M<lbod. for E",mallnl riot; 0( CII<m><a1 l'\ju'l" Hu..... ...t Non·human -. ...t E<ooyo«moEcbl«! h, v. B, V""k. G. C, Butler. D G Ho<l.noj D D, P<aUU~ I"~SCOPE

Mathematical Dose-Response Models andtheir Application to Risk Estimation

David G. Hocl

ABSTRACT

A brief Wo'''W IS presented of ""eral slallsllcal tochniques for eslimatilli riskbased upon math.malical dose-respon.. models. The linear or nJlC·hil model.the multistage model. and Mantel-Bryan apprnach are d'scww:<.!. The import.ance of the m&nJlCr in "'hich lhe back....ound i. incorporated inlO the model i,al.., considered. Time-to-tumour models are c<>nStd~ .lIOn, ,."h IIOme n.wstatislical measure. of potency, The u.. of ....f.ty factors i. compared "'th Ih.mults from dose-mponl<: .stimation ,how,., the pol.nlial '.Inability in lhe....f.1)" factor approach. F,nally. the 'mpact nf kUI.t;os on th. relat,orulupbet"ttn adnuni.t.red and .ffecti,'. dose i. di5Cll5Sed.

I INTRODUCTION

The pressin, need to develop rational eJlposure 1e,'.15 of chemicals in theenVIronment and the work~ has been a serious i.."" for many ,ovemmenlS.The ""onom;o com oh.sulalion, musl somehow be balan<:ed apin5llh. ri,ks tohuman health and ..'On·bein,. On lhe h.alth side ofthe equalion. tbe need is foreslimates of human health .If,,,,,s .II specified uposure levels. The ,deal methodfor nbtaintnll this information i' throusJt quality .",demiolollical 5ludies.Vnfonunalely. useful human data are qu,t. rare, In manyca.... human e~posure

has been limited in >cope or duration, Quil. oft.n espo,ures are complexmixlures. and a.e lhereby .endered unsuitable for mk estlmalion. All1O.hisloncal exposure 1e""ls neceuary for cancer studies a re u.ually .ithe. unknownor una,-..ilabl•.

An alt.mati".IO dependilli upon .p,demiological dala i. the use of 1.1bora loryexpenm.nlS, w'lhexl.apolat,on of the mulls 10 man. Toxicologim believ.thatqualilalive .xtrapolalion. are generally 'alid. The difficully Wilh a toxioeololl>C.llapp.oach li.. ,n qua.nUficalion, "'hich i. lhe ....nce of ri'k eslimation. Inaddition to tile ~peo:ies eilr1opolation problem, lhere IS a nfed 10 ..umale ell",,\>at exposure 1.,..ls w.ll below th. experim.ntal region, In ord.r to eslimate 1"",'·

348 Mellw<b for EJ"ItW/mg RiJk of ClU",,,cDllnjury

dose dfects. malhematical modeb repre§cminllhe d<><e-re$pon§c relationshipba"e b<en de-'eloped. Thtse part~ular teehnlqlln. and Ihtlr Ule In eonjullClionwith oaf.ty facIo". are the topic of thi' r"'iew,

1 DOSE-RESPONSE MODELS

For a number ofyu". matbematicians and ,tati.tlCians have spent conSIderablee\fort on modellmg dOIe-lflponle ...Iatlonships both in toxicology and inepidemiology' Th....\forts h...e also illCluded stlldy of tlm•.tO-lflponlefUllCtions and tbeir dep<nde~ upon do« The... is tbe expectation of beIng ableto predict mults oUl$lde the expenrnental re.,on using mathematical model'.Thi' prediCli'. capabilit) is derived from tbe need to obtain 10w-<lMe risk..timat... For example_ toxlColol'''s produ« tUmOur ir'l<'id.rlCe dala for ,·ariou.do... ofca"inoi<ns admini..ered to rodent. o".r their lif.time:, U.ing tbis data.the statistICian aHemp" to fit one ofa ...net)' ofcu".... "lth the idea that it "illbe pos,ible to ..timat. tumour incidenet at a low .nvironmental dOle Ie>'el w.llout.ide the ••p<rimental regIon, M.ny f..l that by using dMe-re.ponse models.the .nti ...t)' of the experimental data i. used: the...by a sc;'ntificaUy p...ferable..hmat. for ..tabllsblng human exposu... l,mits;s prodt>o:ed. The: ule of dMe­...'pon§c mod.l....m.lo be quit. a .... sonable approacb. Howe,·er. dangers do«i" In u,mg matbematical mod.I,_ Tb. "atl"ician may be misled into beli.,ingtbat he i. producina an ..timate "'ith arealer preci.ion than the matb.maticalmodel is capable of prodl>Cina for a glwn ..t of toxicologIcal data.

Among the , .. riou. types of mathematical models "bicb ha... =e,,-.datt.ntton in 10"-<lMe ri.k e'timation a... tolerance di'tnbutlon mod.ls and'-UtoUS hit mod.ls. The first group offunction. tbat a ... derived from a loleranetdIStribution approach are tbo.. whICh a..ume that each 'nd, ....du.al m thepopulation has a thr..hold dOIe-r..poo.. , It is also a..umed that indi,-idualthreshold I<\'el, \"lr)' withIn a population. accordIng to some SUttistlcaldi.tribution. E""mpl.. of thi' cia" of model. illClude both the 10ll'''1C and theprobil model,. Bolh of th fUrIClions ha... pro"en to be quite useful in term, of..timalina dose-r..pon ffects. particularl)' ,,-itbin the ••perimental reaionMantel and Bl)an (1%1) prodt>o:ed one of the first 10w-<lMe risk "Umlltionprocedures. They cbose to employ' the probit dIStribution as tbe underlyIngdOIe-responle model. Mantel and Bryan a..umed that the probil of tbe lif.timeprobabilil) of tumour IS. linear fUrICtlon of log,. dose, Th.) furtber a..umedthat 'be ,,;alue of tb. slope of thIS hnur relationshIp was equal to ooe "'hich Ihe)'beli.,·ed would pro'ide a con...r....ti'-. o,'erestimat. of tumour illCide~ in tbe10w-<lMe reglon_ Ncxt. they ..kulated the dose ~orrespondinl 10 an .ffect at agi"en 10'" probabilil)' of tumour such a. \0-', Correction for .pontaneoustUmours ,,-as iKWmplished b)' usmg Abbott's formula. It ha' subsequently been'hown that the Mantel-BI)'an approach does not nece....nly prodtlC( acon,,"'altve e"imate of nsk, The "'~aknes.s Ites in a peculiarit)' of the probn

nod<:I. MalhemallCJllly. ,t approaches lhe origin mIlCh more rapidly than any of.he olher dose response fU!\Clions. For lhis reaSOn. it t~nds to esl"nlilt nsk I~vtls

