The Linearized Multistage Model and the Future
of Quantitative Risk Assessment
Kenny S. Crump
ICF-Kaiser, 602 East Georgia, Ruston, LA 71270, USA
The linearized multistage (LMS) model has for over 15 years been the default dose-response
model used by the U. S. Environmental Protection Agency (USEPA) and other federal and state
regulatory agencies in the United States for calculating quantitative estimates of low-dose
carcinogenic risks from animal data. The LMS model is in essence a flexible statistical model that
can describe both linear and non-linear dose-response patterns, and that produces an upper
confidence bound on the linear low-dose slope of the dose-response curve. Unlike its namesake,
the Armitage-Doll multistage model, the parameters of the LMS do not correspond to actual
physiological phenomena. Thus the LMS is "biological" only to the extent that the true biological
dose response is linear at low dose and that low-dose slope is reflected in the experimental data.
If the true dose response is non-linear the LMS upper bound may overestimate the true risk by
many orders of magnitude. However, competing low-dose extrapolation models, including those
derived from "biologically-based models" that are capable of incorporating additional biological
information, have not shown evidence to date of being able to produce quantitative estimates of
low-dose risks that are any more accurate than those obtained from the LMS model. Further,
even if these attempts were successful, the extent to which more accurate estimates of low-dose
risks in a test animal species would translate into improved estimates of human risk is
questionable. Thus, it does not appear possible at present to develop a quantitative approach that
would be generally applicable and that would offer significant improvements upon the crude
bounding estimates of the type provided by the LMS model.
Draft USEPA guidelines for cancer risk assessment incorporate an approach similar to the
LMS for carcinogens having a linear mode of action. However, under these guidelines
quantitative estimates of low-dose risks would not be developed for carcinogens having a nonlinear mode of action; instead dose-response modelling would be used in the experimental range
to calculate an LED10* (a statistical lower bound on the dose corresponding to a 10% increase in
risk), and safety factors would be applied to the LED10* to determine acceptable exposure levels
for humans. This approach is very similar to the one presently used by USEPA for noncarcinogens. Rather than using one approach for carcinogens believed to have a linear mode of
action and a different approach for all other health effects, it is suggested herein that it would be
more appropriate to use an approach conceptually similar to the "LED10*-safety factor" approach
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for all health effects, and not to routinely develop quantitative risk estimates from animal data.
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Introduction
This year (1996) marks the twentieth anniversary of the first published paper on
the linearized multistage (LMS) model for cancer risk assessment.1 For about 15 years
the LMS model has been the default model of the U. S. Environmental Protection
Agency (USEPA) and various other federal and state regulatory agencies in the United
States for calculating quantitative estimates of low-dose risks from exposures to
carcinogenic agents.2,3 However, the USEPA recently issued draft guidelines for public
comment that, if adopted, would mean that the USEPA would no longer rely upon the
LMS for low-dose extrapolation.4 Consequently, this appears to be a particularly
appropriate time for the present paper, which reviews the history of the LMS model and
comments on its future and the future of quantitative risk assessment. Criticisms of the
LMS model by Lovell and Harris5 that appeared in a recent issue of this Journal are
also addressed. Finally, since the LMS model still seems to be poorly understood, the
model is also described and its key features characterized.
Historical perspective
Before 1970, most toxic responses were considered to have an exposure
threshold below which adverse health effects would not occur. Safe levels of exposure
were derived by application of safety factors to an experimentally determined no-effect
level. This procedure is still applied today to non-carcinogens. However, during the
60's and 70's several substances (e.g., asbestos and vinyl chloride) were found to
cause cancer in humans at doses below those that were previously considered to be
safe. Because of that and because most cancers were believed to be mutagenic in
origin, a threshold mechanism for carcinogenesis began to be questioned. A linear nothreshold dose response was considered plausible for a chemical that acted through a
mutagenic mechanism to produce a change in the DNA of a single cell that eventually
led to cancer, since it seemed reasonable that the probability of a mutagenic
transformation would be proportional to the amount of chemical that reached the target
site. Experimental support for this point of view came from apparent linearity of many
dose-responses for mutagenicity, and the linearity of responses seen in some
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epidemiological studies, such as the data on leukemia incidence in Japan among
atomic bomb survivors.6 The 1977 National Academy of Science Safe Drinking Water
Committee concluded, ".. that, if there is evidence that a particular carcinogen acts by
directly causing a mutation in the DNA, it is likely that the dose-response curve for
carcinogenesis will not show a threshold and will be linear with dose at low doses."7 A
more general argument for low-dose linearity was presented by Crump et al.8 and
Peto,9 which stated that the dose response should be linear at low doses whenever the
toxic mechanism entailed an augmentation of a process that was already producing
toxic effects (e.g., tumors) in the absence of exposure.
This was the prevailing view that led to the development and use of the LMS
model.1,10,11,12,13 The LMS model was developed specifically to fill the need for a
statistical procedure for bounding the low-dose slope of a dose-response curve, in
accord with the presumed linearity of the dose-response. The desire for a bounding
procedure resulted from the desire by a regulatory agency to be health protective. a
a
Personal note: My first exposure to risk assessment occurred during a one-year
appointment in 1973-74 as a Visiting Scientist at the National Institute of Environmental
Health Sciences (NIEHS). As I was about to return to my university position, my
replacement, Harry Guess, arrived two days prior to my departure. Harry took
advantage of this brief overlap to quiz me at length about interesting problems I had
uncovered during the previous year. This serendipitous meeting began a collaboration
that lasted several years and spawned what eventually became known as the LMS
model.
