United States
Environmental Protection
Agency
Developing Relative Potency
Factors for Pesticide Mixtures:
Biostatistical Analyses of Joint
Dose-Response

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                                                 EPA/600/R-03/052
                                                  September 2003
       Developing  Relative  Potency Factors
        for Pesticide Mixtures: Biostatistical
         Analyses of Joint Dose-Response
L*                     u-s- Environmental Protection Agency
                      Region 5, Library (PJ.-12J)
                                       12th
                  National Center for Environmental Assessment
                     Office of Research and Development
                     U.S. Environmental Protection Agency
                          Cincinnati, OH 45268
                                              /"T'V Recycled/Recyclable
                                                  Printed with vegetable-based ink on
                                                  paper that contains a minimum of
                                                  50% post-consumer fiber content
                                                  processed chlorine free.

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                                  NOTICE

      The U.S. Environmental Protection Agency through its Office of Research and
Development funded and managed the research described here. It has been subjected
to the Agency's peer and administrative review and has been approved for publication
as an EPA document.  Mention of trade names or commercial products does not
constitute endorsement or recommendation for use.

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                                 FOREWORD
      This report was developed by the U.S. Environmental Protection Agency's (EPA)
Office of Research and Development (ORD), National Center for Environmental
Assessment - Cincinnati Office (NCEA-Cin) in collaboration with EPA's Office of
Pesticide Programs. It contains information concerning biological concepts and
statistical procedures for improving the application of Relative Potency Factors (RPFs)
to pesticide mixtures. This research supports the need for chemical mixtures risk
assessment research as mandated in 1996 by both the Food Quality Protection Act
(FQPA) and  the Safe Drinking Water Act Amendments. Research results are presented
regarding the theoretical basis for RPF-based risk assessments; new quantitative
methods for  applying RPFs are shown. The RPF approach assumes toxicity of the
mixture components can be characterized using dose addition. Thus, the basic tenets
of dose addition, common toxic modes of action and similarly-shaped dose-response
curves among the mixture components, are investigated and discussed. This research
was undertaken to continue exploring and developing cumulative risk assessment
strategies beyond current applications and is intended to improve future applications of
RPF based risk assessments.

      The statistical methods presented in this effort are based on research  conducted
by Jim Chen, Yi-Ju Chen, and Ralph Kodell through an Interagency Agreement
between EPA and the Food and Drug Administration's National Center for Toxicological
Research. An external review was conducted by Drs.  Christine F. Chaisson,  Pavel
Muller, and Walter W. Peigorsch under EPA Contract No. 68-C-02-060/061 with Versar,
Inc.

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                               RESEARCHERS
      This research was sponsored by the U.S. Environmental Protection Agency
(EPA), Office of Research and Development, National Center for Environmental
Assessment - Cincinnati Division (NCEA-Cin). NCEA-Cin researchers collaborated with
scientists from other organizations to conduct this research and to author this report. A
number of other scientists also contributed their ideas, provided discussions and
review, and wrote text toward completion of this effort. These individuals are listed
below.

Authors:

EPA's National Center for Environmental Assessment, Cincinnati, OH
      Linda K. Teuschler
      Glenn E. Rice
      John C. Lipscomb

EPA's Homeland Security Research Center, Cincinnati, OH
      Richard C. Hertzberg

EPA's Office of Pesticide Programs, Washington, DC
      Karen Hamernik
      Alberto Protzel

U.S. Food and Drug Administration's National Center for Toxicological Research,
Division of Biometry and Risk Assessment, Jefferson, Arkansas
      James Chen
      Yi-Ju Chen
      Ralph L. Kodell

Contributors and Reviewers:

EPA's National Center for Environmental Assessment, Research Triangle Park, NC
      Gary Foureman

EPA's National Center for Environmental Assessment, Washington, DC
      Femi Adeshina

EPA's National Health and Environmental Effects Research Laboratory, Research
Triangle Park, NC
      Jane Ellen Simmons
                                      IV

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                        TABLE OF CONTENTS

                                                               Page

FOREWORD  	iii

RESEARCHERS  	iv

LIST OF ABBREVIATIONS 	  vii

KEY DEFINITIONS	viii

EXECUTIVE SUMMARY 	xiii

1.    INTRODUCTION  	 1

2.    DOSE ADDITION CONCEPTS   	 9

3.    RELATIVE POTENCY FACTORS	 12

     3.1.  JUDGMENTS OF COMMON TOXICOLOGIC ACTION  	 15

4.    CHOICE OF DOSE METRIC IN CHARACTERIZING MIXTURE
     TOXICITY BY DOSE ADDITION	20

     4.1.  RPF DOSE ISSUES 	 20

          4.1.1. Administered Dose	 21
          4.1.2. Internal Dose 	 21
          4.1.3. Mixtures Exposures Through Multiple Exposure Routes 	21
          4.1.4. Mixtures Exposures Though a Single Exposure Route in
               Different Species  	 22

     4.2.  CHOICE OF DOSE MEASURES  	 25

5.    BIOSTATISTICAL DOSE-RESPONSE MODELING FOR
     CUMULATIVE RISK 	26

     5.1.  DOSE-RESPONSE MODEL FOR COMBINED EXPOSURES	2

          5.1.1. Constant Relative Potency	 30
          5.1.2. Nonconstant Relative Potency	 31
          5.1.3. Constant and Nonconstant Relative Potencies in the
               Same Mixture	 32

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                       TABLE OF CONTENTS cont.