;1lJ',l'Icanlly lowtr lhan lhost esurnaled by olher procedures.Another mod~1 for Io"'-<lose ~Ilrapolalion which has ~ons>der.blt appeal

"""ause of ito "mplieily is the linear 0. "",,·hil model. In this case, the....umplion is lhal a l"...a. ulrapolalion from a low uptril1>t1llal~ poiol 10the origin (lhe background I",..lj "'ill prod""" an upper bound to the lrue con...xponion of lhe dose response functIon in lhe Io"'-<lose region. Wllh lhe lin~ar

mod~t 0"" IS nol making a Vtry dttalled mechanISm'" assumplion. How",'"" aneslimal~ is produced whieh is. hoptfully. an upptr bound 10 lhe lru~ low-<loserisk. This procedur~ has 1:>o<n tnliclud from lhl! standpolnl lhal illS often 100

con"",·ati... in o.......tirnalin! lhe risk at low dose Ie,'~l•. 11 has been .hown(Crump tI "I" 1976) that for lhose inilances in ,,'hieh lhe carcinogen ilCIS in anaddil;'" fashioo .. ilh regard 10 lhe sponu_ backgJ'OUnd tumours, the dosc~

....ponse fUllCtion is essenlially Ii""ar,n lhe lo"'.(,\ose rtglon. Funlltnnore. il has1:>o<n numerically .ho",n (Hoe!. 1980) lh.al ~...n if only a .mall portion of lbebackground is addil;v~. th~ dose-r~sponse funCtion Still Ixhavts ;n a linearrnann~r in lhe low.(,\ose rtgion. Therdort. the Iin~arity assumplion may nOI Ix asconse",ali,'~ as some Ixlie,·~ it 10 be

The neXI broad "''''lOry of models are Ihose "'h"'h a.., rtf~rred 10 as hilmodel<. This nam< is dtri"':d from tilt COllCq)l thaI lhr ca"'ino~nic procnsconsiSIS of ....ties of somalic mUlations. ~a~h behavinl as lhou'" lhere is aPoi..".. ~"~nl (h'l) lhal ~hara"'rrizco lhe chan~ In lhe «II. By upo:nc:ncin/l; a...rico ofI~ ""~n,,. a cell becomes malignant and is lhen recosniud afl~r alat~lICY or !rowlh ptriod as a c10llt of cane.. ceUs. The , ..'0 modd. "'hlch hav~

receIved ,h~ mOll allen lion a,.. lhe Annila~-Don multista~ model (P~IO. 1977)and lhe 1"= mullihit model d~,..loped 1»' Cornfidd and Van Ryz,n(Cornfitld. 1977; R.i and Van R)"21n. 1979), If the probabilily OflUmOUr at dosed i. denoted as P(dj. then malhemalically lhe mod~ls a", as follo".."

multistage:

mull,hil;

P\dj-I-up[-m.,(a;+b,dj] ",.b,;loO

P(J) J: o'l·-l exp(-6/jdr/r(k)k>O.9>O

The one·bil mod~1. prtviously discussed. is • special case of bolb of th<soomodels. They do dilf~r lhou'" in lhal Ihr multi".,. model .110.... for lhepossibility of a polynomial ",Iallonship in dose. ,,·he..,.. the I"mnu multi hi'model a..umes lhal lhe ",Iallonship is~ 10 a gi...n po....r. S".ce lhe mu1tiSla~

mod~1 aSSullltS a polynomial ",Iallonsh,p in dose. lhe mod~1 is ofl~n ~xpected 10beh.a,'~ a. if il W~rt a hn~ar model in the lo"'-dose "'l,on. Th~ gamma multi hitmood. on llle olher hand. does allow for low-order polynomtallmns: thus. ilfrequenlly produces mucb lo..~r risk esllnutes. For ~xampk. if In lheexperil1>t1l·

talltpOll modenct iJlPW110 IllClWC Ii the mud lJ.'Mtr dose. lhe muluhl1 dcc!;DOl .1Iow for 1M pouoblll1~ of t1lher • lu>u,r or • quadrallC Iffl\'l "beln, prewnl.Tl!eKfo..,. PI <.h>, 'lUI"'", 1M mulillul model produces ••....-y 10'« esum.,e 01rn.t for 10-...... ~'d ..hen compared .,1b the mulliWl,. 1DOdcl. In lhell"'od .,1b the pmma mwllbd modd.. CornDdd aDd Va.a R)........,...... \bal-tvour-l Iwnoun are~I 0' U>duaod. I..-rs aDd lIlCOfl'ORlcd\hom malbtmabCllll) (0110'''1''1 Abbolf. formula. Bnd'ly. AbboIl's fonall1rop'" lhc probabLbl) 0'1_' al dow J .... P (J).,th

"'here 1'(4) Ii lhe modd bmlt ..-d .,th 1'(0) _ 0 aod the spoatallC<MlS rale, ~ P' (0) 011 1M othn lwod, the 1Il1l1usI.,. lIIOdel tiS""'" lhal bKkarOWld Ii

