Document e5j1XgOpBqo2o5xEx7nggjDkM
N ovem ber 16, 196q
M em o to: D r. Kehoe
From : T. Sterling
Subject: D eposition of Dead
The rea n a ly sis of the lead data w as m otivated m ainly by the qu estion w hether one could a s s e s s the am ount of lead d ep osited in the tissu e s of individuals who are exposed to lead in quantities g rea ter than usual. I can now subm it a solu tion to you of w hich I am reasonably certain.
This report w ill deal with the bare bones of the problem , as w e have d isc u sse d it during the yea r. It does not co v er other a s p e c ts of la s t y e a r 's w ork w h ich a r e in v a ry in g s ta te s of c o m p le tio n . F ig u res, tab les, and a lis t of the sym b ols used and their m eanings can be found at the end. I have a lso su m m arized som e p rosp ective w ork w hich appears im m inently n ecessa ry . C onclusions or ap p li cations w ere avoided until further d iscu ssio n with you.
I w ould lik e to add one note of ca u tio n . In com puting actu al valu es I have been w illin g to g en era lize from the ob servation done on s ix su b jects, under som ew hat varying con d ition s. It is encouraging that a fa irly co n sisten t picture em erg es at the end d esp ite individual d ifferen ces, Yet, the calcu lation s a re lim ited by this fact.
F in a lly le t m e acknow ledge the patient help and coop eration that w as given by M r. Shaefer (in gen erou s q u an tities) by M r. C reech , and by John P h air, who w as w illin g to to ss the problem around sfilmoat e n d le ssly .
HE
A U
*} .1
-a 3
'
r b
^ u
2
Some Prelim inary D iscussion
We sta r t w ith a sim p le d e sc r ip tiv e m od el {F ig . 1). In th is m odel lead m ay be in gested or inhaled. E xcretion of lead is through fe c e s and u rin e. Lead m ay be d ep osited or returned to the body fluid com partm ent. Lead m ay a lso be lo st through sw eat or m ay be taken in without being m easu red . H ow ever the statem en t m ay W euristically be m ade that unm easured intake and output cou n ter balance each other. A long term com parison of lead, in and out, a s m ea su red by the exp erim en tal p roced u res show s that over a long duration lea d in {P b l)is app roxim ately equal to lead out (P bo). Table 1 gives the com parison for a num ber of su b jects.
P rim ary Equilibrium
A n e q u a lity o f P b l and PbQ du rin g the p r e - e x p e r im e n t a l p e r io d
m ay be taken to indicate that the subject is in a state of equilibrium
in which the amount of lead excrefced^ p p ^ x im a te l^ equ al^ th e amount
of lead in gested . Since som e of P bl co m es into the body com partm ent
one cannot a ssu m e that the equilibrium is due to d irect elim in ation of
P b l. H ow ever, even if som e of the P bl en ters the body and even if
som e fraction of that am ount is d ep osited it is e till p o ssib le to obtain
such an equilibrium betw een P b l and P bo. The lo g ica l step s by which
such an equilibrium is p ossib le can be conveniently d iscu ssed by a
ph ysical analogy.
-
L et us take a beaker of w ater B in w hich the volum e of fluid V
is kept constant. A sam ple of fluid, S, is rem oved each day and an
equal amount of clea r fluid added.
Case 1 L et u s now add a co n sta n t am ount X of a aolu^ble s a lt, to the
beaker each day. The fir st day som e of the sa lt w ill be rem oved
3
with S. The p recise am ount rem oved w ill be S J or that proportion of X in V that fa lls into S. It is obvious that after adding X once, only a fraction w ill have been rem oved . The le v e l of the sa lt in the beaker w ill in crea se until the am ount rem oved is equal to the am ount of salt added. An equilibrium w ill be reached when
S Case 2
x
( 1)
Let us a ssu m e next that w e disturb this equilibrium by rem ovin g a constant am ount of X, dX^through so m e other channel w ithout m od ify ing V. A gain it is obvious that the p r o c e ss w ill go to equilibrium at a tim e when
S { 1-d ) x
(2)
Case 3
L et us a ssu m e that only part of x is added to B and that the r e m ainder is p laced d irectly in S. W here k x is added to S and k ^x is added to B and w here
kj x + k^x = x so that
k 1 + k2 1 equilibrium when
S ( y ) + ki x = x
'
<3 )
or when the am ount of sa lt rem oved in the sam p le S equals to the
am ount added. Of co u rse in the la tter c a se
HE 001 3058
4
Case 4
L et us a ssu m e that x is not a constant quantity but a random
variab le, taking on valu es betw een x . < x < x
. When
m in.