                                                                 Page

     5.2.   STATISTICAL ALGORITHMS FOR SUBCLASS GROUPS
           WITHIN A MIXTURE	 34

           5.2.1. Top-Down Approach	 36
           5.2.2. Bottom-Up Approach  	 39

     5.3.   CUMULATIVE RISK ASSESSMENT  	42

           5.3.1. Mixtures Reference Dose	 43

6.    CONCLUSIONS	45

7.    REFERENCES 	 48

APPENDIX A: Chen et al., 2001. Using Dose Addition to Estimate
            Cumulative Risks from Exposures to Multiple Chemicals  	 A-1
APPENDIX B: Chen et al., 2003 (In Press).  Cumulative Risk Assessment for
            Quantitative Response Data  	 B-1
                                  VI

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                         LIST OF ABBREVIATIONS

CRA        Cumulative Risk Assessment
DBFs       Disinfection By-Products
ED         Effective Dose
EPA        Environmental Protection Agency
FQPA       Food Quality Protection Act
HI          Hazard Index
ICED        Index Chemical Equivalent Dose
LOAEL      Lowest-Observed-Adverse-Effect Level
MF         Modifying Factor
NOAEL      No-Observed-Adverse-Effect Level
OP         Organophosphorus Pesticide
ORD        Office of Research and Development
PBPK       Physiologically-Based Pharmacokinetic
RAF        Risk Assessment Forum
RfD         Reference Dose
RPF        Relative Potency Factor
TEF        Toxicity Equivalence Factor
TEQ        2,3,7,8-TCDD Toxicity Equivalents
UF         Uncertainty Factor
                                    VII

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                               KEY DEFINITIONS
Absorbed Dose - the amount of a substance crossing a specific barrier through uptake
processes.1

Additivity - When the "effect" of the combination is estimated by the sum of the
exposure levels or the effects of the individual chemicals. The terms "effect" and "sum"
must be explicitly defined. Effect may refer to the measured response or the incidence
of adversely affected animals. The sum may be a weighted sum (see "dose addition")
or a conditional sum (see "response addition").3

Bioavailability - The state of being capable of being absorbed and available to interact
with the metabolic processes of an organism. Bioavailability is typically a function of
chemical properties, physical state of the material to which an organism is exposed,
and the ability of the individual organism to physiologically take up the chemical.1

Chemical Classes - Groups of components that exhibit similar biologic activities, and
that frequently occur together in environmental samples, usually because they are
generated by the same commercial process. The composition of these mixtures is
often well controlled, so that the mixture can be treated as a single chemical.  Dibenzo-
dioxins are an example.3 (Note: this is slightly modified from the original version).

Chemical Mixture - Any set of multiple chemical substances that may or may not be
identifiable, regardless of their sources, that may jointly contribute to toxicity in the
target population. May also be referred to as a "whole mixture" or as the "mixture of
concern."3

Complex Mixture - A mixture containing so many components that any estimation of its
toxicity based on its  components' toxicities contains too much uncertainty and error to
be useful. The chemical composition may vary over time or with different conditions
under which the mixture is produced. Complex mixture components may be generated
simultaneously as by-products from a single source or process, intentionally produced
as a commercial product, or may coexist because of disposal practices.  Risk
assessments of complex mixtures are preferably based on toxicity and exposure data
on the complete mixture. Gasoline is an example.3

Components - Single chemicals that make up a chemical mixture that may be further
classified as systemic toxicants, carcinogens, or both.3

Dose Additivity - When the effect of the combination is the effect expected from the
equivalent dose of an index chemical. The equivalent dose is the sum of component
doses scaled by their potency relative to the index chemical.3

Dose - The amount  of a substance available for interaction with  metabolic processes or
biologically significant receptors after crossing the outer boundary of an organism1.
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Dose-Response Assessment - A determination of the relationship between the
magnitude of an administered, applied, or internal dose and a specific biological
response. Response can be expressed as measured or observed incidence, percent
response in groups of subjects (or populations), or as the probability of occurrence
within a population.2

Dose-Response Relationship - The relationship between a quantified exposure
(dose), and the proportion of subjects demonstrating specific, biological changes
(response).2 U.S. EPA's draft 1996 Cancer Guidelines further state: "Whether animal
experiments or epidemiologic studies are the sources of data, questions need to be
addressed in arriving at an appropriate measure of dose for the anticipated
environmental exposure. Among these are:
            whether the dose is expressed as an environmental concentration,
            applied dose, or delivered dose to the target organ,
            whether the dose is expressed in terms of a parent compound, one or
            more metabolites, or both,
            the  impact of dose patterns and timing where significant,
      •     conversion from animal to human doses, where animal data are used,
            and
            the  conversion metric between routes of exposure where necessary and
            appropriate."