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The USEPA's carcinogen assessment group (CAG) began developing
quantitative risk assessments for carcinogens in 1977.2et al. The earliest assessments
by CAG were based upon the one-hit model, which produced a linear dose response.
However, the one-hit model was criticized because it often did not fit data adequately
and because the one-hit assumption did not appear plausible in many instances. In
response to these criticisms, USEPA selected the LMS over other competing doseresponse methods14,15,16,17 to replace the one-hit model for calculating water criteria for
carcinogens18 and the LMS has been USEPA's default dose-response model for
carcinogens ever since.b,2,3
The selection of the LMS model by USEPA addressed both of the criticisms that
had been directed at USEPA's use of the one-hit model. The LMS was more flexible
than the one-hit model and therefore fit data at least as well, and often better, than the
one-hit model. Also, the LMS had its origin in the Armitage-Doll multistage theory of
cancer19 and therefore appeared, at least, to have a stronger and broader biological
basis. In addition, the low-dose risk estimates obtained from the LMS approach were
usually very close to those obtained from the one-hit model, so little of the conservatism
of the one-hit model was lost by switching to the LMS.
Description of the LMS Model
Estimation of low-dose human risk from animal data requires two major
extrapolations: from high dose to low dose and from animals to man. Although the term
"LMS" has sometimes been applied to both extrapolations, the LMS model performs
bPersonal
note: A small informal meeting was held at USEPA headquarters in
1979 for the purpose of selecting a new cancer dose-response model. Three outside
statisticians with competing models were present: Bernie Altshuler advocating a
Bayesian approach16 John Van Ryzin advocating the gamma multihit model 17 and
myself advocating the LMS model. My impression was that a turning point of the
meeting occurred when Roy Albert asked Van Ryzin point-blank what criticisms he had
of the LMS approach, and Van Ryzin answered to the effect that, whereas he preferred
the gamma multihit model, he had no strong criticism of the LMS. At any rate, soon
thereafter USEPA gave me a small contract to calculate water quality criteria using the
LMS model, which was EPA's first use of the LMS.
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only the high-to-low-dose extrapolation step, and can be used in conjunction with any
animal-to-human extrapolation method, including those based upon
pharmacokinetically-derived estimates of internal dose.20,21
The LMS was inspired by the Armitage-Doll multistage model of cancer19 and is
a generalization of that model. In the Armitage-Doll model a cell goes sequentially
through J stages before becoming cancerous. The times required for a cell to go
through different stages are independent and exponentially distributed, and different
cells progress to cancer independently. If the rate at which a cell goes through the i-th
stage is i + id, where d indicates dose, then in certain cases (see Crump22 and
J
1  exp [ C t
J
 Qi (d)]
i = 1
Moolgavkar23) the probability of cancer by age t is approximately
where C is a positive constant and Qi(d) = 1 + (i/i)d if i > 0, and d otherwise.
The exponent in (1) is a polynomial with non-negative coefficients whose degree,
K, is equal to the number of dose-related stages (which will be less than the total
number of stages, J, if some of the i equal zero). The polynomial contains a positive
linear term except in very special circumstances when the background response is
zero. Unless two or more of the i = 0, the model will contain a linear term. Since i
= 0 implies that a cell cannot pass through stage i unless exposed to the carcinogen,
this model will be linear in dose at low doses whenever there is a positive background
cancer risk.
The LMS model is obtained by dropping the dependence upon age and
generalizing the polynomial in the exponent of (1) to include all polynomials with nonnegative coefficients:
P(d) = 1 - exp[(q0 + q1d + q2d2 + ... + qKdK)],
(2)
where qi  0. This model has fewer linearity-implying constraints than (1) and may not
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include a linear term even when the background cancer rate is greater than zero. The
LMS model (2) is only loosely related to the Armitage-Doll multistage model (1), and
consequently the term "multistage" is something of a misnomer since the parameters of
the model, qi, are not directly related to the stage-specific rates, i + id, in the
Armitage-Doll model. Rather than being interpreted as a mechanistic or biological
model, the LMS model is more accurately thought of as a statistical dose-response
model that explicitly contains a linear dose-response term (q1) as one of its parameters,
which makes it convenient to compute an upper confidence bound on the low-dose
slope, and that can describe both linear and non-linear data sets.