j *** r i &jc.
the am ount of x^ v a r ie s a t random S
w ill vary so that the amount
$i of sa lt rem oved in S, w ill fluctuate betw een x ,
j .Inin
x. m ax.
If the b e s t u n b ia se d e s t im a te of = E (x ) = x^ i s c o n s id e r e d ^
It follow s that over a duration of
n
= E (x) nn
and that
2 V
8
j `j
or that the average output eq u als to a v era g e intake and that the 22
V a r ia n c e o f outpT&l, er , e q u a ls to th e V a r ia n c e of in p u t, <r . &X xi
If the variab le input of x is co n sid ered for C ase 3 it is s till
true that
* /*ix
zi b v +
a
\
vj
n Ex. Jill.
n
but now
2 s.
2 >v
8tf
b ecau se of the add itivity of V a ria n ces. A sim ila r statem en t ap p lies to C ase 2.
KET 00130h9
5
R ecovering Unknowns
L et u s assu m e that one can m easu re only the amount of salt rem oved in the sam ple S, w hile V, and x a re unknown.
F rom o ust d isc u ssio n of C ase 1 it is c le a r that if one p erm its the p r o c ess to go to equilibrium one can then know X, the am ount of salt added. If the am ount of sa lt itse lf is added in variable quantities the average am ount of sa lt rem oved during equilibrium em erg es as the b est estim ate of S (x), the average am ount added. C ases 2 and 3 p resen t d ifficu lties in that two unknowns have to be evaluated from a single m easurem ent. H ow ever, a solution here is p ossib le if we change the lev el e equilibrium . F or in stan ce, in C ase 2 if a known amount A x is added to the beaker a new equilibrium w ill be reached when
C)
in m easu rin g the proportionate in c r e a se of sa lt in the sam p le the c onstant d and the origin al value of x can be c om puted.
In c a s e 3 a sim p le co m p a riso n of two equilibria)' is not su fficien t to reco v er the unknowns and k^. A num ber of m ethods could be u sed h e r e . On cou ld , fo r in sta n ce com p are the rate w ith w hich the n ew e q u ilib r iu m i r e a c h e d fo r a n u m b er o f d iffe r e n t in c r e m e n ts of x- t, A x i,. 2 e t c . It i s o b v io u s th at th e r a te w ith w h ic h the s a lt in S r e a c h e s e a c h now e q u ilib r iu m i s a fu n c tio n o f A x and o f k^ and k^.
The d iscu ssio n of the p h ysical analogue points to the follow ing c o n c lu sio n s:
1. A m odel con stru cted along its lin e s should show the sam e p rop erties p resen tly exhibited by the data.
HE 00130C0
o
2. 13a. a n a l o g u e c a n b e c o - o r d i n a t e d a l m o s t p o i n t f o r p o i n t w i t h a r e a s o n a b l e p i c t u r e o h o w th e escchangs of in o r g a n i c l-ead take a place. a. The soluble s a lt ia v a ria b le q u an tities is eq uivalent to the daily inorganic lead intake. h. The constant volum e in the b e a k e rs is equivalent to the body fluid. c. The sam p le, S, is equivalent to the day by day ex cre tio n . d. C asa 2 d escrib es a deposition p ro cess that m ay possibly rem ove som a of the load fro m the body fluid and sto re it in t i e s u e a n d b o n e . e. C ase 3 da sc rib e s a p ro c e ss that m ay be s im ila r to the d i r e c t f e c a l o x c r e t i o n o f a f r a c t i o n f I s a d i n g e s t e d d a i l y . f. C ase 4 allow s for treatesan t of the data even u ad ar v ariab le conditions of intake and output.