Effective Dose (ED10) - The dose corresponding to a 10% increase in an adverse
effect, relative to  the control  response.2

Exposure - Contact made between a chemical, physical, or biological agent and the
outer boundary of an organism.  Exposure is quantified as the amount of an agent
available at the exchange boundaries of the organism (e.g., skin, lungs, gut).2

Exposure Assessment - An identification and evaluation of the human population
exposed to a toxic agent, describing  its composition and size, as well as the type,
magnitude, frequency, route and duration of exposure.2

Extrapolation, low dose - An estimate of the response at a point below the range of
the experimental  data, generally through the use of a mathematical model.2

Human Equivalent Concentration (HEC) or Dose (HED) - The human concentration
(for inhalation exposure) or dose (for other routes of exposure) of an agent that is
believed to induce the same magnitude of toxic effect as the experimental animal
species concentration or dose. This adjustment may incorporate toxicokinetic
information on the particular agent, if available, or use a default procedure, such as
assuming that daily oral doses experienced for a lifetime are proportional to body
weight raised to the 0.75 power.2
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Index Chemical -The chemical selected as the basis for standardization of toxicity of
components in a mixture. The index chemical must have a clearly defined
dose-response relationship.3

Index Chemical Equivalent Dose -The exposure to a chemical that is expected to
elicit the same response as that of the index chemical, when the chemicals are
administered by the same route, at the same duration and frequency.  The chemical
and the index chemical must share a common mode of action.

Internal dose - A more general term denoting the amount absorbed without regard to
absorption process.1

Independence of Action - Mixture components that cause different kinds of toxicity, or
effects in different target organs; the risk assessor may then combine the probabilities
of toxic effects for the individual components.3

Mechanism of Toxicity or Mechanism of Toxic Action - The set of molecular and
cellular events leading to a toxicologic outcome. [A toxicologic outcome is considered to
be damage to the organism at any level of biological organization (i.e., molecular,
cellular, tissue,...).]4

Mode of Action - The set of biological events at the target tissue or target organ
leading to a toxicologic outcome.  [A toxicologic outcome is considered to be damage to
the organism at any level of biological organization (i.e., molecular, cellular, tissue,...).]4

Model - A mathematical function with parameters that can be adjusted so the function
closely describes a set of empirical data. A mechanistic model usually reflects observed
or hypothesized biological or physical mechanisms, and has model parameters with
real world interpretation. In contrast, statistical or empirical models selected for
particular numerical properties are fitted to data; model parameters may or may not
have real world interpretation. When data quality is otherwise equivalent, extrapolation
from mechanistic models (e.g., biologically  based dose-response models) often carries
higher confidence than extrapolation using  empirical models (e.g., logistic model).2

Physiologically Based Pharmacokinetic  (PBPK) Model - Physiologically based
compartmental model used to characterize pharmacokinetic behavior of a chemical.
Available data on blood flow rates, and  metabolic and other processes which the
chemical undergoes within each compartment are used to construct a mass-balance
framework for the PBPK model.2

Point of Departure - The dose-response point that marks the beginning of a low-dose
extrapolation. This point is most often the upper bound on an observed incidence or on
an estimated incidence from a dose-response model.2

Risk - The probability of deleterious effects on health.1

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Relative Potency Factor Method - A mixtures risk assessment approach used to
assess risks posed by mixture components that exhibit a common mode of action. The
toxic potency of each mixture component is compared to that of an index chemical
generating a measure of potency for each component that is relative to the toxicity of
the index chemical. For application, the shapes of the individual component dose-
response functions must be similarly over the region of the mixture exposure.

Response Additivity - When the response (rate, incidence, risk, or probability) of
effects from the combination is equal to the conditional sum of component responses
as defined by the formula for the sum of independent event probabilities.3

Similar  Components - Single chemicals that cause the same biologic activity or are
expected to cause a type of biologic activity based on chemical structure. Evidence of
similarity may include parallel log-probit dose-response curves and same mechanism of
action or toxic endpoint. These components are expected to have comparable
characteristics for fate, transport, physiologic processes, and toxicity.3

Similar  Mixtures - Mixtures that are slightly different, but are expected to have
comparable characteristics for fate, transport, physiologic processes, and toxicity.
These mixtures may have the same components but in slightly different proportions, or
have most components in nearly the same proportions with only a few different (more
or fewer) components. Similar mixtures cause the same biologic activity or are
expected to cause the same type of biologic activity due to chemical composition.
Similar mixtures act by the same mechanism of action or affect the same toxic
endpoint. Diesel exhausts from different engines are an example.3

Simple  Mixture - A mixture containing two or more identifiable components, but few
enough  that the mixture toxicity can be adequately characterized by a combination of
the components' toxicities and the components' interactions.3

Target Organ - The biological organ(s) most adversely effected by exposure to a
chemical substance.2

Uptake  - The process by which a substance crosses an absorption barrier and is
absorbed into the body.1

Sources

1U.S. EPA.  1992.  Guidelines for Exposure Assessment; Notice. Federal Register.
57(104):22888-22938.

