dosng tms. Ths s ndd for css whn subjct chngs dosg durng thrpy or hs non-gnorbl dffrncs btwn ctul dosng tm nd schduld dosng tm, tc. Th smplfd modl s for th css whn ll subjcts r constrnd to hv xctly th sm multpl dosng schduls wth rgulr dosng ntrvls nd constnt doss. Unlss thr r clr dscrpncs mong dosng schduls, ths pproch for modlng whch gnors smll dffrncs btwn th ssgnd dosng schdul nd th ctul dosng tms s commonly usd n prctc for rly xplortory modlng. Th computtonl tm for th gnrl modl cn b vry slow, spclly for lrg dtsts rsultng from Phs III trls, but t my b ncssry dpndng on th dosng schdul. Undr ch modl, w lso dscuss how to hndl mssd doss. 2

3 In Scton 2 th bsc formuls for sngl nd multpl dos dmnstrton r rvwd n th contxt of oncomprtmnt modls wth frst-ordr bsorpton nd lmnton. Scton 3 xplns SAS progrms for fttng multpl dos phrmcokntc dt. Concludng rmrks r n Scton ONE-COMPARTMENT MODELS AFTER SINGLE AND MULTIPLE ORAL DOSE ADMINISTRATION Th dgrm of on-comprtmnt phrmcokntc modl wth orl dmnstrton s dsplyd n th grph blow. Th drug frst ntrs th gstrontstnl (GI) trct, thn gts bsorbd nto th cntrl crculton, nd s subsquntly lmntd from th cntrl crculton. GI Trct k Cntrl k Wth frst-ordr bsorpton nd lmnton, for sngl dos dmnstrton wth dos mount D dmnstrd t tm 0, th concntrton-tm profl s gvn by: C(t) FDk C( t) ( k k ) V kt kt ( ) whr k s th bsorpton rt constnt, k s th lmnton rt constnt, F s th bovlblty prmtr (frcton of th drug tht rchs th systmc crculton), nd V s th volum of dstrbuton (th volum n whch th mount of drug DF would nd to b unformly dstrbutd to producd th obsrvd concntrton). Not tht n th contxt of orl dos dmnstrton, only th pprnt volum of dstrbuton s dntfd. Th lmnton rt constnt cn b xprssd s blood tht s totlly clr of drug pr unt of tm). Cl V D, Cl Cl V D V / F whr s th clrnc ( s ntrprtd s th volum of For multpl dos dmnstrtons, th suprposton prncpl cn b ppld to drv th concntrton-tm profl C(t). W strt by consdrng th most gnrl cs. Assum tht n ndvdul rcvs m doss t tm t, t 2,..., t m, rspctvly. Dnot th dos mount t tms t s D (,..., m ). To smplfy notton, dfn th dosng ntrvls s τ t + t for,..., m nd lt τ 0 t. Applng th suprposton prncpl, w obtn th gnrl modl: whr C( t) n ( k Dk k ) V D n {,..., m } s such tht n < t tn+ { xp[ k ( t τ j )] xp[ k ( t τ j )]} () j 0 t nd n m f t > t m, l 0 A spcl, yt most common cs s whn D D (,..., n) cs, quton () cn b smplfd to: nd τ (,..., n ). τ Undr ths spcl 3