addllm, aDd for thas rasoa lhe lIlOdeI bcha,·.. as a linear fllllC1lO1l 0'dow Illibe"""'-<10K "'JIOll (Crvmp ., .,. lor.6).. CoruDeIcl (FDA AdvtSOlY CommJu....1911) 'Ill.... one of the &m III iJIo,.- WI ",ilh,n 1M espcn",nl1a! "",on many ofIlles< models Ii' the uptnmonLaI dall tqua.ll) ...-elL )"fl. )'IC1d qWle d,lfercrll 100>.d_ rn.t n1ImaICL The cboo;:c of1M model hcwrntscn'ocal "'hen 0Ilt .UtnlplS10 ... illLOle .If"",s.' 10,", dOlo«, panlC\llarl) II>o<c ..'hlCb .n: .."ll oUlildc 1Muper;menl.l ...,on. T.ble I ,lluslralC$ ,he c",>Cal nalure of model chol« for10\11--<.1_ ri'k C$lIl1LOUon Bolh!he pmma muluh,••nd II>t Am'lll.SC'-DolImullo'lase model ha'e "suable prnnt. In II><" fa,our for l!>t" ,..hd,'y ,ncsumal;n,lo,,--<.Ioot elfeelS In carc,n0l"n",... The examples I"en ,n T.ble Irepresen. some of.he be>! dos.e ,e.pon.. carc,nolenc»' da•• rurrenll} .'"lIll.blefor modellln, p..rpo»e$. Examln"" Ibe mios of Ihe v"lu.lI~·..f.--<.Iost (VSD)",'imaIe. for Ihe '''0 models. one nOli= Ihe app"n:n. problem. "'oth "lnylchloride, This is. special case "'here Ihe pmmo mUllihil "'Iimale,.he lo.. --<.Ioseeffeets qUlle poorl)' Th" ,n.",urlC)' is due 10 II>< <:otlCa"" nalure of lhe dOst­re.ponse cun.'e...hlCh Ii "KIely recoln1U'd a, be>n,.he ......11 of ,,'Il.-allon of.MlCIi,..uon (>f , ..n)-1 chlonde. Ne,enl>tl..,. II>< drama.1C numrnl:al 'tsuh of.difference of<llhl orden ofmaarmudc "bel.'""n !he 1"'0 model. for \lnyl chlondesh.mllcl'mp.... K>cnUm "'"h 1M pos.sobllil}' off.,lure us"',. model on al',,,ndata 5C1. For all the O\her uampln.!he mulllitaae m<Kic1 prodooes Ioo>~r VSD.."mal'" tban lbe pmma mulllhll. As prenoUlJ) dl5nlS>ed. ,Ius Ii dut 10 II><Io---ordcr pol)lMIIrIS&llmIU lhal "'" presen'''' the mlllustaF m<Kic1, bul nol m!he pmma mul"hll model Ofthe 1CCI compouodsO\her lhan 'Ul)lchlondt. "'''obsen~ \bal for four ol'lbcm lbc nr.l>O of the VSD for the 'M) IDOdcls II .,Uun anorder of ........I...x: ltuff oflbml an: bet.""" ODe aod '''''' orden of_pul"",',",ooflbml arc "'sUun '''''' aDd th= orden of mapu.,udo; ODe oflbml dlD"cn by........ than th= orOrn ofmaputudo. n... IIa SomO<II-prollkm.. Ollr 'IIlllIIId IlopoWI ....'CSallUSOlUlble IDOdcls "''C!Uld pr-Dd\IICC est'.....les ",,1!w:I about one order0' maarutudc_ n... II a ,.apIuil: iIlIIsInr.""" of lbc CfrIlJC resullS oblamcd forthese cIcvca cumples lIiIllJ _lied 'reasonable' mocleIs. We~ \bal the

DoH-hspo1lH MO<kI, QN/ tMi, Applka'iM tI> RWr Eslinwtion 3S1

Table 1 Comp,,,uoo of ..limaled Io..'-<!ost n>k ..,,""' ... ""01 ,,,", mulli>"'F aDdgamma muhiba models'

Ealima'ed VSO' vso ",tio fOfo..mma multilu, Multi>"'F ",urulla mull .....ge

N"rilolriacet"aCId (~'TAI "' 1.9 ~ 10-' 4.2 ~ I(l'

AlIat".in 8, 0.28 7.9 ~ 10-' l,S ~ 10'Ethylm<thlOurea '" '" ••2.3.7.8·T<traclllo,,,·d,benzo.p-dio"" J.I~ 10·' 1.6~IO-' "Dim<thylnitrooam'.. 7.7 ~ 10-' 1.9~IO-' 4 I ~ 10Vinyl uhlorO<lc l.9.10-'· 2.0~10·' 2.0~ 10-'

8iICllloroR>eth)'1 01,,",' 17~10-' 4,0 ~ 10-' 9.2. 10Sod,um ,""",hann 1.12 O.lJ "Elhyknnhiou,ca 13~ 10 .,

"O",ldrio 6.3 ~ 10"' 2,2 ~ 10" 2,9~ 10'

DD' U~10" 6.4~1O-· 7,S ~ 10

, ".IU<> _ I""" T.t>Iro I ODd 1 '" 'hr FCI<Iol Sol", CowKiI Il<pon [FDA. I'll,, "",..ny..r._IVSD). oklIn«l.. 'ho' <100< .~"h n ..,.....t«l "'prod._ ""~ '''''''''',~ 0_ hoeqr""nd or 10 "

situation miallt be "''ell wo,... for othercompou!>d> bo<:auso Iho dala is nOllikely10 be of the quahty found ,n the eumples at hand. The nriabihly bel.....,n low_dose r;,k estlmales usin, various m<Xk1s is wen known. Calculalions SU"h asIhose &,"<1I,n Table I aro often used by ""me a, the bas;, for ,uuostin, that 10"'·dose risk eslimation is an unreliable prO<:edure ,,'hieh should be avoided...!len",'« poss'ble. Th,s llOSIIlon can be 1l'mperoo by rtqurnna an appre<:ialion ofIhe ,pee,lin assumption. of,..riou. uontondin, m<Xkl. and a fochna w,lh roprd10 thelt biolol,ual plauSlb,litj' A purely moth.moncal ,.suo 's. how","¢r.",nerally o"orlookod. Thi. issue i' lhe one of incorporation of ba<:kuound inlOlhe model•. In order 10 auh,o"o ,h. d,ffero".,.. .ho...n ,n Table I Or ,n any 01h«such compari""n bet..'..,n modol' one mu,1 u.uall)' assurn< lhal tho baukl'ound,ncidence "md<pcndenl or '. ¢qual to lOrO. If. ho..ovor, any small ponion of theba<:kground can be a..umed to beha'-e in an additive manner lhen tho mood ",11mathomatlQlll)' be linoa. al 10...--dOle lo'-els (.... Hoel. 1980), nu. in tum1)'PlCa)l)' provonts lho ottlllTe""" of lho Iroal nurn<ri<:al differenuos foundbetw«n lhe models at 10'" dose 1",'01., Thu. the major issue of model uhoice canbe sil"ifiantly influencocl by th. quesllon of inckpend= ofbackl'ound. Sinceonly a small amOunt of ba<:kuound need' 10 be addili"e 10 prooUet: hnunly "'0find lhal tho mod.l diff'f<'nu., may be much I... a problom than peoplo presenllybello" •.