The LMS model (2) is fit to data consisting of the dose, number of animals, and
number of animals with the tumor of interest in each of the experimental dose groups of
a bioassay. The fitting is accomplished, and the qi estimated, by maximizing the
likelihood of the data.13 In the earliest versions of the LMS model, the polynomial
degree was not restricted, and some of the first published papers on the LMS model
dealt with the problem of estimating an infinite number of non-negative polynomial
coefficients from a finite number of data points.1,11 Although this version of the model is
interesting mathematically, it has not been used for regulatory purposes as far as I
know. In the development of the theory used to construct statistical confidence limits,
the degree of the polynomial was restricted to a finite number. 10 Simulation results
verified the important result that confidence limits developed using the LMS model must
vary linearly with dose in the low-dose range.24,25 Difficulties with Wald-type confidence
limits led to their early replacement by likelihood-based limits,12,13,26 which have
consistently used by USEPA since adopting the LMS. Other extensions of the LMS
model permit modelling of time-to-tumor data27,28,29 and combining of data from studies
with different background rates.13
Use of the LMS in Risk Assessment
The LMS model is used to estimate the extra risk,
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[P(d) - P(0)]
= 1  exp [ ( q1 d + ... + q K d K )]
[1 - P(0)]
(which is the probability that an animal gets a cancer from dose, d, given that it would
not have gotten a cancer in the absence of the dose), or the VSD (virtually safe dose)
corresponding to an extra risk of , which is defined as the solution to the equation
[P(VSD) - P(0)]
= 
[1 - P(0)]
Regulatory levels are often based on values of  in the range 105 to 106. These levels
are not determined from the point estimate of the VSD derived from the best fit of the
model to the data, but rather from a statistical lower confidence bound, VSD* (typically
of size 95%), on the VSD. Lower confidence bounds on the VSD, as well as upper
confidence limits on extra risk, invariably vary linearly with risk or dose at low doses,
which is the property that gave rise to the name "Linearized Multistage Model". What
this means is that at low doses, the 95% lower confidence bound on VSD, VSD*, is
approximately linearly related to the extra risk, , by
VSD*  /q1*,
(5)
and the 95% upper bound on extra risk at dose, d, is approximately given by the linear
relation
*  q1*d,
(6)
where q1* is the 95% statistical upper confidence bound on the linear coefficient, q1.c
cLovell
and Thomas5 state a corresponding approximation for the theoretical
parameter values (as opposed to the confidence limits), namely P(d)  q1d. However,
this applies to the extra risk, [P(d) - P(0)]/[1 - P(0)], and not to P(d). It is also important
to remember that this approximation is valid only when q1 > 0. Lovell and Thomas
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The likelihood method is the best approach for computing confidence limits for
the LMS model.13,29,30 In this approach the confidence bound for a quantity (e.g., extra
risk or VSD) is calculated as the most extreme value of the quantity that is also
consistent with a certain reduction in the likelihood from its maximum value, with the
amount of reduction defined in terms of a chi-square distribution (see Crump and
Crockett13 for details). Although this method of calculating confidence limits does not
artificially force the linear term to be positive, confidence limits are invariably linear at
low doses. Because of this feature, the USEPA adopted the practice of summarizing
results from applications of the linearized multistate model only in terms of the q1*
values, or "potency factors" as they are also called. Although the linear approximations
(5) and (6) do not hold at high doses, higher order terms typically do not begin to
contribute importantly until extra risks are at least 1% or greater. Consequently, the
linear approximations (5) and (6) are adequate for most regulatory applications. d
Since 1  P(0) is in the denominator of the expression for extra risk, extra risk will
tend to be larger when the background rate is larger. This conflicts with the
toxicological decision to give qualitatively less weight to a tumor endpoint with a high
background rate. Whenever this is a concern, it may be more appropriate to use
failed to note this in their worked example on page 96, and, because q 1 was estimated
to be zero, erroneously claimed that the MLE of the VSD corresponding to a extra risk
of 106 was infinity. They also stated that the software package Tox_Risk "used the
other parameters in the low dose estimation" to achieve a finite VSD. In fact, Tox_Risk
provides the correct VSD for the model fit; Lovell and Thomas obtained the value of
infinity because they were applying the linear approximation in a case where it was not
appropriate.
d
Application of a linear approximation to the dose response at high doses where
it is not valid can lead to nonsensical results, such as the response probability
exceeding 1.0. When extra risk estimated using the linear approximation is greater
than 1%, it is safest to recalculate it directly from the original data, rather than relying
upon the linear approximation. At the very least, the approximation 1  exp(q1*d)
should be used for extra risk in place of q1*d whenever the two expressions differ
meaningfully. Similarly, Ln(1  )/q1* should be used to approximate the VSD*
whenever the value of this expression differs meangingfully from /q1*.
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additional risk, P(d)  P(0), in the LMS procedure instead of extra risk. The linear
approximation to the statistical upper bound on the additional risk is exp(q0*)q1*d,
where q0* is the value of q0 associated with the likelihood-based statistical bound
calculation. Thus, to use additional risk instead of extra risk, it is only necessary to
replace q1* by exp(q0*)q1* in expressions (5) and (6). Additional risk is an option in
standard LMS statistical software packages.29,30
Whenever K is greater than the number of dose groups minus one, without the
non-negativity constraints, qi  0, the LMS model would be over-parameterized and
would fit a data set exactly. However, because of the constraints, the model will not fit
data exactly except in special cases even when the number of estimated parameters
exceeds the number of dose groups. The algorithm for fitting the model to data and
computing confidence limits acually performs appropriately even when the number of qi
is larger than the the number of data points, although in some instances the point
(MLE) estimates of the qi will not be unique.1 The original mathematical development
of the LMS allowed the degree of the polynomial to be unbounded 1,11. Although that
feature is not implemented in LMS software programs such as GLOBAL86 30 or
TOXRISK,29 essentially the same data fits and confidence limits can be obtained in
most cases by selecting a relatively large value for K (e.g., K = 10). In most regulatory
applications, K has been set equal to the number of dose groups less one, which has
the unorthodox effect of making the form of the model dependent upon the number of
dose groups. It is probably a better approach, at least for computing confidence limits,
to simply use the same large value for K (e.g., K = 10) in all applications. This
approach is illustrated in the examples presented in the next section.
Examples of applications of the LMS model
Key features of the LMS procedure are illustrated in the examples in Table 1 and
the accompaning graphs in Figure 1. These examples are based on the experimental
design currently used by the National Toxicology Program: a control group and three
dose groups, 50 animals per group, middle dose equal to one-half of highest dose and
low dose equal to one-fourth of highest dose.
