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OCR for page 185
E
Cancer Models
One ultimate purpose of laboratory investigations of potential carcinogens is
to predict the human health effects of mixtures to which the population is ex
posed environmentally. Mathematical models have been applied throughout
the history of toxicity testing to describe doseresponse relationships that might
allow extrapolating from the high doses used in the laboratory to the low doses
generally encountered in the environment. Understanding of the biologic
mechanisms of carcinogenesis has advanced, and newer models describe dose
response relationships in a manner that reflects newer knowledge of the bio
logic bases of disease.
When the carcinogenic effects of a mixture, instead of single chemicals, are
being considered, the concern is usually over whether interactions might occur
between components of the mixture and yield toxicity greater than what would
be expected, given knowledge of the toxicity of the individual materials.
Models of interaction observed at the high doses used in laboratory bioassays
of carcinogenicity might predict additivity at lower, environmental doses. The
purpose of this appendix is to detail and discuss that phenomenon. When mix
ture components are tested for carcinogenicity in a laboratory bioassay, inclu
sion of highdose multipleexposure groups might yield little useful informa
tion, if human environmental exposure to the mixture occurs only at low doses.
In such a case, the results of this appendix suggest that cancer predicted from
bioassays of the individual mixture components can yield a good estimate of
the cancer risk associated with human exposure to the mixture. The basis for
and qualification of that conclusion are elaborated in the next section.
185
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186
APPENDIX E
IMPLICATIONS OF THE MULTISTAGE MODEL
One approach to estimating the cancer risk associated with exposure to mul
tiple carcinogenic agents at low doses is to assume that the excess risks are
additive, that is, that the excess risk of the mixture is equal to the sum of the
excess risks of the components.
The multistage model, originally developed by BerenblumShubik (1949)
and Armitage and Doll (1954), provides a basic theoretical framework that can
be extended to account for exposure to multiple agents. The following section
describes how the multistage model can be generalized to take into account
exposure to multiple agents. The generalized model is used to analyze the
relationship between the extent of interaction observed in a bioassay and the
departure from additivity at environmental exposures. On the basis of this
examination, general principles concerning the most efficient approaches
for estimating the joint effects of multiple agents at environmental doses are
discussed.
DERIVATION OF THE MULTISTAGE MODEL THAT ACCOUNTS
FOR EXPOSURE TO MULTIPLE CARCINOGENIC AGENTS
The multistage model assumes that a cell becomes a cancer cell by progress
ing through an ordered sequence of stages. The number of stages is assumed
to be some small number k, and the cell is neoplastic once it completes the
kth stage.
Following the approach taken by Whittemore and Keller (1978), we define
Pift) = probability that a cell is in the ith stage at time t, the age of the
organism, and
kift) = transition rate from i to the (ith + 1) stage at time t.
The product of these two terms, [\ift)~[Pift)], is the instantaneous probabil
ity that a cell leaves the in stage for the (ith + 1) stage at time t. The rate of
transition out of the ith stage at time t is defined as dPitt)/dt. This instantaneous
rate of transition is equal to the instantaneous probability of transformation of a
cell from the (ith _ 1) to the ith stage minus the instantaneous probability of a
transformation from the 'ah to the dish + 1) stage. This set of conditions defines
the set of simple differential equations:
dPo(~)/~ = [ Xo(~)][Po(~)]
Pit)/ = [—Jo)] [Pi(~)] + [\i, (I)] [Pi ~ (I)]
~Pk(~)/~ = [yak ~ (I)] [Pk—~ (I)]
Pot)= ~
Pi(0) = 0, i = I, . . ., kl, (E1)
Pk(0) = 0,
which can be be solved to obtain Pift).
To account for exposure to m carcinogenic agents, we assume the transition
rate of a cell from the ith to the (i + 1) stage is
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APPENDIX E
m
Ni(t) = Hi + ~ [9ijxj(t)].
J=1
where Xj(t) = exposure to jut agent at time t,
of = background transition rate for i th stage, and
id = unit exposure transition rate forge agent on th stage.
~7
(E2)
Equation E2 assumes that each of the molecules of the different agents in
the mixture acts independently of the others with regard to the probability of
causing a cell transformation.
