Incompatible Models?
Low-Dose Linearity for Noncancer Risks and
Dose Additivity for Mixtures
Resha M. Putzrath, Ph.D., DABT
Navy and Marine Corps Public Health Center
Resha.Putzrath@med.navy.mil
Presented at
Beyond Science and Decisions:
From Problem Formulation to Dose Response
5 May 2011
The views expressed in this presentation are those of the
author and do not necessarily reflect the official policy or
position of the Department of the Navy, Department of
Defense, or the U. S. Government.
1
Introduction



EPA’s 1986 guidelines and 2000 guidance on
mixtures use toxicological principles to derive
procedures for dose additivity of components
for evaluating noncancer risks.
NRC’s Science and Decisions proposes linear,
no-threshold, low-dose extrapolations for
noncancer risk assessments.
Are these models compatible?
2
Outline

Definitions of “linear” and “dose additive”

Implications of current, regulatory definitions
and current practices

Examples
• Phthalates and Cumulative Risk Assessment. The
Task Ahead. National Research Council, 2008
• Toxicity equivalency factors (TEFs) for PCBs
3
Define “Linear”
Formula for a straight line: y = mx + b
 Formula for low-dose linear per EPA’s
2005 cancer guidelines: y = mx
 Other mathematical definitions of linear

• A linear equation a1x1 + ... + anxn = b,
where a1, ... , an and b are constants and x1,
... , xn are variables.
• A collection of one or more linear
equations (linear algebra)
4
Define “Dose Additivity”
EPA’s 1986 guidelines on risk assessment of mixtures [emphasis added]
“Dose addition assumes that the toxicants
in a mixture behave as if they were
dilutions or concentrations of each other,
thus the true [probit] slopes of the doseresponse curves for the individual
compounds are identical, and the response
elicited by the mixture can be predicted by
summing the individual doses after
adjusting for differences in potency; this is
defined as the ratio of equitoxic doses.”
5
Mathematics of Hazard Index
EPA’s 1986 guidelines on risk assessment of mixtures [emphasis added]
“Suppose that two toxicants show the following log-dose probit response
equations:
Y1 = 0.3 + 3 log Z1
Y2 = 1.2 + 3 log Z2
(4-1)
(4-2)
where Y1 is the probit response associated with a dose of Z1 (i = 1, 2)…
Dose addition assumes that the response, Y, to any mixture of these two
toxicants can be predicted by
Y = 0.3 + 3 log (Z1 + pZ2)
(4-3)
Thus, since p is defined as Z1/Z2, Equation 4-3 essentially converts Z2
into an equivalent dose of Z1 by adjusting for the difference in potency.”
6
Additional Definitions


Adding within the dose-response
function, i.e., response = f(dose1+dose2)
Defining a dose-additive function, and
constructing isobols
• Experimentally defines proportions
that produce the same response.
• May look the same as HI or within
function, but has different properties
7
Definitions for This Analysis
(Unless Otherwise Stated)

“Linear” is a straight line from a point of
departure to the origin, i.e., (0,0).

“Dose addition” is EPA’s Hazard Index, HI,
where
n
HI = ∑(Ei /ALi)
for n chemicals, where Ei is the exposure to the
ith chemical and ALi is the acceptable level of
the ith chemical.
8
Implications of Response = m ∙ Dose

Response is directly proportional to dose
• NOT approximately, but exactly!
• No non-linear processes, e.g., receptor binding (Hill
equation) or secondary metabolism due to
saturation.
• Only valid for some part of the dose-response
function, else response could exceed 100%.


No threshold for any effect for any chemical,
e.g., lethality of distilled water.
Dose addition, response addition, and many
other methods for combining data all work
equally well.
9
Analysis of Incompatible Assumptions

Assume that the more potent chemical is C1.

The less potent chemical, C2, should behave
as if it is a dilution of C1, i.e., C2 = (1/p)∙C1

What are the dose-response functions for C1
and C2 (at low doses)?
10
Possible Low-dose Curves
If the curve is nonlinear, the assumptions
are incompatible by definition.
 The curve could be a straight line with a
non-zero y-intercept, e.g., due to a
background response level. Current
regulatory solution: subtract background
and becomes Response = m∙Dose.
 The curve could be a straight line with a
non-zero x-intercept, i.e., a threshold.