4 kt nτk kt nτk Dk ( ) ( ) C( t) τk τk ( k k ) VD (2) In clncl trls, t s possbl tht subjcts my mss on or mor doss. In ths cs, th smplfd modl (2) dos not pply but th gnrl modl () s stll pplcbl by trtng mssng doss wth dos mounts qul to 0. Altrntvly, undr th spcl cs bov, smplfd formul cn b drvd to ccount for mssd doss nd ths cn sgnfcntly rduc th computton tm f th numbr of mssd doss s smll rltv to th totl numbr of doss. If sngl dos, sy s mssd, thn followng th smplfd modl (2), cn b wrttn s D C(t) m kt nτk kt nτk Dk ( ) ( ) C( t) k k ( k k ) V τ τ D Dk ( k k) VD whr I ( ) s th ndctor functon. I( m n) tks vlu of f m n thr r k mssd doss ( D m,..., D m ), thn C( t) ( k Dk k ( k ) V Dk k D ) V k [ t( m ) ] [ ( ) ] [ ] τ k t m τ I( m n ) (3) D k k t nτk ( ) τk t ( τk nd 0 othrws. Mor gnrlly, f k k [ t( m j ) τ ] k [ t( m j ) τ ] [ ( ) I( m j n) ] (4) j k τk ) n In th contxt of nonlnr mxd ffcts modls (Dvdn nd Gltnn, 995), phrmcokntc prmtrs such s k Cl VD (bsorpton rt constnt), (clrnc) nd (volum of dstrbuton) r ndvdul spcfc; but w ssum tht thy com from common dstrbuton (.g, lognorml dstrbuton). In ddton, condtonl on ndvdul spcfc phrmcokntc prmtrs, th obsrvd drug concntrtons r llowd to vry round so tht C(t) s ntrprtd s th condtonl mn or mdn concntrton (dpndng on th ntr-subjct rror structur). In prctc, dffrnt ntr-subjct rror structurs r fttd to th dt ncludng ddtv, proportonl, nd xponntl (Bl nd Shnr, 998). Not tht whl concntrton s n obsrvd vrbl, th rndom phrmcokntc prmtrs, nd, r not obsrvd; modl s fttd wth dstrbutonl ssumptons for ths rndom prmtrs nd th prmtrs of ths dstrbutons r stmtd,.g., mn nd SD of V D nd. k Cl,, V D k Cl, C(t) k, Cl, In th dscusson tht follows, w dopt prmtrzton of nd wth ndpndnt lognorml dstrbutons for ntr-ndvdul vrblty nd n ddtv rror structur for wthn-subjct vrblty. Lt V (,2,..., m) D b subjct 's bsorpton rt constnt, clrnc, pprnt volum of dstrbuton rspctvly. For ch ndvdul, w us y j nd C j ( j,..., n ) to dnot th obsrvd concntrton nd condtonl mn concntrton (s functon of ndvdul spcfc phrmcokntc prmtrs) t tm t j rspctvly. Thn 2 yj Cj + ε j, ε j ~.. d N(0, σ ), k θ xp( η ), Cl V D θ 2 xp( η2 ), θ xp( η ), 3, V D k, Cl, whr ( η, η2, η3) r ssumd to b ndpndnt cross ndvduls nd follow multvrt norml dstrbuton wth mn (0, 0, 0) nd covrnc mtrx Σ. Th SAS progrms dscussd blow cn b rdly dptd to ccommodt stutons such s ltrntv prmtrzton, dffrnt rror structurs, or mor complx structurl modls. 4

5 3. SAS PROGRAMS FOR FITTING MULTIPLE DOSES PK DATA In ths scton, w dscrb th tmplt SAS progrm for fttng th gnrl nd smplfd modls dscrbd n Scton 2. To ft ch modl, w dscuss n dtl bout ts rqurd nlyss dt structur, how to construct th mn functon, nd how to hndl mssd doss. TO FIT THE GENERAL MODEL Th gnrl modl s pplcbl to mny stutons such s whn subjct chngs dosg durng thrpy or hs non-gnorbl dffrncs btwn ctul dosng tm nd schduld dosng tm, tc. For smplcty, w ssum tht n multpl dos study ch ndvdul took thr 00 mg orl doss, t 0, 24 nd 48 hours. W furthr ssum tht PK smpls wr tkn t hours, 4, 2, 24, 48, 49, 52, 60, nd 72. To construct th mn functon nd llow for th most flxblty, w propos th followng nlyss dt structur (llustrtd for on ndvdul): Tbl 2. Dt Structur to Ft th Gnrl Modl () ID n tm y dos_ dos_2 dos_3 tm_ tm_2 tm_ In th bov tbl, ID s th ndvdul dntfcton vrbl, n s th numbr of doss n ffct, nd tm s th tm of obsrvton. Snc th concntrton smpld t tm 24 ws tkn rght bfor th 2 nd dos, th numbr of doss n ffct ws stll on nd thn n s for tht rcord. Corrspondng to ch dos, w crt two vrbls: dos_ nd tm_. If t n obsrvton tm, th th dos s n ffct, thn dos_ s th dos mount of th th dos nd tm_ s th obsrvton tm rltv to th th dosng tm; f th th dos s not n ffct, thn both dos_ nd tm_ tk th vlu of 0. For xmpl, t hour 4, only th frst dos s n ffct, hnc only dos_ nd tm_ tk nonzro postv vlus. Th smpl SAS cod for crtng ths tm vrbls s: rtn tm_ - tm_3 0; rry t[3] tm_-tm_3; do to n; t()tm-(-)*24; nd; As llustrtd n Tbl, th mn functon s dffrnt ftr ch dos. Instd of usng f/thn sttmnts to spcfy th xct mn functon ftr ch dos, th gnrl modl () cn b drctly ppld wth ths spcl dt structur snc th vrbls dos_ nd tm_ r st to 0 for th rcords bfor th dos ws gvn. k Cl, To us th NLMIXED procdur to stmt th prmtrs ndv, nd thr vrncs s dscussd n Scton 2, rsonbl strtng vlus for ths prmtrs r vry mportnt. Som pprochs for obtnng strtng vlus mght nclud:. Rsults from th frst n mn (FIM) study, or domn knowldg. 2. Th mnng of th prmtrs. For xmpl, k s th bsorpton rt constnt, nd should b btwn 0 nd. Strtng vlu of vrnc for ch prmtr cn b obtnd usng th NLIN procdur. Bsd on th rsults of fttng nonlnr modl for ch subjct sprtly usng PROC NLIN, th smpl vrnc of ch stmtd prmtr cn b clcultd. To mk t sr to pss thos strtng vlus of vrncs nto th PROC NLMIXED, mcro vrbls cn b crtd., 5