Suppose we also assume the following:
· The time required for a cell in its kth or final, malignant transformed state
to grow into a deathcausing tumor is approximately constant and equal to w.
· The probability that a given individual cell causes a tumor is very small.
· An organ contains N cells of a specified type, each capable of being the
origin of the tumor causing death.
· N is vely large.
· Each of the cells acts independently with regard to becoming transformed
and ultimately leading to a tumor.
Then the agespecific death rate due to a specified tumor type in a particular
organ can be expressed, to a close approximation, as
hate = N dot—W) = N[&k_lft—w)][Pk~(t—W)], (E3)
which is the instantaneous conditional probability that deaths from tumor will
occur at time t given that the animal is alive just before time t. The uncondi
tional probability of death due to that tumor by age t in the absence of compet
. · .
ng rest as IS
P(t) = 1—exp {   h(V)dv }
= 1—exp {   N[dPk(V—W)Idv]dv}.
(E~)
We assume that exposures are constant and continuous over a lifetime, as is
often the case in chronic carcinogenesis experiments. That means that Fit) =
xj, j = 1, 2, . . ., m and that the transition rates are constants. In this restricted
case, Equation E2 has the simplified form,
NiLt) =
m
Hi = [se +.Z dijXj]
J=1
(ES)
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188
APPENDIX E
Note that this assumes that the exposure level x; at time t determines \(t), as
opposed to some other value as a part of the exposure history, such as cumula
tive exposure. Substituting this definition for the ithstage transition rate in
Equation E1 and integrating both sides of the equation yields the iterative
solution,
t
Pl(t) = TV= hot
o
P2(t) = 1 \1 ~vdv = \1 ~ 2{
I~ vk—1
Pk(t) = ~ k—1 Ok—2 ·
o
(E6)
. A0 dv = II ~ t
(k—1)! j=o J k!
With these results, the agespecific death rate defined in Equation E3 and the
probability of death due to that tumor defined in Equation E4 are
expressible as
k—1 k—1
in(t) = N t II )
APPENDIX E ~ 89
)\s = tats ski, j
Pi = Hi + aim, (E9)
A = ~j + dj2X2
Substituting these values into Equation E8 yields the expression
Path = ~—exp ~—Nick ~ fIo`~`xi + ~Bj~x~(cej + ~j2x2~/cYjajk! l,fE10)
which may be written in the more concise form
P(t) = 1—exp ~—A(1 + B~x~1 + B2X2~}, (E11)
where
k—~
A = Ntk II oak!, and
s=u
Be = hi~/ai and B2 = dj2/~j are the relative transition rates for the first and
second agents, respectively.
If the risk of death from nontumor causes is negligible to time t, Equation
E1 1 represents the probability of a tumor death by time t. Using this model,
the next section describes the dose dependence of the synergistic effect.
ESTIMATION OF LARGEST SYNERGISTIC EFFECT DETECTABLE
IN 2 x 2 BALANCEDDESIGN EXPERIMENT
The experimental design most often used for estimating the joint effects of
multiple agents is the balanced 2 x 2 form consisting of four experimental
groups: control, a single exposure group for each of two agents, and a joint
exposure group at the same doses of the single agents. With that type of design,
it is shown how large an interaction effect can be measured in a practical bio
assay under the assumption that the doseresponse relationship of two agents
can be represented by the specific form of the generalized multistage model
depicted in Equation E 1 1. That degree of interaction will represent a practical
upper bound on the interaction of two agents at experimental doses, because
greater interaction cannot be measured within the constraints of the chosen
experimental design. On the basis of this degree of interaction, we assess the
rate of error that is introduced by assuming that the joint risk is additive at
environmental exposures. The approach establishes an error rate for additivity
that is as great as can be directly measured from a 2 x 2 bioassay design
defined by the example under the assumption that the multistage model in its
generalized form is a valid representation of the joint carcinogenic effect of two
agents. In the example that follows, it is assumed that there are 50 animals per
exposure group.
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190
APPENDIX E
To obtain an extreme interaction effect, we impose several additional condi
tions on our hypothetical bioassay and use them to estimate values of the
parameters in Equation E1 1:
· The background rate is so low that we expect to see no tumors in the
control group in most experiments.
· The singleexposure doses of each agent are chosen to yield identical
responses in the bioassay. This condition may be expressed within the context
of the model depicted in Equation E11 as BE = B2x2, which, for ease of
notation, we set equal to the constant V.
