The R Package PK for Basic Pharmacokinetics

Transcription

1 Wolfsggr, h R Pacag PK St 6 h R Pacag PK for Basc Pharmacotcs Mart J. Wolfsggr Dpartmt of Bostatstcs, Baxtr AG, Va, Austra Addrss of th author: Mart J. Wolfsggr Dpartmt of Bostatstcs Baxtr AG Wagramr Straß 7-9 Va Austra Summary wo-phas half-lf stmato bcam popular th rct yars but thr ar o stad-alo fuctos popular statstcal stadard softwar. A lot of popl who ar trstd pharmacotc (PK) aalyss hav a mdcal bacgroud but ar lss xprcd mplmtg complx fttg algorthms. o ma R [] mor popular thr commuty a fw fuctos for basc PK aalyss ar mplmtd th R pacag PK []. h avalabl fuctos for two-phas half-lf stmato ar prstd a smulato study. Itroducto A commo modl for dvdual half-lf stmato s th two-compartmtal modl assumg a dstrbutv phas ad lmato phas. h xpctd drug lvl E(y t ) at tm t for a dvdual ca b xprssd as a lar combato of two xpotals t t E( yt ) α α whr ad must b postv to b physcally magful ad > to sur dtfablty. Ital half-lf rprstg th dstrbuto of th drug ad trmal half-lf dcatg th actual dgradato of th matral ar calculatd as BIOMERIE ud MEDIZIN 5/6

2 St 6 Wolfsggr, h R pacag PK t / ; ( ) log( ) ad. log t / ; A commo approach s to us olar fttg wth th last squars crtra to stmat α,, α ad. h stmats obtad by ths approach may dpd o th fttg algorthms ad o th startg valus usd. L t al. [3] prstd a two-phas lar rgrsso modl by fttg two smpl ordary last squars rgrssos o log-lar trasformd data basd o th two sts of pots {t,,t } ad {t,,t }. h mthod of L rsults -3 sts of two rgrsso ls wth rqustg at last data pots th trmal phas. h stmats of th two sts of rgrssos ar ˆ α ˆ, ad ˆ α ˆ,,,-3. h total sum of squars rsduals s dtrmd by ( log( y ) ˆ α ˆ t ) log( y ) ( ˆ α ˆ t ) I addto, th smpl last squars rgrsso quato for all pots s also calculatd wth ( log( y ) ˆ α ˆ t ) If th rgrsso ls for tal ad trmal phas do ot o th trval [t,t ] (.. γˆ t or γˆ t ) whr ˆ α ˆ γ ˆ ˆ α ˆ th st. Amog th -3 rgrsso tupls ad th sgl-phas modl, th o wth th m( ),..,-3 s usd to calculat tal ad trmal halflf. If a sgl-phas modl s slctd (m( ) ) th half-lf so dtrmd ca b utlzd as both tal ad trmal phas half-lf. Btma [4] prstd a smulato study of th L mthod modfd to us th last absolut dvatos, Hubr-M ad o-paramtrc rgrsso ad usg th sum of squard rsduals to slct th bst tupl of rgrsso ls. For xampl, BIOMERIE ud MEDIZIN 5/6