2U.S. EPA.  2003.  Integrated Risk Information System.  Office of Research and
Development, National Center for Environmental Assessment, Washington, DC.
Online-  http://www.epa.gov/iris
                                      XI

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3U.S. EPA. 2001. Supplementary Guidance for Conducting Health Risk Assessment of
Chemical Mixtures.  Office of Research and Development, Washington, DC.
EPA/630/R-00/002.  Available in PDF format at: www.epa.gov/NCEA/raf/chem mix.htm

4U.S. EPA. 2002. The Feasibility of Performing Cumulative Risk Assessments for
Mixtures of Disinfection By-Products in Drinking Water. NCEA-C-1257. Final Draft.
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                           EXECUTIVE SUMMARY








      Cumulative Risk Assessment (CRA) is defined in U.S. EPA's Risk Assessment



Forum (RAF) CRA Framework (U.S. EPA, 2002a) as "the combined risks from



aggregate exposures to multiple agents or stressors." CRA has become an important



research area, reflecting the interest of U.S. EPA's regional risk assessors, program



offices, Office of Environmental Justice, and Office of Children's Risk.  In implementing



the requirements of the Food Quality Protection Act of 1996, U.S. EPA's Office of



Pesticide Programs has developed guidance for conducting CRA's of chemicals that



appear to act by a common mechanism of toxicity (U.S. EPA, 2002b).  Because the



organophosphorus pesticides (OPs) are considered to exert some of their toxic effects



via a common toxicologic mechanism (i.e., cholinesterase inhibition), these compounds



have been the subject of a CRA (U.S. EPA, 2001 b). Additional CRA's may be



performed on  additional pesticide classes (e.g., triazinines, carbamates) and other co-



occurring substances for which a common mode of action can be identified. The risk



assessment method employed in the OP cumulative risk study is the Relative Potency



Factor (RPF) approach (U.S. EPA, 2000). Dose addition is the critical methodological



assumption, requiring the mixture components to act by the same toxic mode of action



and to have similarly-shaped dose-response curves.



      Assessing the cumulative toxicological effects of multiple chemicals has been



addressed from time to time (NRC, 1988; U.S. EPA, 1986, 2000). Methods and data



that can be used to estimate the risk of exposures to multiple chemicals have been



developed. Although U.S. EPA guidance exists regarding the basic theory for RPFs,
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the toxicological criteria for defining and determining a common mode of action among
chemicals continue to need refinement; results on this subject are presented in this
report. Further, results are presented on appropriate statistical methods for CRA,
based on research published in Chen et al. (2001, 2003). Biostatistical approaches are
shown for grouping chemicals identified as having common modes of action, proposing
two classification algorithms to cluster chemicals into subclasses within which
chemicals have similarly-shaped dose-response functions. Chemicals within subclasses
are combined using the RPF method when a constant relative potency among
chemicals exists.  Additional methods are shown to calculate cumulative risks inclusive
of these subclasses (i.e., combining across subclasses for which a non-constant
relative potency exists) using either a joint dose-response approach or by integrating
the concepts of dose addition and response addition.
      An important question in mixtures risk assessment research is how to assess a
mixture containing some chemicals that share a common toxic mode of action and
other chemicals that do not. Current additivity methods have evolved to handle either
the former (dose addition) or the latter (response addition). Alternatively, the risk
assessor may choose to do the assessment based on whole mixture data. The
biostatistical methods developed in this report provide alternative methods to evaluate a
mixture under three scenarios.  The simple case occurs when there is certainty that a
common toxic mode of action is operating, so a dose addition approach can be  applied.
The second case occurs when the mixtures can be divided into independent mode of
action subclasses; dose addition and response addition can be integrated to make the
assessment. The third case occurs when mode of action is uncertain, so a joint
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dose-response modeling procedure is used to create a range of risk estimates. Thus,

these approaches enrich the available library of mixture risk assessment methods

beyond what is currently published by the U.S. EPA (1986, 2000). Further, these

approaches may be useful in future assessments of pesticide mixtures to be evaluated

under FQPA.  Finally, the results presented here are generalizable to assessments of

other environmental mixtures; the risk assessments that support environmental

regulations of important environmental mixtures such as dioxins, polychlorinated

biphenyls, and OPs are based on concepts of additivity (U.S. EPA, 1989b, 2000,

2001 b).

      The research results in this report can be applied to reduce uncertainties in

RPF-based risk assessments of chemical mixtures. These results also show how

mixtures risk assessments can be conducted using additivity concepts. Various sources

of uncertainty exist in  most mixtures risk assessments, including uncertainties

addressed in this report regarding several factors:

            Common mode of action across mixture components (Sections 2, 3)

            Similarly shaped dose-response curves across mixture components

            (Sections 2, 5)

           Value of internal vs. external dose estimates for developing RPFs

            (Section 4)

           Choice of dose metric (moles vs. mass) to use in a cumulative risk
           assessment (Section 4)

           Cross-species extrapolation of relative potency factors (Section 4)

            Estimating risks for a mixture with two or more common mode of action
           subclasses (Section 5).
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      Biostatistical modeling in this report presents ways to combine dose-response

information, partitioning the mixtures into common mode of action subclasses. These

models can then be used to estimate risks for specific exposure scenarios or used to

develop toxicity values, such as a reference dose for the mixture.  Three RPF-based

methods are discussed, reflecting what is known or uncertain about the mixture

toxicology. These approaches can be applied using internal or external doses.