· The doses ofthe single agents are chosen to be as low as possible such that
an observed result that would be equal to the expected value of the response
would indicate a statistically significant increase in tumor rates over the con
trols. That is done so that a response can be measured for each agent, but the
greatest possible leeway is left for measuring the joint effect of two agents.
Assuming zero tumors in the control group, because the background rate is
low, five or more tumors are needed in a test group for the response to be
considered statistically significant at the 0.05 level, according to a Fisher exact
2 x 2 test. To meet that condition, we select our exposure dose so that the
probability that the tumor increase is statistically significant is 0.5. That im
plies that the probability of obtaining four or fewer tumors in a singleagent test
group is also 0.50. The probability that a single animal will develop a tumor at
the singleagent dose under the conditions of the experiment is obtained from
Equation E11 by setting BE = 0 and B2x2 = V, or vice versa. This yields a
response expressed in the form
P=1—expel—A(1+V)~.
(E12)
The probability of four or fewer tumors is obtained from a binomial distribu
tion with n = 50 and P defined as in Equation E12. The probability is set equal
to 0.5 so that the value of A(1 + V) can be estimated. The relationship may be
expressed as
4
~ (50) [1—eA(1 + V)s] eA(l + V)(50  s) = 0.5
s=0
which is solved numerically to obtain the solution,
A(1 + V) = 0.097282.
, (E13)
(E 14)
To estimate values of the individual terms A and V, we must impose an addi
tional condition on the bioassay:
· Interaction cannot be measured quantitatively if the joint response is
100%. To obtain the maximal measurable interactive effect, we assume that
the joint interactive effect will yield a 100% response 50% of the time, so a
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APPENDIX E
191
large interaction effect would be measurable 50% of the time. The probability
that all 50 animals will respond at the joint exposure dose is obtained from the
binomial distribution with P = 1—exp {—A(1 + V)2} and n = 50, where P is
obtained from Equation El l when BE = B2x2 = V. The relationship may be
expressed as
[1—eA(1 + V)23so = 0 S
which has the solution
(E15)
A(1 + v)2 =—in [1—(o.5)l/s°] = 4.28546. (E16)
The numerical values for A and V can be obtained by solving Equations E14
and E16 simultaneously. Dividing Equation E16 by Equation E14 gives the
result
A(1 + V)2
A(1 + V)
= 1 + V= 0 og87524862 = 44.05187.
V= 44.05187  1 = 43.05187,
and
A = 0.097282/44.05187 = 0.0022084.
To obtain estimates of values of the responsemodel parameters B1 and B2,
the forms with which Equation El l is expressed, one additional condition is
needed: exposure doses x1 and x2, which would give the singleagent re
sponses, are required. They represent a scaling factor to redefine the equation
for specific exposure potencies. To obtain a specific equation, we arbitrarily
assume that the lowdose response to each of the single agents resulted from
exposures at x1 = 0.1 and x2 = 0.2; agent 1 is twice as potent as agent 2.
Because BE = B2x2 = V, it follows that
B1 = 43.05187/0.1 = 430.5187,
and
B2 = 43.05187/0.2 = 215.2593.
Substituting the numerical estimates of A, B1, and B2 into Equation E11
yields the following doseresponse model:
P(x1,X2)
or
P(x1, X2)
=
= 1—exp {  0.002208~1 + 430.5187xl)~1 + 215.2593x21}, (E17)
.
1—exp{  [0.002208 + 0.9507x1 + 0.4754x2 + 204.6552xlx2]}.(E18)
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192
APPENDIX E
Given this model, it is possible to estimate the risk at any exposures x~ end x2.
The model derived here predicts an interaction effect that is one of the largest
that can be estimated from a 2 x 2 balanceddesign bioassay with 50 or fewer
animals per group. If designs as assumed here were used to estimate joint
effects, interactions of the magnitude assumed here would be among the largest
observed. *
ESTIMATION OF JOINT EFFECTS OF ENVIRONMENTAL
EXPOSURE WITH PREVIOUSLY DERIVED
NUMERICAL MODEL
The model depicted in Equation E 18 will be considered to be the true under
lying doseresponse relationship. That model has a large interactionsynergis
tic effect built into it; it is about the largest that could be measured with a classic
2 x 2 balanced design with 50 animals per exposure group and that would
allow the statistical estimation of the multistage model.