For lho examples in Table I. lhe "al= liven in tbe Food Safely CoulICil

3S2 M~,1roib fM Eflimalin{} RW: ofCMm;"a!lnjuTY

repon (FDA, 1911) indicate IMtttle estimates from the pmma multihit an: quiteprtti",. For eumplo, with NTA, the e<hmated VSD 15 0.8 while the lower 91 ,5 /.conlidence limit on thi' risk e<timate i. 0.1. Ths is an exccedinslY tiibt "'lUnateand one would as.sUme lluIl " .. are 97,5~. 5ure tlult the virtually safe dose is nolow., tluln 0.1. 00 the other !Land, the muhi'tage model suggests tlull lhevirttally safe dO$<' 10,..1i5 1.9" 10-" whK'h is many orlkrs o( maglllludc: belowthe confidence limit gi,..n by lhe: pmma muhim!. Thi' cloarly illustrate< theconcept tlult conlidence limil5 Ihat are produced ";th these "'Iimate< indicateonly lhe: Stali51ical va...bihly .MOcio.tM "-;th the linin, of tbe partICUlar dose­""pon... model, lbey do not indicate the errors llult may be present in lhem.llht:matical model «(or nlher example<..., Hoc!. 1981). NC\'ertbeks$. ,he~ o(a model ";th un,..ati5tic sta'i5tical confidence interval5 may totally misleadIndividuals whose task il is to set exposure level•.

3 T1M£-TO_TUMOUR MODELS

The next step toward ,eneralization nf dose-.....pon.. model. is tbe ",corpor_alion of lhe additional dimen'ion of tUne. Twn of the earliesl eumple< includethe log-normal dlstnbution u$ed by Blum and Drucke')' (..., Albert andAll5huler, 1913) and Ihe Weibull distribulion, "'hich is related In the Armitage0..11 muhi5tage modeL These d"tribulion5 specify the probabIlity o(a tumour'sbe,n, obsel'\'ed at a ,;,..... age fnr a';'..n dose. In !ovo'-dO$<' extrapolation, tbetime-to-Iumour modeh are pla,ued by tilt same problem. lhal tilt mo,.. ustaldO$<'--=ponsc models have; na"",ly, one cannot a.ily diilinguish statisticallybetween the lo,-normal and tlte Weibull distribulion in lilling a ~ta set, Yet,they prod""" quite different low-dose n5k e<lImate<. For eu.mple. Wh,llemoreand AIl5huler (l916) fil the long_normal and lite Weibull diS1ribulion models totlte Dnlland Hill data on lung cancer in British physician. ""-';-'1'" lheirciprelleconsumption. Whinemore and AI15hulcr Iound that both models iii the ~la

quite " ..11. Yet, if "ne we,.. to eslimate effttt. f"r either a short duration "fexposure or for a low-dO$<' rate, conside.-able differences "'"uld become:apparent, dependIng on which of the Iw" modell were oelectN.

Recent work in ti"""'to-tumour models ha' focused on tlte factorabi!ily of thetime and dO$<' Iuneti"ns. That is, one can ,,'nte the huard Iu""tion 115 a prodl>Ctof tw" functIon" """ Ihat depends solei)" on the dose oI lhe carcin"geo; lhesecond that depends "nly "n the duratIon of exposure Or on the age of lilt..po»ed indlv;dtaL The multistage model is a faet".-ahlc model. For bolhepidemi"logical data and ..penmcotal .an,mal ~ta, one can plol the IOJ "I tltehuard funcllon ,'enuslhe lime function. Through Ihis mcchani.m. il bccomet;

ol»;ous that a simple linear ~latIOn$hlPortco .....ult.. Thi5 "Italion fon""", if"..have.a fact"rable model and the age Iact"r i. represented a, a simple power. Forexample, in the 0..11 and Hill data on tlte British physicians, Whiltem"re andAllshuler e.timated lhat the hazard function could he wrinen .. the prodUCI oI

[){Ju-hJPOllH MD<k1s aM. rltei, Application la RiM E.srinwlioft 3Sl

~onsumpllon ral. and duration of .mokinl, Whal i' Importanl for lhi'factorizalion i' lhc ~nll(al dcpcndene<; on duralion of exposure "nce il i.repre<cnted as a hip po"'.' of lim. a, compared ,,-ilh lh. function of do.. ,Th.refor•. in lhe 'Iud}' of lhe smokinl habtts of an 'nd, ....dual. " boxo"""appar.nt 'hat the a.e at "hich one I><&ins smokln' i' crucial, Tbt modeltollrnal.. lhal lh. duralion i. '0 ,he fourth Or fifth po".r wh.rea, lhe oC1u.alClgarene cons"mplion. the d05C rale. i. to lhe firsl 0' 5CCOnd p""".r,

Tbt r.lalion.hip bctwe<:n Urn. and dose IS ".ry 'mportant,n d.ahng ....'h less'han hf."m. exposuJ<$. ThIS "lu.alion i. frequently encounl.rerl where .1'"demlololi<al data is in"ol,-ed (Day and Bro"'n, 1980). Tbtrefore. in any alterni'll10 quantify ri'k ,,'ilh less lhan lifetim. exposures. il i' neces<ary '0 lake inlo ac­count the posSIble complellty of lhe depend<flC< on dura',on ofexposure. This isillU>lrated. for .umple. in the p.aper by Mesclson and Ru...1l (1977). in "hich mu­tagenIC PO''''''''1 of a compound ,,-as comp.arcd ""h 'IS camnogt1lic poleDCY. Indel.munin,lhe car<:inoccnic pot.ncy il "-..s nece.... ry for M=I""n and Ru...1l'0 es"mal< hfellmt risk from 50mt «1< of expenmenlal data where ,he anImal.~i,'ed le.. Ihan Iif.lim...posurC5, To accomplish lh". il "as aSiumed lhaldepende"", on "me ,,'as of the 'h,rd power, whereas lhe do.. dependence wasa..umed '0 I>< linear.

In lhe" "ork on tnne·lo-tumour models. Day and Brown !Ia"e fQCust<! on Iheissue of .arly'-OlaF " u. laIc-stage .ff""" of a carcinogt1l for mulli.llIgemodel•. If one I><h", lhal a carcinogen aff""IS an early iOlllll"ng slage of ,hecarcillOle1Iic process. Ih.n the "·"h<!ra,,al "f Ihe exposure 5h"uld nol ha"" a"apprtCIabk "ffec' "n fu'ur" ri,k•. Thi' <l<'<"U'" boxau"" crilical cell' would ha".already been inilialed. On lhe otb.r hand. iflhe CamnOI." acto as a promot"r.lb"" w,,!>drawal of an exposure would l.n<! 10 rerluce futult; ",h qUil.dramalically. BYU.inllhesc mulli5tal. mod.b. il i.hoped lhal ".. "-ould !>tabk10 mak" conclusions aboul wbether a partICular mal""al is aCI,nglS an in,l,al"ror a promo'or. There arc probl"m. "ith Ihis approach (B","n and Hot!' 1983),In tbal for laCi. """ ofdata. lilt proctSS ofeslimalma whlCb stage is tb••tage orSlagos affec'ed i. a ' ..ry ""';li,.. procedure and """ can .asily "rr beca.u.. of lhevariability in tb. data.