In Table 1, Data Set I (see also Figure 1A), the dose response is linear and q1 is
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the only qi with a positive estimate. The value of q1* (0.24) is 1.5 times the estimate of
q1 (0.16). As a rule of thumb, whenever the data derived from a typical experimental
design exhibit an approximately linear dose response, q1* will generally be roughly twice
the estimate of q1.
The same linear trend seen in Table 1, Data Set I is also seen in Table 1, Data
Set II (Figure 1B) through the three lowest doses, but the response at the highest dose
increases in a highly non-linear fashion. In this case with a high value of K (K = 10), the
estimate of q1 is virtually unchanged (in order to continue to fit the linear trend at the
lower doses), and a positive estimate of q10 is included in the model to allow it to fit the
very high response at the highest dose. The value of q1* is larger that the value it
attained in Data Set I, and in fact, attains the same value as when the high dose data
are deleted, as illustrated by Data Set III. This is because the very high response at the
highest dose does not provide any additional constraint on the value of the low-dose
slope. This is illustrated in Figure 1B by the fact that, if the low-dose-linear portion of
the bounding curve were extended into the high dose range, it would lie below the data
point at the highest dose. Note also that the bounding curve for Data Set II is not linear
at high doses, which illustrate the fact that the linear approximation is not valid at high
doses.
The value of q1* is lower in Data Set II with K = 3 (0.15) than with K = 10 (0.30).
The fit of the model to Data Set II with K = 3 is marginal and no linear term is estimated
despite the linearity of the data at the lower doses. Consequently, the smaller value of
q1* (0.15) obtained with K = 3 than with K = 10 appears to be an anomaly due to the
artificial constraint of K = 3.
Data Set IV illustrates the fact that for non-linear data, the estimate of q1 can be
zero. Nevertheless, it is possible that the true dose response from which the non-linear
data arose had a small linear component. The q1* value of 0.036 for Data Set IV is the
LMS 95% upper bound on high large this linear component could have been. In nonlinear data sets such as this, the upper bound estimate of extra risk can be orders of
magnitude higher than the point estimate.
Data Set V illustrates the fact that q1* will be positive even when there are no
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responder in any dose groups. Since negative data can arise even when the true dose
has a small positive component, any valid bounding procedure must produce a positive
risk from negative data. Of course, since negative data provide no evidence of any
carcinogenic risk, there may be good reason not to use the data set for performing low
dose extrapolation in the first place.
The small difference between the q1* in Data Set IV (0.036) and the q1* in Data
Set V (0.015) may be surprising. However, it must be kept in mind that since -- as was
also seen in Data Set II -- the high response in the high dose group in Data Set IV does
not constrain q1*, q1* attains the same value in Data Set IV as it would if the high dose
data were deleted. I.e., the upper bound on the low dose slope is the same in Data Set
IV whether or not the data at the highest dose are included.
In Data Set VI the number responders in each dose group is increased by 10
over the corresponding number in Data Set I, and the value of q 1* (0.40) for Data Set VI
is larger than the value (0.24) obtained from Data Set I. There are two reasons for this.
First, when the background response is higher (but still less than 50%) there is greater
statistical (binomial) variation in the response, which causes the confidence limits to be
wider. Second, the definition of extra risk has the term 1 - P(0) in the denominator, and
consequently will be larger when the background, P(0), is larger. As noted earlier,
additional risk, defined by P(d)  P(0), does not have the latter property, and therefore
may be more appropriate for some applications.
As these examples illustrate, the LMS model is flexible statistical model that is
capable of describing both linear and non-linear data. It explicitly contains a linear dose
coefficient (q1) as one of its parameters, which is useful for calculating the largest lowdose slope that is consistent with the data (e.g., the 95% upper bound on the low-dose
slope). Upper confidence bounds on low-dose risks calculated using the LMS model
always vary linearly with dose. If the response at the highest dose is significantly above
the linear trend of the lower dose data, the high dose data do not constrain the lowdose slope, and in such a case the high dose data will have little or no effect upon the
confidence bound for the low-dose slope. If there is no dose-response (i.e, no evidence
of a carcinogenic effect), the LMS procedure will still give a positive upper bound for
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the low-dose slope, as should any valid bounding procedure.
Responses to comments on the LMS model by Lovell and Thomas5
Specific responses to several of the points raised by Lovell and Thomas 5 are
made in the Appendix. In summary, I agree that the parameters of the LMS model do
not related directly to any underlying biological events. This is an important point for
consumers of LMS modelling to understand, since many persons appear to believe that
the LMS model is more close related to the Armitage-Doll multistage model than it
actually is. I also agree (since it was documented in some of the earliest papers on the
LMS24,25) that the point estimate of q1 (and the point estimate of low-dose risk, as well)
is often highly variable. However, these point estimates are generally not used to set
regulatory levels. Further, I consider this variability to be something of a virtue, since a
stable estimate of q1 would falsely imply that the low-dose risk could be estimated with
a high degree or precision. As detailed in the Appendix, I disagree with a number of
other points raised by Lovell and Harris. Some of their conclusions represent
generalizations made from poorly selected or non-representative examples. A number
of their examples involve widely spaced doses which do not characterize the dose
response very well and are not representative of current cancer bioassay designs. In
other examples, they force-fit the LMS to data having a strong negative dose-response
trend, which the LMS was not designed to fit. Some of the examples presented by
Lovell and Harris illustrate features of the LMS that should be considered advantages,
rather than disadvantages. For example, they point out as a "problem" the fact that q1*
becomes larger if the top dose data are excluded. This is exactly what must happen
since statistical confidence bounds tend to be wider when based on smaller amounts of
data. Another of their examples illustrates the "problem" that multiplying all of the
experimental doses by a common factor causes q1* to decrease by the same factor.