In many situations, the results of multipleagent exposure experiments are
not available, so estimates of cancer risk associated with environmental expo
sures must be based on the results of singleagent expenments.
In this section, we evaluate the error that the assumption of lowdose additiv
ity could introduce. For the purpose of the evaluation, the true underlying
doseresponse relationship for a single agent is assumed to be known and to be
the marginals from Equation E18. The term "marginals" refers to the re
sponse obtained from exposure to one agent when there is no exposure to the
second agent. Thus, the excess cancer risk associated with exposure to the first
or second agent is
P*(x~) = P(x~,O)—P(O,O)
or
(E 19)
P*(X2) = P(O,x2)  P(O,O),
respectively, with P(xl,x2) as defined in Equation E18. Assuming additivity,
the estimated joint excess risk associated with both exposures is
P*(x~,x2) = P*(X1) + P*(X21
(E20)
The error introduced by that assumption can thus be evaluated by comparing
P*(x~,x2) with the true excess risk of
P*(x~, X2) = P(X1, X2)—P(O,O).
(E2 1)
*To our knowledge, an interaction of the magnitude of this example has never been observed;
however, an infinite interaction is theoretically consistent with theory underlying the multistage model.
A bioassay design similar to that defined here would not, however, allow the estimation of a multistage
model with more extreme interaction than that given here.
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APPENDIX E
193
The true increased risk associated with exposure to one agent conditional on
exposure to a second agent may be written as:
P*(x1~x2) = P(x1,X2)—P(O,X2),
and
(E22)
P*(x2~xl) = P(xl~x2)—P(X1,°~.
We can evaluate the error introduced by the approximations for various val
ues of x~ end x2. The results ofthis analysis are shown in Table E1. From Table
E1, the following trends are noted:
· The synergistic effect is extreme at high doses (i.e., x~ = 0.1, x2 = 0.21.
Each agent by itself gives about a 10% response, but jointly they yield a 99%
response. In this case, the additivity assumption clearly does not hold.
· If one agent remains at a high dose (i.e., x~ = 0. 1) and the other is reduced
by 2 or more orders of magnitude, the additivity approximation is reasonably
good. The same is true for the conditional risk for the highexposure agent.
However, the conditional risk for the lowexposure agent is 40 times higher
than predicted by the additivity assumption at smaller exposures.
· If exposure to both agents is reduced by 2 orders of magnitude, the addi
tivity assumption is reasonably good. If exposure to both agents is reduced by 4
orders of magnitude, there is virtually no difference between the true risk and
that predicted by the additivity assumption.
PRACTICAL IMPLICATIONS DISCERNIBLE FROM
NUMERICAL INVESTIGATION OF GENERALIZED
MULTISTAGE MODEL
If the previous examples were representative of the types of doseresponse
relationships encountered in practice, the following conclusions can be drawn:
· Agents with high environmental exposure such as background radia
tion, cigarette smoke, and some workplace exposures must be investigated
very carefully, because, if they act on the same cell type in the same organ, the
potential for a strong synergistic effect is great.
· The excess cancer risk at low doses from an agent that acts on the same
cell type in the same organ as another agentts) to which exposure at high levels
occurs (e.g., cigarette smoke) could be seriously underestimated in an animal
bioassay, because the bioassay ignores the effects of the other agentts), such as
cigarette smoke, on the estimated augmented risk.
· When all environmental exposures are 34, or more, orders of magnitude
below that associated with observable effects in bioassays or in epidemiology
studies, additivity assumptions can provide a reasonable approximation of the
. . . ~
joint risk.
OCR for page 185
194
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OCR for page 185
APPENDIX E
195
· Because of the previous result, it might be counterproductive to test the
agents in a laboratory experiment, because interaction observed at high expo
sures could vanish at low exposures. In such cases, little additional information
about lowdose behavior under the assumed model would be gained from joint
exposure experiments, compared with that from singleagent experiments.
The next section presents a more general mathematical treatment of the con
ditions under which the lowdose additivity assumption might be expected to
be accurate.