3. The o b v io u s a d v a n ta g e of the m o d e l ia that s o m e im p o r ta n t c o n c lu s io n s c a n b e r e a c h e d f r o m ksiow lsdg of; a. lead rem oved by excretion b. lead intake c . a d d i t i o n a l l e a d a d d e d i n fcasgvra l a c r m a n i a .
4. The m ajo r co n cern w ill be w ith the consequence of adding an in c re m e n t of lead through' the lungs. Both the in c re m en t sb sJ t h e p o s s i b l e a m o u n t o f t h a t i n c r e m e n t t h a t i s d e p o s i t e d c s s b-* ev alu ate d fro m the lata by u s e of a m@del sim ilar to the aelo g u e .
tf/r 0013061
7
We return now to the flow of lead intake and excretion of F igure 1. The p re-ex p o su re period w ill be evaluated to begin w ith. During this period the su b ject's lead in gestion and excretion is
m easured but he is not yet exposed to an increm en t of lead in a ir.
JLet PbOQ =
PbY) w here the function is lin ear
so that
PbU + P b F * k|PbI + k PbV , + a = Pbo.
J J J ^J
J
(5)
where
PbOj * to ta l e x c r e tio n o f le a d on th e j th day.
PbFj = fe c a l ex cretio n of lea d on the j th day
PbU^ u rin a ry e x c r e tio n of le a d on th e j th day
in g e s tio n of le a d on th e i th day tbssfisr th e j th day of m easurem ent
PhVj 1 am ount of lea d in body flu id on the j th day a * constant, a, may con sist of three parts.
1. (PbY) * a fraction, d , of lead in body fluid that is dep osited .
2. D = d ifferen ce betw een unm easured input and lo ss of lead.
3. e = error of m easurem ent.
n Zc a s s u m in g th at in th e lo n g run 1__ = 0, o r th at th e e r r o r i s not
c o r r e la t e d to an y o f th e o th er v a r ia b le s , th en a = d P b V 4- D. jjt i s assu m ed that PbU, excretion of lead through urine, is sim p ly a lin ear function of PbV and of urine volum e VU.
FbUj * Cj PbVj + c 2VUj +
(6)
w here again, n
2e 1 n
0
The la s t assu m p tion ap p ears to be rea so n a b le, During the
experim ental period both urinary lead (PbU) and blood le v el of lead
9
into the body, so that
L * S + L.
(7 )
The am ount of lead excreted after the fir s t day is given by P b 0]L - k ^ P b l ^ + JLfl) +kJPbVQ + (1-1c 1) L b + L ^ ] + a
The amount of lead excreted after the second day is given by P b 0 2 = k 1(P b I_i + L g ) + k 2 C pbV o + ( l - k 1 ) ( l - k 2 ) L 8 + ( 1 - k ) J . ^
If w . let
b1 b2
u -V U -k2)
+ ` 1- k 2> V V ] + *
P b o 2 = k (Pbl_ + L a ') + k. P b V o + (b.ILs, + L b )<l"t b 2 ) 3 . + a
<i ItfK cl Note that PbV^ stands for the amount^in total body fluid at the onset
of the new additional in crem en t. A fter th e n th day, the amount of
lead excreted w ill be given by the general ex p ressio n {8a)
(8a) PbOn = k ,1{ P b I -1 + L s ) + K2 ~fP b V o + (b ,l La +bL . X l + b_2 +2b \ +____+b"2
but
n n- 1
2 Q l + b 2 + b 2 +____b*J*x ] =
n 1 - b.
1 1 - b.
and
1 - b. l i n t _____ i n -- > oo 1 - b.
1 1 - b.