Actually, any contrary result would be evidence of a very serious problem with the LMS
model, since it would imply that risks estimates depend upon the units used to express
dose.
When evaluating the LMS model, two key issues ought to be considered: "Does
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the LMS do what it was designed to do?" and "Is what it was designed to do what is
needed in a given application?" As explained earlier, the LMS model is basically a
flexible statistical model that produces a statistical upper bound on the low-dose slope
of a dose-response curve. I believe that appropriate statistical methods are used with
the model and, as illustrated in Table 1, produce scientifically appropriate solutions to
this bounding problem. Thoughtful examination of the examples presented by Lovell
and Harris should convince one that this is the case in these examples, also. Whether
a statistical upper bound on the low-dose linear slope is an appropriate basis for lowdose risk assessment in specific instances is a much more difficult question, since there
may be cases in which a linear dose response does not seem plausible, or where the
low-dose slope may not be reflected in the observed data.
Recent developments in cancer risk assessment
Physiologically-based pharmacokinetic (PBPK) models: The development
and use of PBPK models has been a very useful advance in quantitative risk
assessement.31,32,33 PBPK models use physiological information such as organ
weights, blood flow rates, etc., in conjunction with chemical-specific information such as
blood-tissue partition coefficients, diffusion rates, metabolic rates, etc., to model
concentrations of a chemical and its metabolites at target sites within the body as a
function of external exposure scenarios (e.g., time-dependent air concentration
scenarios or oral intake scenarios). PBPK models are useful for both low-dose
extrapolation and cross-species extrapolation. However, in order to develop
quantitative estimates of risk, a PBPK model must be coupled with a dose-response
model that links the tissue dose predicted by the PBPK model to the biological
response (e.g., cancer). If this dose-response model is unreliable then the risk
estimates will be unreliable regardless of the predictive ability of the PBPK model. Also,
a misconception appear to exist to the effect that use of a PBPK model obviates the
need for a "species scaling factor." Since selecting a species scaling factor is eqivalent
to making an assumption regarding the dose measure that produces equivalent risks in
animals and humans, a scaling factor must always be selected when estimating human
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risk from animal data, even when a PBPK model is employed. However, it is true that
use of a PBPK model may permit use of a more appropriate scaling factor.
Mechanistic dose-response models: "Mechanistic" or "biologically-based"
dose-response models use biological information in addition to tumor response data to
relate measures of internal exposure to tumor response. Such models involve
parameters with biological meaning, such as cell proliferation rates, that could in
principle be measured directly, rather than being estimated in a model-fitting process. It
has been hoped that, by incorporating more biology, such models could provide more
accurate estimates of low-dose risks.
The most well-known mechanistic model is the Moolgavkar-Venzon-Knudson
(MVK) two-stage model34,35,36, which expresses the age-dependent probability of cancer
as a function of more basic physiological parameters such as cell mutation rates, cell
death rates, and cell division rates. If one or more of these physiological parameters
are assumed to be functions of the dose of a carcinogen, and mathematical doseresponses are assumed for these parameters, then the MVK model also expresses the
probability of cancer as a function of dose. However, in most applications of the MVK
model to-date,37,38,39,40,41 a functional form was assumed (without biological support) for
the dose response of the underlying physiological parameters. This approach has the
same limitation as approaches based upon non-biological statistical models: Different
assumptions about the dose response for the MVK parameters will fit the tumor
bioassay data equally well but produce enormous different estimates of low-dose
risks.42,43,44 Thus, estimates of low-dose risks obtained in this way from mechanistic
models are subject to the same model uncertainty that is present in estimates made
from statistical models.
The USEPA recently published for review and public comment an MVK model for
dioxin based upon liver cancer and focal lesion data in rats. 41 This model predicts
essentially the same risk as the LMS model, which had previously been applied to the
dioxin data by USEPA. The reason for this is that the mutation rate from normal to
intermediate cells, and also the proliferation rate of intermediate cells, were assumed to
vary linearly with dose at low doses. These low-dose linear responses were converted
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by the MVK model into low-dose linear responses for cancer. I.e., the MVK model gave
the same answer as the LMS model because the dose-response in the MVK model was
assumed to be linear. However, the underlying data are consistent with a linear, nonlinear45 and even a threshold dose-response. Thus, even when using a biologicallybased model, assumptions made about the shape of the dose response at low doses
will still drive low-dose risk estimates.
Biologically-based models have the theoretical advantage over statistical models
of being capable of incorporating data on the dose responses of more basic processes
that combine to produce the dose response for cancer. For example, if a chemical
causes cancer by increasing the rate of proliferation of initiated cells, then doseresponse data from cell-proliferation experiments could perhaps be incorporated into
the development of a mechanistic model. Provided the dose-response for cell
proliferation could be measured accurately at lower doses that the dose-response for
cancer, cell-proliferation data could be used to extend the observable dose-response
further into the low-dose region. Although this approach is conceptually attractive, there
are serious practical problems to be overcome. First, there is the problem of model
misspecification. There will be uncertainty regarding, e.g., the assumption that cell
proliferation is the (only) mechanism leading to cancer, and regarding the specific
assumptions needed to incorporate the cell proliferation data into the model (e.g.,
assumptions regarding the distribution of cell cycle rates, homogeneity of cells,
independence of proliferating clones, etc.). Second, there will also be concern about
the relevance of the additional data incorporated into the modelling. For example, there
will be uncertainty regarding whether the cells for which cell proliferation data are
available are the cells that are actually at risk of progressing to malignancy. There will
also be uncertainty regarding an assumption that the cells capable of progressing to
cancer are homogeneous. Even if these problems could be satisfactorily overcome in
certain instances, it is unlikely that the necessary data and mechanistic knowledge will
ever be available to allow this approach to be used routinely. And even when such data
are available, they may not extend the dose response far enough into the low-dose
range to be useful in regulatory decisions.