6 Lt pkdt.ss7bdt b dtst hvng th sm dt structur s prsntd bov. Assum tht th strtng vlus for V, k nd Cl r 20, 0.8, nd 0.6, rspctvly. Blow s th smpl cod for obtnng th strtng, vlus of vrncs usng PRON NLIN: /****to gt strtng vlus for vrnc usng PROC NLIN****/ proc nln dtpkdt; by d; prms v20 k0.8 cl0.6 ; kcl/v; cstk/(v*(k-k)); modl y cst*dos_*(xp(-k*tm_)-xp(-k*tm_)) + cst*dos_2*(xp(-k*tm_2)-xp(-k*tm_2)) + cst*dos_3*(xp(-k*tm_3)-xp(-k*tm_3)); ods output prmtrestmts stmt ANOVAANOVA; run; Usng th strtng vlus of thos prmtrs nd thr vrncs nd followng th sm nottons s n Scton 2, th smpl cod for fttng th gnrl modl () usng PROC NLMIXED s blow: /****run PROC NLMIXED to ft th gnrl modl ****/ proc nlmxd dtpkdt mthodgauss; prms tht20 tht20.8 tht30.6 sgm_&sgm_ sgm_2&sgm_2 sgm_3&sgm_3 sgm&sgm ; v tht*xp(t); k tht2*xp(t2); cl tht3*xp(t3); sgm2_sgm_**2; sgm2_2sgm_2**2; sgm2_3sgm_3**2; sgm2sgm**2; kcl/v; cstk/(v*(k-k)); prd cst*dos_*(xp(-k*tm_)-xp(-k*tm_)) +cst*dos_2*(xp(-k*tm_2)-xp(-k*tm_2)) +cst*dos_3*(xp(-k*tm_3)-xp(-k*tm_3)); modl y ~ norml(prd,sgm2); rndom t t2 t3 ~ norml([0,0,0],[sgm2_,0,sgm2_2,0,0,sgm2_3]) subjctd; run; In th xmpl bov, w prsntd th gnrl cs wth constnt dosg nd nomnl tm ponts. Ths mthod cn b sly modfd to ccommodt dffrnt stutons. In th cs of chngng dosg durng thrpy, crt th dos_ vrbls to ndct th corrct dos mount for ch dos. If ctul tm s usd n th nlyss nstd of th nomnl tm, th vrbl tm_ my b slghtly dffrnt from th schduld post-dos tm, but th dt structur nd th PROC NLMIXED sttmnts ndd sty th sm. Th gnrl modl cn hndl mssd doss sly. For xmpl, f subjct mssd th 2 nd dos nd th 5 th dos n 7-dy onc-dly study, smply ssgn dos_2 0 nd dos_50. TO FIT THE SIMPLIFIED MODEL 6