4 GENERAL TOXICITY

Altbou&h malhematical dose-....pon.. model. ha been .mploytd primarily inlhe area ofcarcinol.nesi., ther" has been 5Orne: int" t ,n applyma lhem 10 Olllt-rtoxiccloJicaI en<lpo,nlS. Of p.arti<ular inl"rC51 are appl""lio"" 10 mUlal.....isan<! 1er1l'ol"1l".i" allhou&h lhere ha"e been 50mt applical,on, wilh regard 10general toxicily. For .nmp,"" in Ihe Food Safety Council repon (FDA. 1971),the gamma mUl'ihil model was~pphed lodal~ on death due to boluhsm ,n miceafl.r Ibe adminisll'lllion of bolUhnum loxin lype 1\; also in euc~ of death due 10cardiac lesions in rats after adminiSlralion ofspan 011. The mam differmce ,,-hich

3S4 Muhodsjor Ullmallll/} Ruk ojClwmorollnjl<r)'

U'SIS beh.·..n camnogenesis da,a and olh., Iypes of 10xlCOlog;cal data is lhalmany beli••,. that Ihreshold. do not exiSl for IhQS¢ matmals Ihal arecarcinogenic and are believed to result from a l\<"oQ(oxic mocltanism. On th.OtM' hand, there i. no doubt a th'eshold for tOX;c .ffoclS atlributable 10 agen ..such as botulinum toun. In thi. ca... one m.a~ prefe, to u.. a Iype ofm.athetrnllical modellhat does nOI beba"e in a linear fashion in tbe l..,..-doseregion. The basic probl.m Ibat rema,n. IS Ibe conc<pt that 'fone,s 10 alrapola'e.ffoc'. oUlside Ih••xpen1lll'ntal region. Ihm the choICe of ,be mathetrnllicalmodel ",'ill ""ally alfoct Ihe quantitative esUmoles ob,alOed from us,nl .""b ap,ocedu,e. It is 'mperal;'" that caulion be ...rcised, and that 'eliance on Ihepoinl ,i.k eslima'es and thei' correspondinll confidence inle""als Ix l<1npe,e<I""ith lood judgment.

5 POTENCY 1ST1:'IATlON

The area ofpotellC)' eslimation has been the object of altenlion r=nt1~ becau..of in,eresl in COmpannll results f,om "anOus I>>," of toxicolol"'al assays. Inpart,cular. there i' moretban pa"inll intere.t in tb. quantitati.,. predictabihly ofsbon'I'no teslS '" hicb are presumptive foro;arc,nogenicuy. Fu,'h.noore. 'here isa need 10 make quantitatlV. comparisons both bet"'·..n Sl'"";.••ucb a, the mou..and rat and bet""..n "ram. wub,n 'peaes ,n rodenl b,.........y•. In order '0acccmplish Ihisg~1. SOme meaSure ofcarcinogenic poI.ncy i' required lhal doesno, noc....rily io,·ol-.. any lo.....,jose eXlrapolahon. Po,ency does. how.ver.depend upon a matb.malkal d....ripuon of ",'bat is Ixh"'ed 10 be ,heappropria'e dQS¢-'esponSoe relauonsh,p. Tlme.tn-lumour OCCUrrence withr.gard to the less than lif.lime .xposure .ituation i. also an importanl faClor, InordOT for an 'n,-.shil'O' 10 be able 10 u<t: mo,e of the .xperim.ntal data "'allablein the lit.ralure. he must Ix able loextrapolate I.... than Iifetim. data to alifetimetumour incidence,

In lhe area of camnOle".<l.S, SaW)"e' til 01. (1984) have rea:nlly d",.loped aprocedure for"'lIJrulling ",·Itat is r.ferred to as TO,•. Th. TO,. relales 10 IUmou'data and corr.spond, 10 tb. tradi'ional LO,•. Thi' procedure ..Iima'es ",·Ita,dO«': would produce 50\ ,umOU/$ in a hfeume Sludy amonl animal. whi<h arelumour-f..... The malhematic. for tbi. procedure bas been d......loped and 11 isIxinl applied '0 large .." of .xperimental carcinog.nesi, data. Hopefully. amu,nmgful quanlita"'.. analysis of Ibe ,'anallon to .arc,nOlenlC ,..poe<t:observed amonl mains and ,pecies ""i,h,n .hemical classes ",ill be able to Ix""'de based on these po,ency ."'mates.

In Ibe area of,bort·teno leslS. special "",pita... Itas been placed on lhe AmesSol_eU(1 'esl'. Ik,n".,n ~I 01. (1981) have d..-.loped a method fo' .."malingpotency in ,he Salml>lt.lla a......)' ""h'ch allempts '0 incorporat. compe,ing,oxicil)' often ob<t;"'ed ,n the Aoms te". Usinllhe Ames 501_dl(1 dlla and itspolency ",umat"'. one can obse"'e what sorts of quanlital,ve .orrela,ion. ex"l

DO#-R~.pt)lIS~ MQlkI. and 'Mi' App/~a'itm 10 Risk EJ';malitm 355

belween Ihis a....)' and ,h~ rodenl c.""nogcne..s .....y. The ....orx on po'~lIC)

and associal..:! mathemaucal ~I< i' quite imporlant from a sci.mificSlandpo,nt. In u<m& malhemalical mod.b 10 impro". our und.rsland,n, of th.quantitali"e relationShIps bet"'een 'he ,.,,"ou< a,say'. "'. fonuna..lydo no' hasemany of 'he ""riou. problem. tbat .~ilt in low-dose nlk ",'imalion, Bothapplicalions. ho"'~-er. u"" ,be sam. speculat,,'e ma'hema'ical model. for ,bedose-.....pon"" fUllClion.