EFFECT OF BACKGROUND TUMOR RATE ON
ADDITIVITY ASSUMPTION
Background tumor rate has an effect on the accuracy of the additivity as
sumption. For the multistage model, the larger the background rate, the faster
the doseresponse relationships approach linearity (Hoer, 19801. As a result,
the additivity assumption is more accurate for higher background rates at levels
of exposure that yield equivalent responses in the laboratory. The example
discussed in the previous section had a background rate of only 0.2%, which
would tend to give a less accurate approximation of risk than examples with
higher background rates. To demonstrate the effect of background rate on the
additivity assumption, consider the following example. Assume that agent 1 at
exposure x and agent 2 at exposure x2 produce equal effects on transition rates
between stages and that background transition rates are equal for each stage.
These conditions yield a model that changes rapidly with increasing dose, so
the deviation from additivity is at the maximum for all models ofthis class. The
model for which these conditions hold may be expressed as
P(x~,x2) = 1—exp ~—A(1 + Sx~1 + Sx2~} = 1—exp ~—A(1 + Sx)2},
where x~ = x2 = x, where the background tumor rate is 1—eA.
For all cases, risk at x = 1 is set equal to 0.5. The extent of deviation of the
true cancer risk from that predicted with the assumption of additivity is shown
in Table E2 for doses of x = 0.01 or 0.001 and various assumed background
rates.
Note that the accuracy of the additivity approximation decreases as the back
ground rate decreases. However, for an exposure that is 3 orders of magnitude
lower then that which yields a response of 0.5. the accuracy is very good for the
full span of background rates.
~ ~ , ~ ~ ~
ADDITIVITY OF EXCESS RISKS AT LOW DOSES
Interactive effects at moderate to high doses have been demonstrated in toxi
cologic experiments and observed in epidemiologic studies, but existence of
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196
APPENDIX E
TABLE E2 Effect of Background Rate on Accuracy of Additivity
Approximation Under the Assumotion of the Multistage Model Defined in
Equation E1 1
a
Deviation of Additivity
True Background Exposure Level Approximation from True Risk
Rate for Both Agents after Subtracting Background, %
0.1 0.01 0.73
0.001 0.10
0.01 0.01 3.39
0.001 0.34
0.001 0.01  1 1.23
0.001  1.18
such effects at low doses is not amenable to direct investigation, because of the
low response rates generally associated with low doses. This section identifies
conditions under which the additivity assumption can be expected to hold at
sufficiently low doses.
MULTISTAGE MODEL
Consider first the multistage model in the case of joint exposure to two
chemicals Cat and C2 given in Equation E8. The probability that a tumor will
occur by time t after exposure tom and C2at doses off end x2, respectively, is
given by:
k k—~ ~
P(x~,x2,t) = 1—exp —Nk! ,.IIo Nii
~ Nflk!t ~ Ail,
for small values of xl and x2, where
(E23)
2
Ai = Hi + [3ijXj
J=1
(E24)
denotes the transition intensity function for the ith stage (i = 0, 1, . . . , k—14. At
any fixed time t, it follows from Equation E23 that the excess risk at low doses
may be written as
H(xl,x2) = P(X1,X2)—P(O,O)
~ clx1 + C2X2,
(E25)
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APPENDIX E
where
/97
cj = (Ntk~lk! ~ (H Pi )3ij > 0
' ° Hi
(E26)
Because the excess risk for chemical Cj is approximately CjXj for sufficiently
small xi, it follows from Equation E25 that the excess risks for C ~ and C2 are
additive at these doses with the multistage model in Equation E23. Note that
the same result also demonstrates that the excess risk associated with each
chemical will be nearly linear in dose at these doses.
