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Sheet 29of 97DESCRIPTION OF LEADPHYSIOLOGICAL MODE L

W.F

. Bille r

August 1983 P. 2 

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DESCRIPTIO

NO F LEA D PHYSIOLOGICA

L MODE L Th e following i s a descriptio n o f the physiologica l mode l whic hwil l be use d wit hthe NAAQ S Exposur e Mode l (NEM ) t o estimat e the bod yburde n o f lead in population s expose d t o lea d compounds . NE M applie d t o estimating lead exposure s wil l trac k the dail yexposur e t o lea d o f cohort s fro mth e tim e o f birt ht o th e sevent hbirthday . Thi s exposur e wil l includ e airborn e lead breathe d int o the lung s an d lea d comin g int o the mout hfrom ingestio n o f food , drin k an d othe r material s (e.g. , soil , dust) . Th e detail s o f th e NE Mmode l applie d t o lea d ar e give n i n a separat e repor t (1) . Give n thi s dail yuptak e o f lead b ya cohort , an d certai n physiologica l parameter s characterizing the cohor t i t wil l b e possibl e t o estimat e fo r ever yda yi n th e firs t seve n year s o f th e cohort s lif e th e amoun t o f lea d i n variou s part s o f th e body . Th e physiologica l mode l fo r performin g thes e computation s i s base d o n the compartmenta l analysi s mode l develope d b yRabinowit z an d other s (2,3) . Th e compartmental-typ e mode l i s wel l suite d t o us e wit hNEM . Th e compartmenta l analysi s approac hdivide s the bod yint o a numbe r o f wel l define d part s calle d compartments . I t i s assume d tha t lead brough t int o the bod yeithe r throughthe lung s o r the mout hi s dynamicall ydistribute d among thes e compartment s an d ultimately , give n sufficien t time , excrete d from the body . Figur e 1 show s the compartment s whic har e include d i n the physiologica l model . I t i s see n tha t ther e ar e four compartments :

1 . Bloo d 2 . Bon e 3 . Sof t tissu e 4 . Teet h Followin g the assumption s o f Rabinowitz , i t i s assume d th e lead comin g int o th e lung s and mout hwhic hi s eventuall ytake n up b yth eP. 4

abov e compartment s firs t come s int o blood . Th e arrow s i n the diagra mar e mean t t o indicat e tha t ther e i s a dynami c an d continua l exchang e o f lea

d betwee n the bloo d compartmen t an d eac ho f the othe r thre e compartments . Ther e i s n o direc t exchang e betwee n an y o f the othe r compartments . Al l lead entering th e bon e mus t com e t o i t fro mth e bloo d and al l lea d leavin g th e bon e mus t firs t pas s int o the blood . Th e sam e i s tru e fo r the teeth . I n the cas e o f th e sof t tissu e al l lea d enterin g th e sof t tissu e enter s fro m the bloo d compartment . However , althoughmos t o f the lead leaving th e sof t tissu e doe s s o b ypassin g int o th e bloo d som e als o i s excrete d fromthe bod yi n swea t o r passe d int o th e fingernail s an d hair . Lead i s als o remove d fromthe blood b ythe kidney s an

d secrete d i n the urine . A s suggeste d i n Figur e 1 , no t al l the lead breathe d int o the lung s enter s the blood . Som e lea d i s no t capture d o n the wall s o f the lun g an d therefor e i s exhaled . Mor e important , remova l mechanism s operativ e i n the lung s bring lea d int o the throa t wher e i t ma yb e expectorate d o r swallowed . I n th e latte r cas e i t ha s anothe r chanc e t o be absorbe d int o the blood during passag e through the digestiv e tract . Furthermore , onl yabou t on e tent ho f the ingeste d lea d passe s int o th e bloo d a s i t move s throug hth e digestiv e tract . Sinc e NE Mwil l provid e the mode l wit hthe dail y quantit yo f lead take n int o the lung s an d mouth , the physiologica l mode l wil l adjus t thes e input s t o accoun t fo r th e los s o f lea d fro m th e lungs , th e reroutin g o f som e lea d fro mth e lung s t o th e digestiv e tract , an d the incomplet e absorptio n i n the digestiv e tract . Whil e entranc e o f lead int o the bloo d compartmen t throughthe route s mentione d abov e i s quit e rapid , the exchang e o f lea d betwee n t e compartment s an d it s excretio n onc e i n on e o f the compartment s i s a relativel yslowprocess . Th e overal l half-lif e o f lead i n th e bloo d i s abou t 3 0 days , i n the sof t tissu e i t i s abou t 4 0 days , whil e i n the bon e i t appear s t o be i n exces s o f 5 0 year s (2) . I t i s assume d tha t the processe s governing the exchang e o f lea d betwee n the compartment s an d excretio n ar e linear , tha t is , theyca n be describe d b ya systemo f linea r differentia l equations .P. 5

Th e rat e a t whic hlea d move s fro mCompartmen t # 1 int o another compartmen t i s proportiona l t o th e lea d conten t o f Compartmen t #1 . Th e proportionalit yconstan t i s dependen t o n th e directio n o f th e flow , th e compartmen t fro m whic hth e lea d i s exiting , an d th e departmen t i t i s entering . Th e constan t fo r th e flo wfro m compartmen t i t o compartmen t j i s define d a s k ^ ^ . Th e studie s cite d abov e suppor t a linea r model . Th e compartmenta l syste m illustrate d i n Figur e 1 i s describe d byth e followin g equations : (la ) dx x /d t = A- k 1#1 x x+ k 2fl x 2 + k 3fl x 3 + k 4fl x 4 (lb ) dX 2 /d t = + k 1 2 *i- k 2 2X2 + 0 + 0 (lc ) dX 3 /d t = + k 3 -JX- L + 0 - k3 3 X 3 + 0 (Id ) dX 4 /d t ***