6 SAFETY FACTORS

Although ",m. use!la. been mad. in ri.x ...imalion of dose-re,pon"" moddsfor 10Ue .ndpoim. olh~r ,han ca""no&.n"",. Ihe ulual approach is to .mplo)'safOly factors, SafOly faclors ".•'" u.ed e...n,i'-ely in ,h. mid-19~in tbe ."'. offood addi,i,'",. This m~'hod 's lllli in uS<' today for many form' oftoxici'y. Safel)'fac'ors au.mptto S<" an alto,,'.ble human daily intax. I~,.l (ADl) of a 'oxicchemic.l. Thi' I~'el is d.t.rmined by diVldins th. no-<>b""n.'able·dfcc, I"'el(NOEL). ""ablished in chronic .n,maltoxici'y <tudies. by ",m. safely factor"'h",h re1I«tS Ih. 'ype of toxicil)' obsoon.'..:! .nd ,he amoun, and qualil)" ofanllable <Xpe"menlal data. Asaf.ty factor often chosen for th. cakulation ofanAD[ is 100. This "~aloe is a oomhinallon oft",o ""para« fa,tors: a ",me"hata,bltrary factor of 10 to ",Oeclth. pos,ible increased ""nsi'ivily ofman relati'-e to,he laboratory t"" an,mals. and an add,tlonal safety factor of 10 '0 accounl fortil< het<rogcn.ity of ,h. hUmlln popula'ion. The safely factor approach bascerla,n appeal m 'ha' pcopl. ar< altrac,ed to its simplicity and ar< no'lik.ly '0 bemisled by Ibe .upposed scien'ific precision ofa panicular math.ma,ical functioo,,'b;,h has been filled to somedo""-re<pon"" data, On th. othor haod, tho use ofa...fely factor a..umes some sort of threshold or practicalth.....hold beha"iour,From ,h,. st.ndpoin,. Oll<' mlly' no, ha,-e much .ppreciauon ofthoerrors Inheren,in appl)'inS a saf.ty factor of 100

Bec.use of lhe "al)'ln& qualn)' of da'a, m a rOC.o, .tudy by ,h. NauonalAcad.my ofSci.nr« Comm"tee on d"oxioS "'aler safet)' (NAS. 1980). il ....asdecided that three Ie"el. of safe')" factors should be u.ed, If the NOEL ISdel<muned from quality human chronIC exposure dala. on. "'ould need onl)' anutlC<naint) factor of 10. A factor of 100 i' used .. hen human data" 1Im"ed orincondo';"e bu, 'here" reliable 10n&_..rm animal da'a for on. or mor< ,peci...An utlC<rtain'y fac'or of 1000 i' uS<'<! "hen no long-,orm Or aCul< humao <!a'a isa~allable and th...p<nmental ammal data is limited All of thi' is, of course,quit< subjoc,iR A major objecllon to applyin, an UtlC<lUmtl' faclor 10 theNOEL i' 'ha' ,h. d.t<rmma'ion of the NOEL i' made by SImply usm& ,....0­

sample stalistical tes" of ,he eXp<llInen'al coo"ol i1foup "o"us th. ~ariou,

treated groups. This procedur-edearly d'p<nds upon 'he study sample d",;&n. Onth. o,her hand. if 0<>< uS<', a matbemallcal dose-re<poosc model. lh••mire <!a,aS<'t i' used and biases due '0 sample SIU consIderationsar-e nol a probl<m. The'"

356 Mnhodsfor Esrimlllillg Risle ofCMmkol I"jury

illm ObV10\!l :iCiemific appeal of \!ling aU of the d.ala amI nm Ignoring the slope;of th~ d"",~respon.. CUI........hich th~ sar.ty f""tot approach unfortunat.l)' d.,..._

To compar. th. effects of using the sarety factor approach. the Safe DrinkmgWat.r Committee (NAS. 1990) .xamine<! a number of compounds for ...h,chthey had camnog.n"i! dat.a, The Ume data "..re e>ttapotated to 10"..-<1"",Ie,..". The NAS Committee 10Clk the lo"'"t d"'" at ... hICh some ca""no~nic

effect ...u obset\'ed aDd apphed a saf",y factor of 500 to 'ha' d"",. This diffe..frl>tn the traditional con>truct;on ofan ADl,n tbat they applied the uncertaintyr""'N to ,he 10.....' observed effect d"'" as opposed to the highest d"'" at ...bicbno effect i. obset\'ed. The re,ults of thIS procedure are reproduced in Tabk 2

Tobk 2 Eo,,_ted bulllOJl can'." ,i<l<. Lbat "",,uld be uoociI'ed ~i,h kIlO"'n <>t

.uspected cam".,,.... ""01' ·..f<ty fact<><' 'Wood>'

Compound

._BHC~_BHC

y-BHC.mCuboD t<trachlorid<~.~

ChloroformDND..k1rinnuHepta.eblo,,,­K'T ricbloroctbyleneVinyl <hlond<

lo_I minimum_effeel dose,mlll,'dayO

1.0. 10..0. 108.0. to1.0.10'5.0 x 10.,6.0 x 104.0.10-'lO. 10·'12" 10'

"u,"U.!O-'I.l. 10

Do"", 6en.-cdfrom ..fety

facto, .pproach,mila-dar

1.0.10 '1.0.10-'1.6.10-'2.0.10·'1.6.10-'1.2 x 10-'1-2xO"

8.(1.10-'4_0.10-'"l"IO-'Hx 10"2.2.tO-·1.0" 10-'l3x 10"3.3.10-'

Upper 9S·.

~--.." ....'" of hfe""'"",k from do.. io

column 2'

1.0.tO-'l3,,10-'t.(l.tO"1.7.10-'1.2" 10-'I.S"IO-'1.4" 10"~.l"tO·'

7.3,,10-'~.6"10-'

l.~" 10-'6.8.10-'l3,,1O-'1.8"10"1.1"10"

• Volt... "" adopt<d r_ T.l>!< llI-o oronD""I",,,,,.nd H..lth. Vol. 1 (SA$. 1910)., The ",""",,,,,,..tr,,,,__.... <ha' "h",h prod......... uJ".t.;a., <blf<KnO< (P _ O~jb«~"",

'ho: ...~""-at>d til< ""'!«II<. For~.lb< "'"""'''''' elf"" ....... o.Il imw:.- ....-, l1laI_ilt<l<n"'" r""" llI<.PP"'"'IOO 01'...r"1_ 01'.1000", It.< 10-.,_.,__d"" (""""'j .... otIoe<..-..i \l '!Iloo ddren 1_.. AD! ID ~hIl:h ...r"1_ ...ppbod In lh<lupot _ .1 .._ "" dO<'< iot -.--.. u..... cbIJereol ..r"1 facIO< "'OlIlII _ cllao. lh<