Development of approaches based on internal doses may reduce some toxicokinetic

uncertainties associated with RPFs based on administered doses. In the Chen et al.

papers (2001, 2003) in Appendices A and B, external doses were used to develop

statistical methods for grouping components into common mode of action subclasses.

The next step in this process is to use RPFs based on internal doses and compare

subclass groupings and modeling results with those developed using external doses.

Recommendations for future RPF research on pesticide mixtures are listed here.

      1)    Develop kinetic models for pesticide mixtures in rodents.

      2)    Using experimental cholinesterase inhibition measures, determine RPFs
            based on both external and internal dose estimates for the rodent.

      3)    Determine if the RPFs based on internal dose estimates significantly differ
            from RPFs developed from external doses for the rodent.

      4)    Apply the biostatistical methods for grouping by common dose-response
            curves using RPFs based on internal and external doses and compare the
            groupings that result.

      5)    Develop kinetic models for pesticide mixtures in humans.

      6)    Estimate  human risks using rodent cholinesterase inhibition responses,
            RPFs based on rodent internal doses, and human internal dose estimates
            using the three approaches presented in Chen et al. (2001, 2003), as
            appropriate.
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      7)    Compare subclass groupings and human risk estimates for all scenarios
            of internal and external RPFs.

      8)    Ealuate the toxicity of different human exposure scenarios with the RPF
            models developed.

      This research was undertaken to continue exploring and developing cumulative

risk assessment strategies based on dose addition concepts beyond current

applications and is intended to improve future applications of RPF based risk

assessments.
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                              1.  INTRODUCTION








      The U.S. Environmental Protection Agency (U.S. EPA) and other regulatory



agencies use risk assessment to evaluate the risk posed to humans through chemical



exposures to contaminants in food, drinking water, or environmental media. Risk



assessment for toxic agents is often conducted to evaluate the potential risks from



exposure to a single toxic agent through a single route of exposure. Although it is



important to evaluate individual toxic agents, people frequently are exposed to many



chemicals simultaneously or in sequence by different exposure routes. These



exposures to multiple chemicals through various media could cause unexpected



cumulative effects. The combined risk from such exposures may be greater or less than



what would typically be predicted from data on individual chemicals. Assessing the



cumulative toxicological effects of multiple chemicals has been addressed from time to



time (NRC, 1988; U.S. EPA, 1986, 2000). However, new methods and improvements



to existing approaches are still needed to estimate risk from exposures to multiple



chemicals.



      Cumulative Risk Assessment (CRA) is defined in U.S. EPA's Risk Assessment



Forum (RAF) CRA Framework (U.S. EPA, 2002a) as "the combined risks from



aggregate exposures to multiple agents or stressors." CRA can include both chemical



and non-chemical stressors, multiple-route exposures, population factors that



differentially affect exposure or toxicity, and community based assessments. CRA has



become an important research area, reflecting the interest of U.S. EPA's regional risk



assessors, program offices, Office of Environmental Justice, and Office of Children's

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Risk.  In 2002, U.S. EPA's Office of Research and Development (ORD) jointly

sponsored a workshop with U.S. EPA's Regions to discuss current case studies,

methods and research needs regarding CRA (U.S. EPA, 2003a). Regional scientists

are confronted with conducting community-based CRA's (e.g., assessing risks from

multi-media, multi-stressor exposures to a population in a specified geographic area).

Successful completion of such assessments require development of new data,

methods, and guidance.

      U.S. EPA's Program Offices generally conduct CRA's on a select group of co-

occurring chemicals, and set broad national standards. Examples of programmatic

interests include:

      The Office of Water needs to conduct chemical mixtures research to support
      requirements of the Safe Drinking Water Act Amendments of 1996 (U.S. EPA,
      1996).

      The Office of Air Quality Planning and Standards has used a CRA approach in
      conducting the National Air Toxics Assessment of 33 air pollutants (a subset of
      32 air toxics from the Clean Air Act's list of 188 air toxics plus diesel particulate
      matter) (U.S. EPA, 2001 a).

      The Office of Solid Waste and Emergency Response assesses contaminant
      mixtures at Superfund Sites (U.S. EPA, 1989a) under the Comprehensive
      Environmental Response, Compensation, and Liability Act (U.S. EPA, 1980).

      The Office of Pesticide Programs has conducted a CRA on organophosphorus
      pesticide  (OP) mixtures (U.S. EPA, 2001 b), under the Food Quality Protection
      Act (FQPA) of 1996 (U.S. EPA, 1997). Case studies may be performed on
      additional pesticide classes (e.g., triazinines, carbamates) and other co-occurring
      substances for which a common mode of action can be identified.


