1 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 [email protected]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.
Transcript
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Incompatible Models?Low-Dose Linearity for Noncancer Risks and
Dose Additivity for Mixtures
Resha M. Putzrath, Ph.D., DABTNavy and Marine Corps Public Health Center
From Problem Formulation to Dose Response5 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.
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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?
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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
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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)
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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 dose-response 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.”
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Mathematics of Hazard IndexEPA’s 1986 guidelines on risk assessment of mixtures [emphasis added]
“Suppose that two toxicants show the following log-dose probit response equations:
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.”
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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
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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.
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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.
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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)?
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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.
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Assume a Straight Line
For C1, R = m∙D + b. What is the dose-response 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 non-zero dose,
R = mC1∙DC1 + bC1 = mC2∙DC2 + bC2
or R = mC1∙D + bC1 = mC2∙1/pD + bC2
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Solve First Equation for Ratio of Doses
Equation ismC1∙DC1 + bC1 = mC2∙DC2 + bC2
Unclear how to obtain the relative potency, DC1/ DC2
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Wait! We Are in Probit Space
From EPA’s 1986 guidelines:
Y1 = 0.3 + 3 log Z1 Y2 = 1.2 + 3 log Z2
orR = log(m∙D) + bC1 = log(m∙1/pD) + bC2 (Eqn. 1)
andR = m∙ log(D + 1/p∙D) + bC1 (Eqn. 2)
What is the response, R, at zero dose?
Hint: It is not zero.
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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.
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Dose-response Curve for C2
Since p = (mC2∙D) / (mC1∙D + bC1 - bC2)
Under what conditions is “p” a constant?
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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
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Why Does This Matter?
For noncancer risk assessment, a dose-response 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.
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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.
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Isobols in NRC, page 79
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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.
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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.”
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NRC Dose Addition Within D-R Curve
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Howdeshell et al. on Dose Addition
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Nonlinear Dose-response of Mixture
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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.
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Whole Mixture or Dioxin-like TEFs?
PCBs similar to commercial mixture, e.g., Aroclor 1260
Other polychlorinated dibenzodioxins and dibenzofuransCongener-specific TEF ± order of magnitude
Dioxin-like PCBs resets the index chemicalPCB126 TEF = 0.1 (± order of magnitude?)
1.00
0.30.1 1
0.10.03 0.3
Other dioxin-like PCBsCongener-specific TEF ± order of magnitude of PCB126
0.03 = 0.1PCB126 ÷ 30.01 0.1
0.1 = PCB126
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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.