2
-
.o that (8b)
b +L
PbOo = k l ( P b!_i+ L j)+ k2 Q pbV o+ -
] +a
( 8b)
The increm ent in excreted lead is now given by the difference b e tw e e n the am ou nt of le a d e x c r e te d w ith o u t the in c r e m e n t L., or PbOo', and the am ount o f le a d e x c r e te d on th e n th day,' o r P b o n, ( f )
KJE" 0053064
lo
APbO = PbQn - PbOo
(9)
=k
.V > =M
but
1 ~b2 " * 2
so that
APbO
=
kL la
+
(1
-
k
1
) '
JL a
+
L, t>
L + L. = L sb
J , CKck So that a fter n day s, the inc r e m e n t of lea d c x crcted Ms equal ex a ctly
to the a d d itio n a l in c r e m e n t of le a d ta k en in r c<vcl<
If now conditions of C ase 2 hold and eom e of the lead p resen t in the body is stored , it follow s that the in crem en t of ex creted lead is g iv e n by ( J.Q )
& P bO = (l-d)L w here d = constant of deposition.
(10)
To find the constant of deposition one sim p ly takes
1--
APbo L.
=d
or 1
Pb.O n__-__P_b_( 'L
=d
(11a) (lib)
where
PbO * the d a ily total le a d e x c r e tio n a fter eq u ilib riu m has been reached
PbO L
= the daily total lead excretion of the p revious eq u ili
brium , before the in crem en t, L, w as given daily.
* * *.
* '
'
= the daily increm ent of lead
d = rate of deposition of L.
tfg 0013065
U se of the D eposition T heorem in a P ra ctica l Solution
B eca u se of the v a riab ility of actu ally m easuring intake and output of lea d for hum an su b jects, the th eorem has to be m od ified to conform to conditions of C ase 4 .(12) and (13)
&PbQ = E (PbQn ) - E {PbOQ)
( 12)
d and where
E(PbO)
E( APbQ)
E( L)
m Z PbOi. 1
m m Z JL.
(13)
E ( L ) = ----m--------------
or the arith m etic average of daily intake or daily output, m easu red
during p eriod s of eq u ilib riu m . One could now obtain the n e c e ssa r y
estim a te s to so lv e for the conditions of the lead cham ber exp erim en ts
by m easuring excretion of lead previous to the experim ental exposure
and after the exposure period has gone on for a su fficien t tim e span to
perm it a new equilibrium .
In p ra ctice such an attem pt w ill not w ork, unfortunately. A ll su b jects show a change in food intake during the exp erim en t w hich, w hile p o ssib ly due to changes in daily regim en, m akes it im p ossib le to u se the m ean ex cretio n during the p re-ex p erim en ta l period as an e stim a te of E(PbO ). T his d ifferen ce is show n in T able 2. We cou ld
o' u s e a n o th er a p p ro a c h to g e t an e s t im a t e o f PbOQ. It i s r e a s o n a b le to a ssu m e that input and output b a la n ce. In fa c t, the th eorem sta te s ex p licitly that the am ount of lead in gested and absorbed from other
001 3 ut>o
12
s o u r c e s and not m e a s u r e d in the e x p e r im e n t is c o u n te r b a la n c e d by lead lo st through other so u rces and not m easu red by the exp erim en t (su c h a s s w e a t y One could^take the a v e r a g e in ta k e m e a s u r e s during e q u ilib r iu m p e r io d s fo r th e b e s t e s t im a te of PbOQ. The v a lid ity of th is p ro ced u re is te s tifie d by fig u r e s in T able 1. In sh o rt, one can take the average am ount of lead ingested^during the exp erim en tal period/jto estim a te the average am ount that would have been excreted due to in g ested lead . T able 3 g iv es the b e st estim a ted of Pbo , Fbo ,
o' n7 as w ell as the b est estim a te of L, as m ea su red d irectly from the am ount of air resp ired by each subject and from the cham ber conc antration.'- The con stan t of dep osition , d, can now be r ec o v e re d by the follow ing exp ression
E (d) = 1
_E_(_P_b__O n*)__-__E__(P__b_O_o ') E(L)
The calcu lation s and resu ltin g proportion of storage are given in Table 4. (The values of L w ere obtained from m easu rem en ts done by M r. Shaeffer^
Lawful relation betw een d and L
It is un likely that d is a constant for any am ount of lead absorbtion. A number of factors appear to'peak against such an assu m ption. F or in stan ce, the lead and tissu e m ay in teract as a
'JCCvWi' t fu n c tio n o f c o m b in e d s u r fa c e areaV" D e p o sitio n of le a d m a y in c r e a s e the lik elih ood of additional d ep osition . D eposition of lead in som e t i s s u e s m a y in h ib it the e lim in a tio n of le a d k*Abody flu id .