""""'- - "" .-. <umpIa• ~I&d """I 'ho "1'1"" </5',; __ ,ol<Nll ......... 01' lir<time """""" ruk II_ ../)...,0111., w""'"_Ht<1I,. fNA$.197l.p. 190, ~Ol'"''''''.~"'''"'' >dolt ",,""01'10 's, ,t.<mit lor,-rn"'ch~"n,!o<i&oQ.OOJ) ...."'1 do, • 10kl " 1000" • 1" to' • U )< 10-' 10 kl"_O"lb<_lht or. .......7" to·' iotlb< "C~ ",lculat<4 ,I",,,,, ruk on n-doy.nd lOOO""'=un'"~

DoK-~Sp()n.U MoMlJ IVlJ 1M" ApplicalHm II) Risk Gslimaritm 3S7

eKepl Ihal a safely factor of .sooo InSlead of 500 wa, used. a. sUUC'lcd by Weil(1972), For Ibe li'l ofcompounds conSidere<.! by lhe NAS Commmcc. lhoc lo"'eslobserved camnogenle effecl le"el i' produud in column 1 and Ihe ADl doseden"cd from the safely f""tor approaob is gl~cn in column 2. An enlry in column2 <on,"I. of lhecorresponding '-aille in column I di,idw b)'.sooo, In column 3.Ihe multistage model for <a~lnogen..is dose-response <Ia,a is applie<.! 10 lhedala ieI. An upper 95 ~/. confidence eSlimale is made for Ihe ca~inogenle

response allhe dose Ie"el in column 2, ,,'hich is Ihe ADI den.'w ""nglhe safelyfaclor approach, Upon exam,nlng lhe ~alucs in Table 2, lv.-o importanlOMervallons should he made, The risk eslimales in column 3 suggest Ihal lhocADl 1e,..1allo"-, for an esllmate<!l,fellmc probabllny ofcancer as hip as 1i. fory.BHC ThIS is a conSlderabl)' Veal« risk I."..llban lhe usually discussed I.,,'elsof 10 - , '" 10-·, The second obscr..-alion is thallhoc ,'alu.. in column 3 ~ary from6.7 ~ 10-' (DOD 10 1.0 ~ 10-' (1-BHCj. If Ihe ADl procedure and Ibemullislagc nik ..limal< procedure ga.-e 'imilar ,..ulls_ one would expecl lhoc"alues In column 3 to be approx.imalel~ cons1&nt. Instead V.'e obsc".. lhallhey.-ary o'-er 1"'-0 to Ihree orders of magnilude. QUII< dearl)' _lhe \lie of ADI Ie,'el.when lhreshold' are not p"""nI can lead 10 high and .'ariable risks.

7 KINETIC MODELS

Pre';ously. it was mentioned lhat the rat <Iala relatinll dow: ohin)'l cbloride v.;tbincidence of anlJlo<areomas is conca.-e, whi<:h r..ulled In Ihe failure of lhocgamma muillhil model 10 pro.'ide se"'ible risk ..limat... When .'in)'l chloride isadminmered to tbe ral il is melabolically con"ened Inlo ils carcinogenic form.Thi' actl\'allon has been sbown to become salurale<.! al cxpcnmcntal dow: Icvels.whlcb a<:<:ounl$ for lhe <onca"e lumour rcsponle funclion, Using cxpcnmmtallydetermined "alues for lhe simple MlChaelis-Menlen kinelic model G<hnnll"aJ (1978) ,h"",-cd Ihallhoc 'effmi"e' 0' ""I".ted dose "'as a<luaJly hnear ".-ilhrespeclto lumour incidence, This imponanl example indicate' Ihe need 10 obtaincbemical kinelic information 10 supplemenl Ibe ba,i< tumour data heforeSlalisti<a] eu,,'e filling e.e~i.... can he meaninJfully underlaken.

Cornfield (1977) con'idered a kmelic model ,,'hi<h eStablished a lhresholdunder sleady-slate conditions. ThIS i' th<oreli<;ally Impossible. since SOmeexposure IS al....ay, possible prior 10 ""hie.'ing a sleady-'lale conditIon. These(OncePlS how••-." ,uUC't lhal non_linear kinel", models.;an produ<c dosc­response funclions ....hICh ha~e lhe appearance of a Ihreshold. Recenlly. Hoe! ~I

al. (1983) discu»ed thoc mod.lling implicalion, of assumIng tumOur TC$pOn.. a,helni relaled to DNA add""l levd, rathocr Ihan 10 lhoc applied dow: of acarcinogen. Of particular inler..t is lhe possibilily lhal SImple m«:ld, dcscnbingthe kin<:tics of addua formauon ma)' exhibit non-linear 'lhresbold' likebeba"l(/Ur. ThIs non-linearity mal ~ull from UlurtllOn of detoxification o!DNA repair processes. The u,ual dOle-response mod.ls for lowodose eshmation

358 Me'hod.< fo, B'imming Risk 0/ Ciu'mieal Inju,y

ha\~ lhe potential for o\'~r~limaling nsk by several ord~rs of magI1ilUd~ whmnon.lin.ar kinetic' are present.

8 CONCLUSIONS

Th.tt are ",v.ral conclu,ion. about dO$<'-r.sponse relati"nships and methods ofe'tirnat;n\; 'hem whICh >Cern reasonable, Fim. th. u'" of mathematical moo.1s ofd"se-re,pon.. r.lat;omh,ps in toxic"I"iY is a useful and imponant tool. It.nabl... indi"iduals '0 make p",jections, .stimation. and finally decisi"ns ...ith",m. d.g= of scientific ..alidity. Apphca""n of math.matical model' t" tb.p",blem "f 10w-d0$<' ris' ei1nnati"n is r.a",nable only a, loni a. t~ moodassumption. are clearly Slated and woll understood. Confid.nce interval•.how..,'"" are often misin,.rpreted because many beli.v. incorrtttly that ,h.confidence inte"'al incorporates poten,ial bioloiical.rror in model ,pecification.On the other hand f.1y fa<tN applICation. are appealing because th.""mpl;city is ....11 und tood, How....er. th.y ar. quitecapabl. ofereatini a fal..sen.. of security by proo",,;ng ..f.ty levels w'hich are too high. r«ulling inpotentially d.let.rious high-ri,k .ituations.