      The FQPA is the most specific act regarding CRA, requiring EPA to consider the

potential human health risks of multiple route exposures to multiple pesticide residues

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and substances that have a common mechanism of toxicity (U.S. EPA, 1997).1 The

first pesticide group to be evaluated (U.S. EPA, 2002b) is the organophosphorus

pesticides (OPs), a group of closely related pesticides that affect nervous system

function.  They are applied to many food crops, as well as to residential and commercial

buildings and lawns. The many uses of this class of pesticides result in frequent and

consistent human exposures. The acute and chronic effects of OPs in humans, wild

animals, and test animals are well known.  OPs are neurotoxic because they bind to

and phosphorylate the enzyme acetylcholinesterase in both the central (brain) and

peripheral nervous systems, reducing the ability of the enzyme cholinesterase to

function properly in regulating acetylcholine, a neurotransmitter.  Acetylcholine is a

critical factor in the transfer of nerve impulses from a nerve cell to a muscle cell or

another nerve cell. If acetylcholine levels are not properly reduced by cholinesterase,

the nerve impulses or neurons remain active longer than they should, overstimulating

the nerves and muscles and causing toxic effects at many sites, including

neuromuscular junctions and synapses of the central and autonomic nervous system.

       As part of the implementation of FQPA, U.S. EPA's Office of Pesticide Programs

has developed guidance for conducting cumulative risk assessments of chemicals that

appear to act by a common mechanism of toxicity (U.S. EPA, 2002b). Because the

OPs are considered to exert some of their toxic effects via a common toxicologic
       'The terms mechanism of toxicity (or mechanism of toxic action) and mode of action represent a
continuum of understanding regarding a toxicodynamic process (U.S. EPA, 2002c). A toxicologic outcome
is considered to be damage to the organism at any level of biological organization (i.e., molecular, cellular,
tissue,...). Knowledge of a chemical's mechanism of toxicity or mechanism of toxic action implies that the
molecular and cellular events leading to a toxicologic outcome are described and well-understood.
Knowledge of a chemical's mode of action implies a general understanding of the key toxicodynamic
events that occur at a tissue level, but not a detailed description of these events at the cellular or
molecular level. Mode of action is defined as the set of biological events at the target tissue or target
organ leading to a toxicologic outcome.

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mechanism (i.e. cholinesterase inhibition), these compounds have been the subject of a

CRA (U.S. EPA, 2001 b). The risk assessment method employed in the OP CRA and

likely to be used in future pesticide CRA's is the Relative Potency Factor (RPF)

approach (U.S. EPA, 2000). This report examines the theoretical basis for the RPF

method, providing useful information to improve and enhance such future applications.

      The RPF approach is appropriate under FQPA because dose addition is the

critical RPF methodological assumption; implementation requires that the mixture

components act by the same toxic mode of action. As explained in Section 2, a

theoretical consequence of this assumption is that the components should have

similarly-shaped dose-response curves between the response threshold and the

maxima. To summarize the procedure, doses of mixture components are scaled by

their potency relative to a well-studied component of the chemical mixture (referred to

as the index chemical) using scaling factors called RPFs. The product of each mixture

component's dose and its RPF is considered to be its equivalent dose in units of the

index chemical. These dose equivalents of all the  mixture components are summed to

express the total mixture dose in terms of an Index Chemical Equivalent Dose (ICED).2

The risk posed by the mixture is then quantified by comparing the mixture's ICED to the

dose-response assessment of the index chemical. To implement this approach, the

index chemical must have an adequate toxicologic dose-response data set.

      U.S. EPA (2000) characterized the RPF methodology as a generalized form of

the toxicity equivalence factor (TEF) methodology that has been used to assess risks
      2The ICED has the same mathematical interpretation as the dioxin toxicity equivalents (TEQ).
TEQ refers to the quantification of dioxin concentrations based on the congeners' equivalent 2,3,7,8-
TCDD toxicity (U.S. EPA, 1989b). ICED is applied to mixtures other than dioxins.

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posed by some dioxins (U.S. EPA, 1989b). The TEF approach uses a single TEF for



each dioxin congener, applying this same TEF to all exposure routes, health effects,



and exposure durations. The RPF methodology was developed for application to a



broad set of chemical groups whose data sets are either less complete than the dioxins



or indicate more variation in mode of action across route, effects and duration. The



significant generalizations in the RPF methodology include the following:



1.     RPFs may be developed to assess risks for a subset of the health effects



      caused by a mixture's components. For example, the same mixture



      components may be shown to cause both  hepatotoxicity and renal toxicity in



      bioassays. Different RPFs may be developed to address the risk of each type of



      toxicity following human exposures. Mixture Component A may exhibit greater



      hepatotoxicity than Component B when compared to Index Chemical C; to reflect



      this, the RPF for the hepatotoxicity of Component A should be greater than the



      RPF of Component B. However, mixture Component B may exhibit greater renal



      toxicity than Component A when compared to Index Chemical C and, to reflect



      this, the RPF for the renal toxicity of Component A should be less than that of



      Component B.



      Note that some mixture  components may act through multiple modes of action



      on different target tissues. It is conceivable that several RPFs may need to be



      developed to adequately address the risks posed by human exposures to the



      mixture.  Thus, the membership of component chemicals may differ across



      groups of RPFs and may also overlap.