V ariation s in d valu es for d ifferen t su b jects b ecam e orderly when, at the su ggestion of John P h air, the im p act of L on d w as c o n sid ered . T able 5 show s that th ere ap p ears to be an ord erly and la w fu l r e la t io n b e tw e e n d and L . Xnstfo&ees In L.a*o usaoctv&iad quite ob viou sly w ith exponential in c r e a s e s in d. With one sin g le
0 01 3 U6 i
13
excep tio n , the d v a lu es fo r a ll su b jects fa ll on the equation (12).
-5. 17(L) + 1.4
d = e "C
(12)
An independent ch eck e x ists p a rtia lly for the valid ity of this e x p r essio n . When L equals zero, or b efore the individual sta rts
h is d aily exp osu re in the cham b er the value of d, referin g now sim p ly t- j
toM n gested le a d , sh o u ld eq u a l to e * o r . Q&' * T h is v a lu e is v e r y c lo s e to the actual ob served d ifferen ce betw een daily intake and excretion of lead for those subjects for whom Pblo > Pboo
Additional E m pirical V alidation
N either the rea so n a b len ess of the m odel nor the sa tisfa cto ry com putation of the deposition constants are by th em selves com pletely sufficient to validate the con stru cts here developed. A very sim ple experim ent could decide this issu e sa tisfa cto rily .
This experim ent w ill u se 5 groups of an im als, each group
co n sistin g of 4 to 5 su b jects. A con trol group w ill be fed ad libitum ,
and the only m easu rem en ts m ade w ill be of the lead c o n s u m e d and
excreted . The other 4 groups w ill be given, as a dietary supplem ent,
sp ecified increm en ts of lead. D epositions con stan ts, d, w ill be com
puted as w as done here and v erified by co m p letely ashing the an im als
and m ea su rin g th eir total lea d co n ten t. In ea ch c a s e , the d iffe r e n c e s
betw een control and experim ental anim als should be an amount of n
le a d equal to dZL. an d the d 's th e m s e lv e s sh ou ld show an exp o n en tia l
a sso cia tio n to the varying in crem en ts.
If th is experim en t has the p red icted outcom e, it w ill be d e s ir able next to esta b lish d as a function of body s iz e , blood le v e l of lead, and of other and related variab les.
Kg' 001 3063
14 Since the m odel should apply to o ther inorganic su b stan ces as w ell, another, and p erh ap s m o re easily m easu rab le agent could be used.
KB 001 3 ') 0 j
15
1 O b se rv ed a v e ra g e daily in tak e (P b l) and output (PbQ) of le a d d u rin g the la s t 22 w e e k s of the c o n tro l p e rio d p rece ed in g e x p o su re to le ad in the c h a m b e r's a i r .
(In mg . of lead )
Subject
C reech exp. 1 C reech exp. 2 B arber exp. 1 B arber exp. 2 B la ck sto n e S v e tlik
E (P bl ) o
for each day
E (PbO o ) for each
day
D ifference E{PbIo )-E(Pbo o
. 329 . 145
. 24o . 198 . 188 . 171
. 294 . 141
. 229 . 241 . 203 .169
+ - 03S + .004 '
+ .011 - .043 - . 015 + . 002
M e a n D if f e r e n c e = . 001
0013070
16
Table 2
O bserved a v erage daily intake E (P b l), during the la st 22 w eeks of control and experim ental periods.