7 As dscussd n Scton 2, for th spcl yt vry common cs wth constnt doss nd nomnl tm, th smplfd modl (2) cn b usd for nlyss, whch cn sgnfcntly rduc computtonl tm. To ft th smplfd modl (2), th rqurd nlyss dt structur s much smplr thn th on for th gnrl modl: vrbls dos_ nd tm_ for ch dos r not ndd. Usng th sm xmpl, th tbl blow llustrts th rqurd dt structur. Tbl 3. Dt Structur for Fttng th Smplfd Modl (2) ID n tm y dos II Th vrbl, II, s th ntr-dos ntrvl. In ths xmpl t s 24 hours, whch mns tht dos s dmnstrtd vry 24 hours. Usng th bov dt structur, th smplfd modl (2) s drctly ppld. Th smpl cod blow prsnts th mn functon nd th modl sttmnt n PROC NLMIXED. cstk/(v*(k-k)); prd cst*dos*((-xp(n*k*ii))*xp(-k*tm)/(-xp(k*ii)) -(-xp(n*k*ii))*xp(-k*tm)/(-xp(k*ii))); modl y ~ norml(prd,sgm2); In th css of mssd dos(s), th smplfd modl (2) s not pplcbl, but ts drvd modl (3) or (4) cn b ppld. Th smplfd modl (3) s for th cs wth on mssd dos, nd th modl (4) s for th cs wth multpl mssd doss. It s mportnt to not tht, s th numbr of mssd doss ncrss, th computtonl ffcncy of (4) dcrss wth ncrsd progrmmng complxty, comprd to th gnrl modl (). Prctclly, on or two mssd doss r not uncommon. In ths ppr, w only consdr th cs of on mssd dos to llustrt th computtonl dvntg of (3) nd (4). Hr w ssum tht Subjct mssd th 2 nd dos n th bov smpl dt. To us modl (3), crt vrbl dosm to ndct th dos mount mssd (s Tbl 4). In ths cs, dosm s 0 for ll rcords bfor th schduld tm for th 2 nd dos snc th mssd 2 nd dos hs no ffct thn. Th vrbl dosm s 00 for ll rcords ftr th schduld tm for th 2 nd dos, snc th mssd dos hs ffct ftr ths tm pont. If nothr subjct mssd dos too (not ncssry th 2 nd dos), th vrbl dosm cn b popultd smlrly: 0 for ll rcords bfor th mssd dos, nd 00 for ll rcords ftr th mssd dos. For subjcts who ddn't mss dos, dosm s 0. Tbl 4. Dt Structur for Fttng Smplfd Modl (3) ID n tm y dos dosm II

8 Th smplfd modl (3) contns two trms: th frst trm s th sm s th modl (2), nd th scond trm ccounts for th mssd dos. For th subjcts who ddn't mss dos, th mn functon s ctully th frst trm of th modl (3). For th subjcts who hd mssd dos, th mn functon s th frst trm of th modl (3) for th rcords bfor th mssd dos, nd th mn functon s xctly th sm s th modl (3) for th rcords ftr th mssd dos. But wth th dt structur spcfd bov, th modl (3) cn b ppld to ll rcords bcus dosm s 0 for th rcords wthout mssd dos ffct nd thn th scond trm s 0. To ft th smplfd modl (3), th sttmnts for th mn functon nd th modl sttmnt r th followng: cstk/(v*(k-k)); prd cst*dos*((-xp(n*k*ii))*xp(-k*tm)/(-xp(k*ii)) -(-xp(n*k*ii))*xp(-k*tm)/(-xp(k*ii))) -cst*dosm*(xp(-k*(tm-ii)) - xp(-k*(tm-ii))); modl y ~ norml(prd,sgm2); 4. CONCLUSIONS In ths ppr, w llustrtd SAS tmplt progrm for mnpultng dt nd thn constructd mn functons for fttng on-comprtmnt modls wth PROC NLMIXED to phrmcokntc dt from multpl orl dos studs. Comprd to th spcl cs wth constnt dosg nd rgulr dosng tm, th rqurd dt structur for th gnrl css s mor complx nd th computton tm cn b vry slow, but ths mor complx dt structur my b ncssry dpndng on th dosng schdul. Although only on-comprtmnt modls for orl dos dmnstrton wr dscussd n th ppr, th dt structur nd smpl cods cn b sly dptd for modls wth ltrntv rout of dmnstrton nd mor complx structurl nd stochstc componnts. REFERENCE Bl, S.L. nd Shnr, L.B. (998), NONMEM Usr's Guds, Vrson V, Unvrsty of Clforn, Sn Frncsco. Dvdn, M. nd Gltnn, D. (995), Nonlnr Modls for Rptd Msurmnt Dt, Nw York, NY: Chpmn nd Hll. ACKNOWLEDGMENTS W r grtful to Amy Gllsp nd Dbb Pnbnco for thr crful rvw nd vlubl commnts. CONTACT INFORMATION Your commnts nd qustons r vlud nd ncourgd. Contct th uthors t Jng Su Mrck & Co., Inc. UGCD-4 Po Box 000 North Wls, PA Eml: jng_su@mrck.com Png Sun Mrck & Co., Inc. UGD-44 Po Box 000 North Wls, PA Eml: png_sun@mrck.com Xodong L Mrck & Co., Inc. UGD-44 Po Box 000 North Wls, PA Eml: xodong_l@mrck.com 8

9 Aln Hrtford Mrck & Co., Inc. UGD-44 Po Box 000 North Wls, PA Eml: SAS nd ll othr SAS Insttut Inc. product or srvc nms r rgstrd trdmrks or trdmrks of SAS Insttut Inc. n th USA nd othr countrs. ndcts USA rgstrton. Othr brnd nd product nms r trdmrks of thr rspctv compns. 9