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Futhermore, It is questionable as to how much effort is even warranted in refining
risk estimates from animal data when the goal is to predict risk in humans. Suppose,
for example, that the mechanistic model for dioxin developed by the USEPA met with
broad scientific acceptance, to the extent that there was general agreement that the
model provided accurate estimates of low-dose risks. What this would provide would
be accurate estimates of the risk of one type of cancer (liver) in one sex (males) of one
strain (Sprague-Dawley) of one species of rodents (rats). It would not provide
estimates of risk of any other types of cancer caused by dioxin and would not
necessarily provide reliable estimates of liver cancer risk in other species or even
females of the same species. Since the dose-response curve shape could well be
qualitatively different in humans and rodents, it can in fact be argued that even knowing
with certainty the risk of liver tumors in male Sprague-Dawley rats would be of little help
in developing more accurate quantitative estimates of human risks.
Although our understanding of cancer mechanisms has progressed
tremendously in the last twenty years, the types of mechanistic information presently
available generally inform low-dose extrapolation, if at all, only in broad terms (e.g., by
suggesting a linear or threshold mechanism). Although such information can be useful,
it does not provide direct input to quantitative models.
Proposed USEPA Guidelines for Cancer Risk Assessment: The USEPA has
recently released new draft guidelines for cancer risk assessment 4 that depart markedly
from the 1986 guidelines.3 These new guidelines call for use of a biologically-based
model for low-dose extrapolation whenever such a model is available. e In the absense
eThe
definition of a biologically-based model in the new USEPA guidelines is so
restrictive that I seriously doubt whether such a model will be available for any chemical
in the forseeable future. The new guidelines define a biologically-based model as one
whose parameters are calculated independently of curve-fitting of tumor data.
Presumably, the tumor data would be used only for model validation. No such model
has ever been constructed, much less validated, as far as I know. It is not clear to me
why such a restrictive definition is needed, since using the tumor data to estimate
"nuisance" parameters (i.e., parameters that are not dose-related) would not lessen the
ability of a model to reliably reproduce the dose-response.
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of a biologically-based model, a statistical model is fit to the tumor data and used to
calculate a LED10* (95% statistical lower bound on the LED10, the dose associated with
a 10% increase in response. The LED10 is equivalent to the benchmark dose that has
begun to replace the NOAEL (no-observed-adverse-effect-level) used by the USEPA in
non-cancer risk assessment.46,47,48,49 If an agent's mode of action is consistent with a
linear dose-response, a no-threshold straight line from the origin through the point
defined by the LED10* is used to extrapolate to risks below 10%. If the mode of action
is consistent with a non-linear dose response, quantitative estimates of risk are not
employed; instead, safety factors are applied to the LED 10* to arrive at human exposure
limits. If the mode of action is uncertain, a linear mode of action is the default position.
Thus, under these guidelines the LMS model will no longer be the default lowdose extrapolation model. However, the LMS could be used to calculate the LED 10*,
and if it were, the resulting low-dose risk estimates obtained from the straight line
extrapolation specified for use when the mode of action is linear would generally be
indistinguishable from straightforward application of the LMS. (One of the reasons the
USEPA cited for use of the straight line extrapolation instead of the LMS was that the
LMS "gave an appearance of specific knowledge and sophistication that is unwarrented
for a default.") Moreover, many reasonable models and statistical bounding methods,
will produce about the same LED10* as the LMS method.f
Discusion and Recommendations
Although the LMS model is a generalization of the Armitage-Doll multistage
model of cancer, the LMS is not a "biologically-based model," since the parameters of
the LMS do not represent specific biological phenomena. Use of the term "multistage"
in connection with the LMS has probably fostered the mistaken impression that the
fHowever,
some unconstrained models that allow the low-dose slope to achieve
the biologically unreasonable value of infinity can, in some instances, produce LED 10*'s
that are lower by orders of magnitute from those produced by models, such as the
LMS, that are constrained so that they cannot achieve infinite slopes. 49 Such models,
which include the unconstrained logistic and Weibull models, should not be used to
calculate LED10*'s.
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LMS model incorporates more specific biological assumptions than it actually does.
The LMS model is best described as a flexible statistical model that permits calculation
of a statistical upper bound on the low-dose slope of the dose-response curve.
Whenever the true dose response is linear at low doses and the low-dose slope
extends into the experimental range, risks predicted by the LMS will generally be within
as factor of two of the true risks (to the experimental animals). However, if the true
dose response is non-linear, then the LMS approach may overestimate the true risk by
orders of magnitude. This problem cannot be overcome by selecting a more flexible
model that incorporates non-linear dose responses. Because true dose responses that
are linear at low doses can give rise to data that appear to be non-linear, use of a more
flexible model could cause the low-dose risk to be underestimated by many orders of
magnitude in some cases. Until the underlying biology is sufficiently well understood
that the low-dose curve shape can be predicted accurately, biologically-based models
will also have this limitation. Thus, it does not appear possible at present to develop a
quantitative approach that would be generally applicable and that would offer significant
improvements upon the crude bounding estimates of the type provided by the LMS
model.