(In m g s . )
Subject
C reech exp. 1 C reech exp. 2 Barber exp. 1 Barber exp. 2 Blac kstone
S v e tlik
E(PbI ) o
for each day
E (Pbln)
for each day
D ifference E(PbI }-E(PbI
o:
. 329 . 145 . 240 . 198 . 188 . 171
. 2c6 . 138 . 182 . 125 . 186 . 220
+ . 113 + . 007 + . 058 + . 073 + . 002 - . 049
M ean D ifference =
+ .034
KE 0013071
17
Table 3
O b s e rv e d v a lu e s of d aily food in ta k e d u rin g the la s t 22 w eek s of the experim en tal period. T hese values a re taken as the b est e s tim a te of E (PbO ). O b s e rv e d v a lu e s of a v e r a g e d a ily to ta l e x c r e t i o n cS{lPb(iJ1)^ i(3 ld o f - i n h a l a t i o H o f l e a d y i Ev(L) d u r i n g th e s a m e period
Subject
E ( P b I n ) = E ( P b O Q) E ( P b O n )
E(L)
C reech exp. 1 C reech exp. 2 B arber exp. 1 B arber exp. 2
B lackstone
Svetlik
. 2058 . 1384 . 1819 .1253 . 1862 . 2201
. 3290 . 3205 . 2347 . 3006 . 3336 . 3932
. 1488 . 3392 . 0627 . 2402 . 3552 . 2706
K r oo 307
Table 4
C om p u tation fo r \ (PbO) and d, b a se d on the f i g u r e s given in Table 3.
A (PbO) = E (PbQn ) - E (PbOQ)
1 - d = & ( PbO) E(L)
Subject
E(PbOn ) - E(Pbo o )
E (L )
( 1- d) ( proportion of
L excreted each day.)
d ( proportion of
L deposited each day)
C reech exp. 1
.3290 - .2058 . 1488
. 8280
. 1720
C reech exp. 2
. 3205 - . 1384 . 3392
. 5368
. 4632
Barber exp. 1
. 2347 - . 1819 . 0627
. 8421
. 1579
Barber exp. 2
. 3006 - . 1253 . 2402
. 7298
. 2702
B lackstone .
. 3336 - . 1862 . 3552
. .4150
. 5850
S v e tlik
. 3932 - .2201 . 2706
. 6390
. 3410
19
Table 5 O b s e r v e d r e l a t i o n b e t w e e n t h e L. a n d d of e a c h s u b j e c t
Subject
B la ck sto n e
C reech exp. 2
Svetlik
B arber exp. 2
C reech exp. 1
B arber exp. 1
Rank 1 2 3 4 5 6
L . 3552 . 3392 . 2706 .24q 2 . 1488 . 0627
D . 5850 . 4632 . 3410 . 27o2 . 1720 . 1579
- 5 . 17 (L ) + 1 . 4 e d=e
He 0 0 1 3 0 7 4
20
A bbreviations and Sym bols:
Pbl. x
p bj
In gested lead on the i th day, in m g s . Total ex cr e ted lea d on the j th day, in m g s .
PbF. 3
PbU. J
P b Vj
L ead e x c r e te d in f e c e s in the j th day, in m g s. Lead ex creted in urine on the j th day, in mgs . Am ount of lea d in body flu id on the j th day, in m g s .
vu.
3 L
V olum e of u rin e on the j th day, in gm . C onstant in crem en t of lead taken in through the atm osph ere
L P art of L, reaching the body via the stom ach
0
P art of L reaching the body d irectly through the lungs.
AP b o i& fference betw een ex cretio n s o le ly due to P b l and excretion due to Pbl + L
k P rop ortion of P b l rem oved d irectly in fe ces 1
k2 proportion of lead in body fluid that is rem oved through fecea and urine
d Constant of deposition: P roportion of L that is deposited in tissu es
b1 - t 1 - k ,| b2 = ( l - k 2)
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