ADDITIVEBACKGROUND MODELS
For exposure to single chemicals, Crump et al. (1976) have established other
conditions under which lowdose linearity will obtain. The same argument
may be invoked in the case of joint exposure to two substances C, and C2, to
demonstrate lowdose linearity and hence additivity of the excess risks associ
ated with C, and C2 separately. Specifically, suppose that the joint response
rate can be expressed as
P(x,, x2) = H(x, + at, x2 + 62),
(E27)
where H is an increasing function of both x~ and x2 with both first partial
derivatives Offhand D2H strictly positive. Here, b~ > 0 and 62 > 0 denote the
effective background doses of x~ end x2, respectively, leading to a spontaneous
response rate P(O,O) = H(~,621. A Taylor expansion of P(x~,x2) shows that,
for sufficiently small x~ and x2,
Taxi, X2) ~ c~x~ + C2 X2,
(E28)
where Ci = DiH (~ ,62) > 0 The model in Equation E27 has been termed an
additivebackground model, because the doses x~ and x2 of Cat and C2 are con
sidered to act in an additive manner in conjunction with the effective back
ground doses b~ and 62
MIXEDBACKGROUND MODELS
For single chemicals, the result of Crump et al. (1976) was further general
ized by Hoel ( 1980) to include combinations of both additive and independent
background. In the present context, combinations of independent and additive
background may be represented by the model
P(x~,x2) = ~ + (1—y)H(x~ + 6,,X2 + 62),
(E29)
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198
APPENDIX E
where 0 < By ~ 1 denotes the probability of a spontaneous tumor's occurring
independently of those induced by Cat or C2 or the effective background doses
b~ and 62. The same argument as used for the additivebackground model in
Equation E27 may be used to show, for sufficiently small x~ and x2, that
Il~x~,x2) ~ C/X + c~x2
under the mixedbackground model in Equation E29, with cj*
j= 1,2.
(E30)
= (1
MULTIPLICATIVERISK MODELS
In some cases, the interaction between two substances might be well de
scribed by a multiplicative model in which the relative risk associated with the
mixture is the product of the relative risks associated with the components. Let
pa = P(x~, 0) and pet = P(0, X2) denote the respective probabilities of tumor
occurrence when chemicals Cat and C2 are administered separately, and let
p00 = P(0,0) denote the spontaneousresponse rate. The corresponding relative
risks associated with chemicals C ~ and C2 are thus given by rut = p ~ 0/p00 and rot
= p0~/p00, respectively. Letting pi ~ = P(x~ ,x2), the relative risk rat ~ = pi ~/p00
when both C, and C2 are administered simultaneously is given by
rat = rare
(E31)
with the multiplicativerisk model (Siemiatycki and Thomas, 19811.
With this model, the excess risks associated with C, and C2 will be approxi
mately additive when both r,0 and rO, are sufficiently close to 1. To see this,
note that, from Equation E3 1,
r,,  1 ~ (rut  1) + (rot  1)
because log r ~ r—1 for r near unity. Thus,
(E32)
PI ~ Poo ~ ~ Pro—Poo) + ~ Pot—poo), (E33)
when the relative risks rut and rO, are close to 1. This implies that, for two
substances whose interaction follows a multiplicativerisk model, as with can
cer of the oral cavity induced by smoking and alcohol consumption (Rothman
and Keller, 1972; Tuyns et al., 1977), the excess risks will be effectively addi
tive at doses where the relative risks are sufficiently small (e.g., less than
1.051.
PREDICTION OF RISK AT LOW DOSES
To predict the potential carcinogenic effects of joint exposure to two or more
substances at low doses, it is necessary to extrapolate downward from higher
doses that induce measurable rates of response (Brown, 1984; Clayson and
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APPENDIX E
199
Krewski, 19861. When data on the effects of joint exposures are available, that
may be done by modeling tumoroccurrence rates as a function of x~ and x2 of
chemicals Cat and C2. The response surface described by the fitted model may
then be used to extrapolate to lower doses.
The preceding section identified models under which the excess risks associ
ated with simultaneous exposures to C ~ and C2 would be additive at sufficiently
low doses. When those conditions are satisfied, it is sufficient to predict the
excess risks associated with Cat and C2 separately, because the excess risk
associated with joint exposure to the two substances is well approximated by
the sum of the excess risks associated with exposures to C, and C2 alone. That
may be done with separate doseresponse curves for C, and C2, so that the
prediction of lowdose risks does not require mapping the response surface
associated with joint exposure to the two substances in this case. When joint
exposure data are available, they may be used not only to identify interactions
that can exist at moderate to high doses, but also to assess the extent to which
such interactions can be reduced at lower doses. To make full use of joint
exposure data in predicting risks at low doses, we require a model for the
probability of tumor induction P(x~, x2) by time t after exposure to chemicals C ~
and C2 at constant closes x~ end x2. For example, the multistage model in Equa
tion E 1 1 is of the form
P(x~,x2) = 1—exp ~—A(1 + B~x~1 + B2x2~}
= 1—exp ~—(GO + Next + e2x2 + e3x~x21), (E34)
where the His are subject to the nonlinear constraint e0e3 = Aim, as well as
the linear constraints Eli ' O (i = 0,1,2,3~.