I believe that quantitative risk estimates of cancer risks have been
particularly useful when derived from human data and when they are used to estimate
risks on the order of 1/1000 or higher, as is often the case when used in occupational
settings. However, the benefit of providing quantitative estimates of exposures
corresponding to very small risks (e.g., 1/1,000,000) is questionable, particularly when
estimates are based upon non-human data. Although such estimates are perhaps best
interpreted as crude estimates of relative risk, this interpretation has not been always
been the one reported in the popular press or the one used by policy-makers. For
example, one of the provisions of the U.S. Clean Air Act Re-authorization uses a risk of
1/1,000,000 as a decision point, with no accompanying guidance on how that risk is to
be estimated. Since estimates of the human exposure corresponding to a risk of
1/1,000,000 is determined in very large measure by the assumptions employed by the
risk assessor, this requirement in the Clean Air Act results in policy being set by risk
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assessors in the guise of science.
If no acceptable biologically-based model is available, the proposed new USEPA
guidelines will provide quantitative estimates for carcinogens considered to have a
linear mode of action, but other carcinogens will be handled in a manner similar to the
way that non-carcinogens are now treated  without resorting to quantitative risk
estimates. Such a fundamentally different risk assessment approach for carcinogens
with different perceived modes of action may hamper debate regarding appropriate
relative levels of exposures to these two groups of carcinogens, just as I believe the
current approach of developing quantitative estimates for carcinogens but not for noncarcinogens has hampered comparisons of risks of carcinogens and non-carcinogens.
The current dual approach for carcinogens and non-carcinogens arose twenty
years ago out of the belief that carcinogens and non-carcinogens had fundamentally
different dose responses.g However, this view is less widely accepted today. One
reason for this is that it now appears that many carcinogenic responses occur
secondarily in response to more basic non-carcinogenic pathology. Harvey Clewell, Mel
Andersen and I recently proposed harmonization of risk assessment approaches for
cancer and non-cancer endpoints.50 Although USEPA's new cancer guidelines move in
this direction, we believe that they do not go far enough. Because of the inherent
uncertainties in estimates of low-dose human risks, we recommended a default
approach for all health effects that would not involve quantitative low-dose extrapolation
of animal data. Instead, it would involve determining a risk-specific exposure in animals
g
In addition to differences in approaches to dose-response, differences also
exist in the USEPA's treatment of carcinogens and non-carcinogens in other steps of
the risk assessment process that do not appear to be well-grounded in science.50 For
example, in animal-to-human extrapolation of non-cancer responses from oral dosing,
animal doses are converted to equivalent human doses on a body weight basis
(mg/kg/day) basis, and an additional uncertainty factor of 10 is typically applied. In the
proposed new USEPA guidelines for carcinogens, this same conversion is performed
using mg/kg/day to the 3/4 power and an additional uncertainty factor of 10 may be
applied for carcinogens having a non-linear mode of action, but not for carcinogens
having a linear mode of action. However, the argument used to support use of the 3/4
power21 is just as applicable to non-cancer as to cancer.
D:\81898707.DOC
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(i.e., NOAEL, ED10 or benchmark dose46,47,48,49), use of pharmacokinetic principles and
data to determine an equivalent human exposure (e.g., an exposure that would produce
the same tissue concentration of the active toxin at the target site in humans) and
application of an appropriate composite safety factor that reflects the severity of the
endpoint, mechanistic information impacting the dose-response, human variability, and
other uncertainties (e.g., inadequacy of database).
Our recommendation for a safety factor approach to risk assessment of both
carcinogens and non-carcinogens was not based on a judgment that thresholds exist
for most effects. Rather it was based on the conviction that low-dose risks remain
largely unknown for both carcinogens and non-carcinogens. Similarly, we view the
selection of a safety factor as involving, not a scientific judgment as to where a
threshold lies, but a combination of scientific judgment about low-dose risks and,
perhaps more importantly, policy considerations regarding how much safety is
appropriate.
I believe such an approach would have several advantages. It would more
clearly distinguish between what is known and what is being assumed in risk
assessment. Current reliance upon uncertain risk estimates developed from
questionable assumptions tends to blur this distinction. I believe our recommended
approach would make it easier to compare the potential risks posed by chemicals that
cause different health effects or have different modes of action, and consequently could
lead to better balance in the level of protection afforded by regulation and in the costs
of regulation. With this approach quantitative dose-response models such as the LMS
model would still be used for modelling the experimental range, but not for low-dose
extrapolation.
References
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APPENDIX
Responses to Specific Comments on the LMS Model by Lovell and Thomas5
..although the LMS model has some degree of biological underpinning, there is no
specific biological interpretation of any of the parameters estimated by the LMS model,
or direct correspondence with measures of the underlying biological events, such as
cell mutation, differentiation, or death rates.
I agree and think it is important for users of the LMS to understand this point.
The LMS model does not use information on the animal's lifespan or whether any
tumors identified have a context of observation, i.e., whether they are considered fatal
or incidental.
Although this is a true statement insofar as it is applied to the basic LMS model, several
extensions of the model have been developed that can utilize information on lifespan
and context of observation.27,28,29 The USEPA has utilized time-to-tumor versions of the
LMS model in some instances.
Although [time-to-tumor models and the biologically-based Moolgavkar-VenzonKnudson (MVK) models] are less conservative [than the LMS] ...