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2.    RPFs may be developed to assess risk for a single route of exposure.  For
      example, the same mixture components may pose risk through inhalation and
      oral exposures. Based on differences in the relative toxicity of the components
      measured in inhalation and oral bioassays, different RPFs may be developed to
      address the human health risks following inhalation or oral exposures.
3.    RPFs may be developed to assess risks for different durations of exposure.
      The toxicity of a group of mixture components may change relative to each other
      depending on the duration and frequency of the exposures. Different RPFs may
      be developed to address the human health risks following different exposure
      frequencies or exposure durations (e.g., different RPFs may be developed for
      exposures that achieve steady-state tissue concentrations of mixture
      components than for those exposures that do not result in  steady-state tissue
      concentrations of the mixture components over the duration of the experiment).
4.    RPFs may be developed to assess risks within a restricted range of dose
      levels of the mixture's components. The toxicities of different chemicals
      relative to each other may change with dose. For example, at higher dose levels
      where significant adverse responses are observed,  an assumption of additivity
      may not be appropriate (i.e., observed effects may be greater than or less than
      those expected under an assumption of additivity).  Thus, it is appropriate to
      restrict the dose range of the components in two ways: limit the range to levels
      for which additivity is an appropriate assumption and, ensure the range reflects
      the exposure levels of interest to the risk assessment. Different RPFs may be
      developed to assess risks to humans for these different ranges.
                                      6

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These generalizations of the TEF methodology allow RPF development to be limited to



specific aspects of mixture toxicity and exposure, allowing the RPF approach to be



more broadly applied.



      An identified research need for the RPF methodology is continued development



of appropriate statistical methods to support the assumption of a common toxic mode of



action. One way to examine this assumption is to evaluate the similarity of the dose



response curves across the mixture's components.  Components with similar dose



response curves can be grouped together into a mode of action subclass for which an



RPF-based risk assessment can be developed.



      Chen et al.  (2001, 2003) present biostatistical approaches for grouping



chemicals suspected to have common modes of action, proposing two classification



algorithms to cluster chemicals into subclasses within which chemicals have similarly-



shaped dose-response functions. Chemicals within subclasses are combined using the



RPF method when a constant relative potency among chemicals exists.  Additional



methods are shown to calculate cumulative risks inclusive of these subclasses (i.e.,



combining across  subclasses for which a non-constant relative potency exists) using



either a joint dose-response approach or by integrating the concepts of dose addition



and response addition.



      Users of the RPF approach should appreciate that this model of mixtures toxicity



is actually a fairly simplistic depiction of the risk posed by the mixture.  Theoretically, the



number of mixture components that can be included in an RPF-based approach is



unlimited, as long  as each component is truly a toxicologic clone of the index chemical.



Pragmatically, there are a number of limitations including the availability of relevant

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toxicologic data upon which to base the RPFs. The Chen et al. biostatistical methods
were developed for pesticide mixtures.  Pesticide mixtures are unusual among
environmental mixtures because component toxicologic data are often available due to
the laws that govern U.S. pesticides. These approaches can likely be used on as many
as 30 or so individual components. The key limitations are having data describing the
dose-response function for each component and toxicologic evidence that each
component shares a common toxic mode-of-action. A statistical  issue is caused by the
toxic potency weighting of the exposure levels. If a poorly studied (high uncertainty)
chemical has high potency, its equivalent dose is high with no discounting for the
uncertainty. As the number of components increases, there may be an increased
likelihood of such a dominant uncertainty.  This emphasizes the need for careful
discussion of uncertainties: their sources and impact on the final  risk assessment.
      This report presents research results regarding the theoretical basis for RPF
based risk assessments and presents quantitative methods for applying RPFs. The two
basic assumptions of dose addition, common toxic modes of action and similarly-
shaped dose-response curves, are investigated and discussed.  Research results
produced by Chen et al. are presented, showing the integration of this research with
applications of the RPF approach.  This research was undertaken to continue exploring
and developing  cumulative risk assessment strategies beyond current applications and
is intended to improve  future applications of RPF based risk assessments.
                                       8

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                        2. DOSE ADDITION CONCEPTS



      U.S. EPA guidance documents on chemical mixtures risk assessment (U.S.

EPA, 1986, 2000) recommend no-interaction approaches underdose addition for the

risk assessment of mixtures of toxicologically similar chemicals. Assuming the

chemicals in a mixture are noninteractive and elicit a common response through similar

actions on a biological system, the chemicals are then assumed to act as if one is a

simple dilution or concentration of the index chemical, and, by extension, each other.

The joint action of the chemicals, then, can be described by "dose addition" (Finney,

1971).

      The fundamental assumption of dose addition is that the components of a

mixture exhibit a common toxic mode of action, underlying the addition of scaled doses.

Research issues include the development of meaningful toxicological criteria for

identifying a common toxic mode of action and the application of these criteria to

evaluate and identify mixture components that share a common toxic mode of action.3

      A theoretical consequence of this assumption is that the dose-response

functions of the components exhibit similar shapes. Theoretically, mixture components

sharing a common mode of action act as either concentrates or dilutions of each other.

The components interact with a common toxicological target, eliciting the same

response. Because the chemicals act as concentrates or dilutions of each other, the

number of organisms within a dose group responding to the same dose of different
      3There are other mixtures approaches that are based on dose additivity. The hazard index (U.S.
EPA, 2000), for example, provides a quantitative method that indicates whether a mixture may pose risk
or not. The hazard index method may be used when detailed toxicity data are not available; for example, a
hazard index can be developed from exposure estimates and Reference Doses.
                                       9

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chemicals should differ in a consistent manner across doses. The consistent



differences in the responses across dose groups will yield similarly shaped response



functions, sometimes referred to as "constant relative potency." For example, if



chemical 2 is one-half as toxic as chemical 1, then, at the same dose, chemical 2



should elicit a response in half as many test organisms as that dose of chemical 1.