3 Wolfsggr, h R Pacag PK St 63 usg th sum of absolut rsduals for stmato of rgrsso ls but usg th sum of squard rsduals for slcto of th bst tupl of rgrssos. Hr, th L mthod s modfd for ordary last squars, last absolut dvatos, Hubr-M ad o-paramtrc rgrssos usg th sam crtro for rgrsso l stmato ad slcto of th rgrsso tupl. A furthr modfcato was to rqur dcrasg rgrsso ls tal ad trmal phass to sur physologcally magful rsults. Mthods Compard. Bxpotal modl as mplmtd R fucto SSbxp (pacag stats) usg stmats for ad obtad by xp(cof(ls(coc~ssbxp(tm, a, b, a, b)))[c(,4)]). whr th maxmum ad th mmum of th two stmats obtad by th abov fucto call wr usd as ad, rspctvly.. Bxpotal modl (Bxp) fttd by th last squars crtra usg th Nldr-Mad algorthm as mplmtd R fucto optm wth th paramtrzato ' ' ( xp( ) xp( δ )) t ( xp( ))t E( y ) α α t whr xp( ) ad xp( )xp(δ) to sur > >. Curv plg as suggstd Foss [5] s usd to gt start valus for olar modl fttg. Wh o adquat start valus (.. < <) ar dtrmd by curv plg, a sgl xpotal modl s fttd wth start valus obtad from a OLS rgrsso o atural log trasformd valus E( y t ) α ( xp( '))t whr xp( ) to sur >. h half-lf so dtrmd ca b utlzd as both tal ad trmal phas half-lf. 3. L mthod usg th ordary last squars (OLS) rgrsso to stmat rgrsso ls wth BIOMERIE ud MEDIZIN 5/6

4 St 64 Wolfsggr, h R pacag PK BIOMERIE ud MEDIZIN 5/6. ad 4. L mthod usg th last absolut dvatos (LAD) rgrsso to stmat rgrsso ls wth. ad 5. L mthod usg Hubr-M rgrsso (Hubr-M) to stmat rgrsso ls wth ( ) ( ) ( ) ad ρ ρ ρ whr (). or < < ρ Hubr M-stmats wr calculatd by o-lar stmato usg th Nldr- Mad algorthm as mplmtd th R fucto optm, whr OLS rgrsso paramtrs wr usd as startg valus. h fucto that was mmzd volvd.5*.483*mad, whr MAD was dfd as th mda of absolut dvato of rsduals obtad by a last absolut dvato (LAD) rgrsso basd o th obsrvd data. h tal valu of MAD was usd ad ot updatd durg tratos as rcommdd by Hollad ad Wlsch [6]. 6. L mthod usg o-paramtrc (NPR) rgrsso as suggstd Brs ad Dodg [7] to stmat rgrsso ls wth ( ) ( ) ( ) ra ra ra ad

5 Wolfsggr, h R Pacag PK St Bxpotal modl fttd o th log-scal (LogBxp) usg R fucto ls (pacag stats) ad usg th procdur prstd [5] to gt start valus for o-lar fttg. Estmats for ad wr obtad by cof(ls(log(coc)~log(a*xp(-b*tm) a*xp(-b*tm)), startstart, cotrolls.cotrol(maxtr))). h maxmum ad th mmum of th two stmats obtad by th abov fucto call wr usd as ad, rspctvly. Smulatos st data wr gratd udr assumpto of log-ormal dstrbutd y t s, whch s a plausbl statstcal dstrbuto for drug lvls. Drug lvls blood caot b gatv, whl th uppr d s op ladg to a o-symmtrcal dstrbuto. A dscusso o th dstrbuto of drug lvls cludg mor complx rror dstrbutos ca b foud [8]. h followg two modls wr usd for smulatos. α 5, /, α 4 ad /. α 5, /4, α 4 ad /8. Wth ach smulato ru, thortcal half-lf of tal ad trmal phass wr dtrmd aftr uformly radom varato of paramtrs (α,,α, ) ad tm pots (, /,, 3, 6,, 8, 4, 3, 48) by ±%. hortcal coctratos E(y t ) wr vard by y t E( y ) t ε t log( E( y t )) ε t whr ε t s a ormal dstrbutd rror wth E(ε t )-V(ε t )/ ad V(ε t )(. )log²(e(y t )) rsultg ubasd log-ormal dstrbutd y t wth dtcal pr tm pot varablty of % (o th log-scal). Wth a probablty w, thortcal coctratos wr cotamatd by a cauchy dstrbutd ε t wth as locato ad V(ε t ) as scal paramtr. h followg dvato log c ( ) log( ) log( ) log( ) ˆ btw thortcal ad stmatd paramtrs was calculatd wth ach smulato ru ad probablty of cotamato. h prctag of smulato ˆ BIOMERIE ud MEDIZIN 5/6