* + k 4fl X x+ 0 + 0 - k 4f4 X 4 wher e X ^ i s th e tota l quantit yo f lea d i n compartmen t i , Ai s th e tota l quantit yo f lea d enterin g th e bod ype r uni t time , an d k^ ^ i s th e su mo f al l th e proportionalit yconstant s representin g remova l processe s fo r compartmen t i , tha t is : (2a ) klfl

- klfQ+ k l f 2 + 1*1, 3 + k l r 4 (2b ) k 2 ^ 2 = k 2 ^ 0 + k 2 # 1 (3c ) k 3 f 3 = k 3 / 1 (4d ) k 4 f 4 = k 4 f l Sinc e lea d uptak e i s constantl yvarying , differenc e equation s provid e a convenien t approximat e solutio n t o th e syste mo f equation s (la ) t o (Id) . W e ca n write : (3a ) X x ( t i+1 ) = A(ti ) + X 1 (ti ) - k-^iX^ti ) + k 2fl X 2 (ti ) + k 3,l x 3<i > + k 4,lX 4 ( i ) (3b ) X 2 ( t i+1 ) = X 2 ( t i ) + ki^Xl^i ) " k 2,2 X 2*.ti ) (3c ) X 3 ( t i + 1 ) = X 3 ( t i ) + k 3#1 X 1 ( t i ) - k 3/3 X 3 ( t i ) (3d ) X 4 ( t i+1 ) = X 4 (ti ) + k 4fl X 1 ( t i ) - k 4f4 X 4 ( t i )

A s writte n th e tim e differenc e t^ + ^-t ^ i s equa l t o one . Notic e tha t i t i s assume d tha t Avarie s wit htime . Thus , i f tim eP. 6

i s divide d int o interval s whic har e smal l compare d t o the tim e constant s i n th e system , and th e X ^ ar e al l know n a t th e star t o f an interval , Equation s (3a-3d ) ca n be use d t o calculat e the X ^ a t th e en d o f the interval . Th e smalle r the tim e interva l compare d t o the tim e constant s k the mor e accurat e the approximation . Give n tha t th e smalles t half-lif e i s abou t 3 0 days , a on e da ytim e interva l ha s bee n estimate d t o yiel d a reasonabl yaccurat e approximation . Furthermore , i f A(t ) i s known , th e Xj(t^ + ^ % concentration s can be substitute d int o the righ t sid e o f the equation s t o yiel d the concentration s a t t 1 +
. Thi s proces s ca n be carrie d ou t ove r a

n entir e time domai n fo r whic hA(t ) i s known . Th e error s i n thi s approximatio n d o no t accumulat e ove r time . An alternativ e approac hwoul d be t o assum e tha t A(t ) i s accuratel yapproximate d b ythe averag e uptak e rat e pe r dayan d us e the exac t solution s t o Equation s (1) . Thi s woul d resul t i n exponentia l term s i n plac e o f the simpl e linea r term s i n Equation s (3) . Give n the larg e numbe r o f individua l computation s the mode l wil l mak e i n following a larg e numbe r o f cohort s ove r seve n years , th e adde d calculation s tim e wit hth e exac t solution s i s probabl y no t wort hthe gai n i n accuracy . Th e exposur e mode l wil l allo wthe computatio n o f A(t ) fo r eac hda yi n th e lif e o f eac hcohor t u p t o seve n years . Eac hcohor t wil l hav e it s ow n se t o f tim e constant s k 1 '^ . Furthermore , thes e tim e constant s wil l chang e a s the cohor t ages . Followin g the premise s o f the exposur e model , i t wil l be assume d tha t the tim e constant s for a cohor t ar e constan t ove r thre e mont hintervals . I f the computatio n i s t o includ e th e unbor n fetus , uptak e wil l be fromth e mother s bloo d an d change s i n th e tim e constant s wil l hav e t o be a t a ver yfrequen t basi s i n the earl ystage s o f feta l development . I f the computatio n start s a t birth , a se t ofinitia l condition s fo r the quantit yo f lea d i n eac ho f the bod ycompartment s wil l hav e t o be assume d a t the tim e o f birth . WilliamF . Bille r 8/29/8 3P. 7

REFERENCE S

1 . W . F . Biller , "Descriptio n o f Lead Exposur e Model" , Documen t submitte d t o Standard s an d Strategie s Divisio n o f the Offic e o f Ai r Qualit yPlanning an d Standards , Environmenta l Protectio n Agency , Researc hTriangl e Park , NC , Augus t 29,1983 . 2 . M . B . Rabinowitz , G . W . Wetherill , and J . D . Kopple , "Kineti c Analysi s o f Lea d Metabolis mi n Health yHumans" , Journa l o f Clinica l Investigatio n 58 , pp . 260-270 , Augus t 1976 . 3 . S . R . Bernard , "Dosimetri c Dat a an d Metaboli c Mode l fo r Lead" , Healt hPhysic s 32 , pp . 44-45 , 1977 .