To avoid nonlinear constraints in fitting models of the form of Equation
E34, the linear constraints Eli ' O may be invoked, as has been done in fitting
the multistage model to data involving exposure to only a single substance
(Krewski and Van Ryzin, 1981; Armitage, 19821. The linear constraints repre
sent a broader class of models than the nonlinear constraints arising in the
derivation of the model, but greatly simplify the statistical problems involved
in fitting the models to experimental data. (Note that, although the linear con
straints admit the possibility that Ed or e2 is zero, upper confidence limits on
values of these parameters will necessarily be greater than zero, so lowdose
linearity is preserved.)
More generally, we consider the model
P(x~, x2) = 1 —exp ~—~ eO + ~ (ei~xt~ + e2 x2)
k k ~
+ ~ ~ e3ixi~xi2i, (E35)
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200
APPENDIX E
where the values of parameters are constrained to be nonnegative. To fit this
model to data on a response surface P(x~, x2) that is highly curvilinear, it might
be necessary to use a value of k in excess of 1 or 2. The model involves (k + 1~2
unknown parameters, so experiments with at least this many exposure groups
are necessary. With k = 3, for example, at least 16 treatment groups would be
required.
The process of describing the response surface would be simpler, if a suffi
ciently flexible model involving fewer parameters were used. For example, the
fourparameter logistic model
P(x~,x2) = [1 + exp ((GO + Text + e2x2 + e3x~x2~i (E36)
might be used to describe the response surface P(x~, x2) within the obse~vable
response range. In analogy with the simple linear extrapolation procedure pro
posed by Van Ryzin (1980), excess risks at low doses could then be predicted
by extrapolating along straight lines joining the response surface at points
P(xt,x1) and the origin P(O,O), where (xt, x~ is chosen so that the excess risk
7r~xt, x1) = 7r*, where 7r* is chosen to be as small as possible, yet still within
the experimentally measurable response range (e.g., 0.01 c or* c 0.101. The
points P(xt,x!) may be estimated by fitting a model, such as that in Equation
E36, to experimental data involving joint exposures to C, and C2.
SUMMARY
Empirical tests of joint action in laboratory settings, which typically are
based on high doses, provide little insight into the corresponding effects at low
environmental doses. The information available on the mechanisms of tumor
induction by chemicals strongly suggests that their relationship with dose is not
linear. Quantitative models of the carcinogenic process, such as the multistage
model of Armitage and Doll, also reflect such nonlinearity in the response to
mixtures. Although a multiplicative exposure effect sometimes dominates at
high doses, further exploration of this model indicates that the joint effect will
be additive (that is, close to the sum of the individual effects) at sufficiently low
doses. A newer model, that of Moolgavkar and Knudson (1981), is more bio
logically specific and yields essentially the same conclusion. (The conclusion
depends on the assumption that the augmented risks of the chemical are small,
with respect to the natural background rate of the tumor.) Additivity at low
doses was also demonstrated under a general class of additive background
models and under the multiplicative risk model when the relative risk for each
component in the mixture is small.
In addition, it can be shown for a broad range of mathematical doseresponse
models that the joint risk associated with a complex mixture can be determined
on the basis of the background risk and the risks associated with the individual
components. The models for which that is true include many widely used dose
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APPENDIX E
201
response models, such as the linear, multiplicative, logistic, and multistage
models. The results suggest the utility and desirability of identifying appropn
ate models.
Prediction of lowdose risks generally requires the extrapolation of results
obtained at higher doses that induce measurable tumorresponse rates. When
the excess risks associated with exposures to mixture components can be rea
sonably considered to be additive, that can be done by downward extrapolation
of the doseresponse curves for the individual components. Otherwise, it will
be necessary to model the tumorresponse surface associated with the joint
exposure to two or more carcinogens. In general, the use of jointexposure data
is to be preferred, in that it provides some empirical evidence on the magnitude
of any interactions that might be present even at lower doses.
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