6 St 66 Wolfsggr, h R pacag PK rus whch a spcfc mthod achvd m(c) was rportd. Smulatos wr prformd wth smulato rus pr probablty of cotamato. Rsults abl Proporto of Smulato rus a Spcfc Mthod Achvd m(c) Mthod Modl w SSbxp Bxp OLS LAD Hubr-M NPR LogBxp % % % % % % % % % % % % % % h proporto of smulato rus whch a spcfc mthod achvd m(c) s summarzd tabl. Mthod ad wr fror trms of m(c). Both mthods wor wll udr assumpto of ormal dstrbutd drug lvls wth dtcal varacs for ach tm pot. h LogBxp mthod showd bst rsults for modl followd by th L mthod usg LAD rgrsso for all rats of cotamato. Usg th spcfcatos of modl, th L mthod usg LAD rgrsso was supror trms of m(c). h scod bst mthod was th L mthod usg OLS rgrsso followd by th LogBxp mthod for low cotamatd data. For hghr cotamatd data, th LogBxp mthod was bttr tha th L mthod usg OLS rgrsso. For w75% th L mthod usg NPR rgrsso was supror to th LogBxp mthod. BIOMERIE ud MEDIZIN 5/6

7 Wolfsggr, h R Pacag PK St 67 h sub optmal prformac of th LogBxp mthod udr spcfcatos of modl (vry small dffrc btw tal ad trmal half-lf) may b du to covrgc ssus of th o-lar fttg approach. abl prsts th proportos of smulato rus whr th LogBxp mthod dd ot covrg. Udr o-cotamatd (w) codtos th LogBxp mthod dd ot covrg.3% ad 43% for modl ad, rspctvly. Udr hghcotamatd (w75%) codtos th LogBxp mthod dd ot covrg 5% ad 46% for modl ad, rspctvly. abl Proporto of Smulato rus Wthout Covrgc of Mthod LogBxp Prctag of cotamato Modl % % % 3% 4% 5% 75% Dscusso ad Cocluso I summary, th o-covrgc of th LogBxp mthod a hgh prctag of smulato rus udr spcfcato of modl s a dsadvatag practcal mdcal rsarch as th complt mthod has to b spcfd advac. I cas of o-covrgc fac of th actual data aalyss, a postror chag of th statstcal mthodology s cssary (.g. usg a dffrt approach to gt start valus l grd sarch algorthms). o avod a possbl postror chag of th statstcal aalyss whch may ma th rsults lss covcg to rgulatory authorts, th L mthod usg LAD rgrsso ca b cosdrd as a good altratv for two-phas half-lf stmato udr log-ormal dstrbutd coctratos. Rfrcs [] R Dvlopmt Cor am. R: A laguag ad vromt for statstcal computg. R Foudato for Statstcal Computg, Va, Austra, 5. ISBN ; URL: [] Wolfsggr MJ, Ja. PK: Basc pharmacotcs. R pacag vrso.3, 5. [3] L ML, Poo W-Y, Kgdo HS. A two-phas lar rgrsso modl for bologc half-lf data. J. Lab. Cl. Md 99, 5: BIOMERIE ud MEDIZIN 5/6

8 St 68 Wolfsggr, h R pacag PK [4] Btma B, L ML, Schroth P. Robust rgrsso mthods for bologc halflf data. Royal Statstcal Socty Itratoal Cofrc, Radg, Utd Kgdom,. [5] Foss SD. A mthod for obtag tal stmats of th paramtrs xpotal curv fttg. Bomtrcs 969, 5: [6] Hollad PW, Wlsch RE. Robust rgrsso usg tratvly rwghd last-squars. Commu. Stat.-hor. M. 977, 6: [7] Brs D, Dodg Y. Altratv mthods of rgrsso. Joh Wly ad Sos, Nw Yor, 993. [8] PharmPK. PharmPK Dscusso Lst Archv last vstd at 5--3; URL: BIOMERIE ud MEDIZIN 5/6