11
Assume a Straight Line
For C1, R = m∙D + b. What is the doseresponse curve for C2?
 It must also be linear at low doses, i.e.,
to act like a dilution of C1.
 To have the same response for any nonzero dose,
R = mC1∙DC1 + bC1 = mC2∙DC2 + bC2

or
R = mC1∙D + bC1 = mC2∙1/pD + bC2
12
Solve First Equation for Ratio of Doses

Equation is
mC1∙DC1 + bC1 = mC2∙DC2 + bC2

Unclear how to obtain the relative
potency, DC1/ DC2
13
Wait! We Are in Probit Space
From EPA’s 1986 guidelines:
Y1 = 0.3 + 3 log Z1
Y2 = 1.2 + 3 log Z2
or
R = log(m∙D) + bC1 = log(m∙1/pD) + bC2
(Eqn. 1)
R = m∙ log(D + 1/p∙D) + bC1
(Eqn. 2)
and
What is the response, R, at zero dose?
Hint: It is not zero.
14
Incompatibility



The response of log(dose) where dose = 0 is
undefined. Therefore, if dose addition is
defined in probit space, then the response to a
mixture at no exposure can not be defined.
If C1 and C2 are low-dose linear in normal
space, the dose-response functions must both
intersect the origin.
This is not an issue if the functions have a
threshold, because
• the response at zero dose is not defined, and
• a range of doses produce zero response.
15
Dose-response Curve for C2

Since
p = (mC2∙D) / (mC1∙D + bC1 - bC2)

Under what conditions is “p” a constant?
16
Straight Line to a Threshold

Compatibility can be resolved by a
straight line from a point of departure to
a threshold (Putzrath. 1997. Regul. Tox. Pharm. 25:68-78).

If bC1 = bC2 > 0
log(mC1∙DC1) + bC1 = log(mC2∙DC2) + bC1
mC1∙DC1 = mC2∙DC2
mC1/mC2 = DC2 /DC1 = constant
17
Why Does This Matter?

For noncancer risk assessment, a doseresponse curve that is a straight line at low
doses is compatible with EPA’s hazard index
if and only if the chemicals have a
threshold, which is not consistent with a
straight-line extrapolation to the origin.

Other definitions of “linear” and “dose
addition” include additional complications.
18
Example: Phthalates

NRC report on phthalates uses three
definitions of dose additivity:
• EPA’s Hazard Index,
• isobols, and
• addition within the dose-response function.


If all low-dose linear, the results at these
doses are equivalent.
If the differences are retained, the result of
estimating the effects of a mixture of
phthalates differs.
19
Isobols in NRC, page 79
20
The Formula Looks Like EPA’s HI, But …



The denominators are the levels that produce a specific effect,
i.e., EDx. In contrast to HI that uses RfD/Cs, no uncertainty
factors (UFs).
The model is defined, not derived. Chemicals are dose
additive if and only if they obey the properties of the model.
Therefore, the model has fewer restrictions (see next slide).
The model does not predict effects of other mixtures.
• It is used to establish mixtures where dose addition has
been observed experimentally.
• From these data, isobols can be constructed, and these may
be used to infer other mixtures that are likely to be dose
additive.
• The likelihood of such predictions is dependant on the
amount of data used to construct the isobols.
21
Some NRC Assumptions
(pages 80-81, emphasis added)

“First, the dose additivity of a particular mixture does not imply
the dose additivity of other mixtures of the same components.”

“Second, conclusions about dose addition, synergism, or
antagonism …may not be the same for different levels of effect
even for similar mixture ratios…”

“Third, the doses DA, DB, DC., ... vary with the effect level…”

“Fourth, with the definition of dose addition stipulated by
Equation 1, the evaluation of dose-additivity or nonadditivity is a
matter entirely for observation using measured dose-response
curves; no consideration of mechanism of action is required.”
22
NRC Dose Addition Within D-R Curve
23
Howdeshell et al. on Dose Addition
24
Nonlinear Dose-response of Mixture
25
PCBs and TEFs

Dioxin-like PCBs are integral to the
commercial mixtures of PCBs.

EPA’s 1986 guidelines on mixtures says the
preferred method is to use data on the mixture
or a sufficiently similar mixture. Absent that,
evaluate the components.

TEFs are based on similar mathematical
assumptions to the HI.

Additional issue of potential for double
counting.
26
Whole Mixture or Dioxin-like TEFs?
PCBs similar to commercial mixture, e.g., Aroclor 1260
Dioxin-like PCBs in a dioxin-equivalent TEQ
Dose
for
Cancer
Dose for
Reproductive
Effects
Either PCB mixture or dioxin TEQ, but not both
27
Uncertainties in Use of TEFs
Dioxin (2,3,7,8-Tetrachlorodibenzo-p-dioxin)
TEF = 1.0
1.00
Other polychlorinated dibenzodioxins and dibenzofurans
Congener-specific TEF ± order of magnitude
0.1
0.3
1
Dioxin-like PCBs resets the index chemical
PCB126 TEF = 0.1 (± order of magnitude?)
0.03
0.1
0.3
Other dioxin-like PCBs
Congener-specific TEF ± order of magnitude of PCB126
0.1 = PCB126
0.01
0.03 = 0.1PCB126 ÷ 3
0.1
28
Parting Thoughts

Before evaluating concepts like “linear” or “dose additive”
ensure everyone is using the same definition. Be especially
cautious of quotations out of context.

“Low-dose linear, no threshold” appears to be compatible
with dose additivity only in the most biologically unlikely
case, i.e., when response is exactly linearly proportional to
dose.

Carefully consider the assumptions that are the foundation
of the models before using them together. If the
assumptions are not compatible, the result may seem
reasonable, but will be difficult to interpret accurately.
29