Although these models could provide smaller risk estimates than the LMS in a specific
case, they are not inherently less conservative than the LMS method. In fact, dose is
not even incorporated into the standard formulation of the MVK model.36 Thus, it is
always possible to paramaterize dose in such a way that the MVK model is either more
or less conservative than the LMS model.
[The point estimate of] q1 is unstable and can be zero.
Presumably, the reason an unstable q1 estimate is important is that it results in unstable
D:\81898707.DOC
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point estimates of risk. This feature of the LMS was well documented over 20 years
ago, as Guess et al.25 and Guess and Crump24 presented results of simulations that
showed that MLE estimates of low-dose risk obtained from the LMS model could differ
over many orders of magnitute whenever there was no strong linear component to the
true dose response. Whereas Lovell and Thomas apparently consider this feature to
be a limitation of the LMS model, I consider it to be a virture of sorts, since a stable
estimate of q1 would falsely imply that the low-dose risk could be estimated with a high
degree of precision. It should also be noted that this level of instability does not occur
with the bounding estimate, q1*.
The change .. from [a zero to a positive estimate of q 1] is abrupt and can depend upon
the identification of one more or less tumor.
Since the data used in fitting the LMS model are discrete count data and do not vary
continuously, parameter estimates from any model will change discretely in response to
discrete changes in the data. If a parameter can be zero for some data sets and
positive for others, it is obvious that changing the response in a single animal will in
some cases change the estimate (abruptly) from zero to a positive value or vise versa.
It is not clear why Lovell and Thomas consider this to be a problem.
q1* is invariant despite the data.
Lovell and Thomas support this point by presenting a number of examples (their
Tables 5 and 6) which show different patterns of tumor responses that produce similar
q1* values. It is unfortunate that the doses were all spaced a factor of ten apart in all of
these hypothetical examples. The dose response is very poorly characterized when
doses are so widely spaced, which accounts in considerable measure for the similarity
of the q1* values they obtained. By comparison, the examples presented in our Table 1
are based on doses spaced a factor of two apart (which is the NTP default), and q 1*'s
calculated in these examples show far more variation than those presented by Lovell
and Thomas.
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Also, in their Table 6, Lovell and Thomas show that for a range of hypothetical
data sets the correspond range of values of q1* is "just over 100-fold." This 100-fold
range is strictly determined by the data sets Lovell and Thomas chose to model, and
does not represent any sort of bound on the range of values of q 1* that can be obtained
from this experimental design. In fact, using the experimental design chosen by Lovell
and Thomas (same doses and numbers of animals), one can posit tumor responses
that the LMS model fits adequately and for which the q1*'s differ by more than 20,000.
Furthermore, their Table 6 includes a number of data sets that display strong
decreasing trends. The LMS model was not designed to fit negative trends (and it is
not clear why anyone would want to apply it to such data). When it is forced upon such
data, the result is a flat dose response and a corresponding very poor fit.
q1* is closely related to the top dose
This point is illustrated using several examples (Lovell and Thomas' Table 7),
some of which are particularly ill-chosen. Three of the comparisons in this table
illustrate the fact that increasing all of the doses by a common factor causes both q1
and q1* to decrease by that same factor. Actually, for the model not to have this feature
would be evidence of a very serious problem, since that would imply that risk estimates
from the model would depend upon the units used to express dose. In all the remaining
examples the dose response has a un-typical supra-linear shape, which the LSM
model, like most conventional dose-response models, will not fit very well. If Lovell and
Harris had examined other experimental designs and other tumor response patterns,
they would have found that q1* is not so closely related to the top dose. For example, in
our Table 1, Data Set II, q1* is not affected at all by the top dose, since it does not
constrain the low-dose slope.
q1* becomes larger if the top dose data are excluded.
Although this does not occur in every case (For example, it does not occur in our
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Table 1, Data Set II), it does occur in many cases. However, it appears to me that this
is exactly what should happen. Confidence bounds tend to become wider as data are
eliminated; hence the upper bound, q1*, tends to get larger as data are dropped from
the analysis -- this holds not for just the top dose data, but for data at any dose.
The correlation pointed out by Lovell and Thomas between the maximum
tolerated dose (MTD) and the q1* is due primarily to the more basic correlation between
the MTD and the TD50, which has nothing to do with the LMS model. This correlation is
caused by the wide range of MTD's of known carcinogens, and the biological fact that
the TD50 for most carcinogens is relatively close to the MTD.44
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Table 1 Data Sets that Illustrate Properties of the LMS Procedure
Dose Levels
0
Ex. #
0.25
0.5
1.0
# of responders
q1*
Positive MLE estimates
I
0
2
4
7
0.24
q1 = 0.16
(K = 3 or K = 10)
II
0
2
4
40
0.30
q1 = 0.16, q9 = 0.34, q10 = 1.1
0.15
q3 = 1.43
(K = 3)
(K = 10)
III
0
2
4
---
0.30
q1 = 0.16
(K = 10)
IV
0
0
0
20
0.036
q10 = 0.51
(K = 10)
V
0
0
0
0
0.015
VI
10
12
14
17
0.40
(K = 10 or K = 3)
q0 = 0.23, q1 = 0.19
(K = 3 or K = 10)
q1* is the 95% upper bound on the linear term, q1, in the LMS model (2). Each dose
group contains 50 animals.
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Figure 1
Fits of LMS model to Two Data Sets from Table 1
Figure 1A
Graph of LMS Fit to Data Set I, Table 1
Response Probability
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Dose
Figure 1B
Graph of LMS Fit to Data Set II, Table 1
Response Probability
1
0.8
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