Future pr",pects for ImprovIng low -<lose risk ...timation Ii. more wi'h bIologythan WIth statistics, Ther. is only a very hmlted amount of inf"nnation '0 beobtained from any careinogen..is bioa,..y. It pro..;d.. po,.nq data and a crud.indication of dose-re.ponse shape. Thi' information. bow.,...r limited ;nsci.ntifiC cont.nt. " ...ential and hopefully can be augmented with diff.rentkind' "f biological data ill ord.r that low-<lose e'timates might be improved. F"rexample. infonna'ion "n the ,hape ofth. do..-re,ponse curv. at low-<lose lev.l.could be oblained from DNA'addUCl m.asurements wh.n this i' the appropriateeareinogene,;, mechan"m. ThIS approach and man}' oth.rs yet to be d.veloped'hould ittatly impro,'e the quality and scope or 10w-<l0$<' ri.k estimation,

9 llEFEREl\CES

Alben. R. E.•nd Altshuler, B. (l~13), Con,ide"uon, "laun; '0 the forrnula'ion ofhm Lts fo' "navoid.ble popul.tion ••peou,"" to <nviron"",n..l• ..-cillOaen•. tn Sanders.C. L. Busch. R H., 8<lllou. I. E.. .nd M.h1wn, D D. (Ed<,) ~IldtCa,eilo<>g<""u, pp. "33-213. AEC S)mpo>ium Series. mls CONF·120SOI. NauonalToohnic.1 tnforrnat'on Senice. SprinJfteld. VirJioi.a.

Bem""o. L.. Koldov, 1.. McC.nn. '-,.nd Pike, M C (11lS1l. An empincal.ppr<W:h '0'I'Ie 'talo'Ioe.1 .nal)~i. of mUlOiCll<.i. dot. from the Salmonc:U. ,..,. Mu'a' Po,,-. 91.26J_281.

Brow·n. K B., arid Hoet. D. G. (1983). Multi"_ pred;ct;on of cancer in ,"naUy dl>5edanImal, wi'h .pplica'ion '0 'I'Ie ED", otud)' frmJam. "Ppl, Ta•. , 3••,0-411

Cornfi.ld. J. (1911), Carcino;en;c rl.k .....'ment. Sc;.",<:<, 1118. 693--699Crump. K. S.. Hoel, 0 G.. LanJley. C. H...nd Peto. R. (1976), Fundamental

carciooaen;c proceo....nd 'heir implicauon. for io,,'dose ri'k ......m.nL C""ar Po",.36. 2913-2979

Dos~-R~$p(»JU Mi'M/s alld ,Mi' ApplkmWrr 1<> RisJc £,slimalion 3S9

Day. N E.. and B,mo·n. C. C. (1980) Mul".....10 modol. a'" pnm,or)" ~nt",n ofcar>c<r. J. ""'n, C"""', Ins" 601,911_989

FDA (1971), Food .nd DruJ Adm,n"t"'''on Ad,""''Y Com",m« OIl Protocols fo'sarti)" E"al""",,,•. 1',,,,,1 on C.""nolO'"'''' Report On C.nc<:r T..1l011n til<- Sofet)E,'.IUMn nf Food Add"i,'" .nd P<"leid.., Tox. apI'/. 1'/10",,<>(.. 20. 419-43B

Hod. D. G. (1980) lncorpora"oo of ba<kl'o"nd tn d"..-rospon.. modols, F,dn 1',0<,J9,1J-1$

Hoel, D G. {19$1). Cornm<nl: caraooJ<nle ri'k R.... ,1na/y."'. 1,63-64Hoel, D G .. K.plan. N. L., and Atide""". M w. (1983). Imphc."on of nonti",..

k,oetle' on ",k "l;mallon;n ca""no,........., So"""", 219. 1032 _1031.G<hnnl, P J. Watanabe, P G.,'''' Park, C. N. (1918). R...,lut;"" or d"..-rospon..

'0''''''' d.to for c"'m",.11 "Gumnl ",...bol", ""u'''t;on: eumpk-,'inj" chloride.Tox. appi. PM''''''' __ 4-4. IBI-$91

M.otd. N. 0'" Bryon. W R (1961). 'Sore,)' l..t,ol or",,,,,noseo,,, 'IOn". J. "",n.Cantn ''''' .. 27,4'' 00

M...I",o. M .. and R l1. K. (1977). Comparison' of ara"""o'" a'" mu....'<nlepol<ll<')'. In Hian, H. H.• Wotson. J D '" Win"en. 1. A. (Ed.,) O''IIW e/ H"""'"Cant", Book C Human Risk ,1 """', PI', 1413-1481 Cok! Sprm, Harbo'Lobora,o,)'. Cold Spnol Horbor. Ne.. Yo'k.

NAS {19'7). Dr"",bog Wortr oN H.ol,A, Vol. 1 N'''e,..1 Acodem)' of S<1e-nce>.W.wnllon. DC: 939_.

NAS (1980). DrinJril'll Wain and H«>II~, Vol. l. NOliotl.1 Academ)' P...... W.ill,nI1Oft,DC: 41$ pa",.

Peto, R. (1977). Epidem;olOJY. mullL'''I' mod.I•••nd .Iw,"'.rm m"tol,n","y ,..". InH,.n. H. H .. Wat""', I. D".nd Win"... J A. (Ed•.)O,~""e/Hwnan Cant", BookC H_1Wk ,1..."""n, pp. 1403-1428. Cok! SpriOJ Harbor Lobo""ory. ColdSpnOJ Horbor. New York

R... K V.n R)"ZIn. J (1979). R,'" .....'''''nl of to>", en'''Oft...ntol ,uWon«>based on ",,.Ioud mul';-hi' model In 8re.Io... N...nd Whill""O". A. (Ed •. )E""91 oN H~(J}IA, Pr· 99-117. SOC;tI)' of Industri.1 ond Applied M.,hemalics P....,Phil.delph ...

Sa..)-.r. c.. Peto, R.. Bern,..,n, L. and PIke, M. C. (1984), Calcula,ioD of can::inogen'"",'<Dey from IoOJ·lerm animal ca..'no"........penmen". 8_"1<> , l1-4Q

Wed. C. S (I')J2). S"';>1ics ....ret) rKlon.nd ><><o,ilic judp><n' ID ,be lualioD or..ftl)' fo' moo. rox. appl. PM' 21. 4S4-46J

Whl1temore. A.••nd AI1>lIul... B. (1976). LuOJ c.nc<:r ioc'der>c< ,n c'prell' >mok....,fun"', .naly;i. or Doll .nd Hill', dot. for Bri",h ph)'"can•. B_rri<~, 32, 80$-816