This pattern should persist with  increases of dose until a maximum response is



achieved (e.g., 100% response). The similar shapes should also persist as doses are



diminished until a response threshold is observed or until one molecule of chemical 2



elicits an observable response.  Between the toxicity threshold and the response



maximum, similar shapes of the dose-response curves should hold.



      In practice, toxicological assays of chemicals having a common mode of action



may not exhibit similarly-shaped dose-response functions. Differences in the observed



dose-response function shapes between chemicals that share a common mode of



action may result from toxicokinetic differences or toxicodynamic differences. Other



factors could include differences in age or gender of the animals tested in the bioassay,



differences in animal stress status either within or across studies, and differences in



whether or not the test animals  were naive to the chemical prior to testing. Random



errors of response may also explain differences in shape. These random errors



describe, from a biostatistical perspective, the distance that an individual's response



may be from the population mean response at a given dose. These differences in the



observed dose-response functions may result in different maximal responses as well as



different thresholds of response within the exposed population.
                                      10

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      These differences in shape of the dose-response functions may preclude



application of a dose-additive model. If the dose-response functions exhibit different



shapes and the resulting risk estimate predicted by dose addition is quite different from



the expected joint mixture response, then scaling the toxicity of one chemical by that of



the other may be an inappropriate means of estimating a mixture's risk. The RPF



mixture risk model may be rejected under these circumstances, even if the chemicals



exhibit a common toxic mode of action.



      On the other hand, these differences in shape of the dose-response functions



may not preclude application of a dose-additive model. If the components exhibit



similarly shaped response functions over the relevant range of doses, as judged by the



exposure assessment, then the use of dose addition may be valid. This relevant range



includes the  range of exposures to the individual components and extends to the range



of the additive dose (i.e., the total mixture dose in units of the index chemical).



      Dose-response modeling research for dose addition includes assessing what is



meant statistically by a "similar shape" (see Section 5), including approaches to quantify



the amount of uncertainty potentially introduced  in the risk estimate when the slopes



are dissimilar.  Methods and criteria are needed to "determine" when a group of



components share a common dose-response function. To conclude, both common



mode of toxic action and similarity shaped dose-response functions are prerequisites



for valid application of dose addition to a chemical mixture.
                                      11

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                      3. RELATIVE POTENCY FACTORS








      U.S. EPA (2000) developed the RPF approach to assess risks posed by



mixtures that are comprised of chemical components exhibiting a common mode of



action for a toxic effect. The RPF approach is based on the concept of dose addition.



Mixture components are grouped using scientific judgment into subclasses called "RPF



Sets" using data on characteristics such as membership in a chemical class (relating to



observed toxicity), and commonality of toxicologic effects, exposure routes, exposure



durations, or dose ranges. To implement the approach, the exposure level of each



component of an RPF Set is scaled  by a measure of the component's toxicity relative to



a selected index chemical (a toxicologically well-studied component of the RPF Set).



This scaling factor, the RPF, is based on a comparison of the component's toxicity with



a similar measure of toxicity for the index chemical (e.g., a ratio of equally effective



doses of the component to the index chemical). The product of the measured



administered dose of each mixture component and its  RPF is defined as an Index



Chemical Equivalent Dose (ICED). The ICEDs of all the mixture components are



summed to express the total mixture dose in terms of an equivalent dose of the index



chemical. The risk posed by the  mixture is quantified by comparing a mixture's total



ICED to the dose-response assessment of the index chemical. [The mathematical



formulas for the RPF are detailed in Text Box 3-1.]
                                     12

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       Appropriate application of the RPF

method requires a judgment that the

mixture components share a common

mode of toxic action and evidence that

the components have similarly shaped

dose-response curves. Evidence that a

chemical class fulfills one of these

requirements does not necessarily imply

that the second requirement is fulfilled.

For the first assumption, the term,

Common Mode of Action, implies that

chemicals in a mixture exhibit  a common

toxicologic outcome when tested and that

the principal toxicodynamic events

leading to this common outcome after the

chemicals reach the target site and the

sequence of these events is understood,

but many of the details are not known.

Because detailed toxicodynamic data are

not abundant for most chemical mixtures

and their components, analysts typically
              Text Box 3-1

Mathematical Representations and RPF Formulas

d, = dose of chemical 1 present in a mixture (units
not specified)

d2 = dose of chemical 2 present in a mixture (units
not specified; must be consistent with those of d,)

pot, = potency estimate (e.g., a slope factor) for
chemical 1 (risk per unit of dose specified for d,)

pot2 = potency estimate (e.g., a slope factor) for
chemical 2 (risk per unit of dose specified for d2)

ICED = index chemical equivalent dose based on
relative potency estimates (units consistent with d,
and d2)

f,(*)=dose-response function of the index chemical
for the response(s) common to chemical 1 and
chemical 2 (units consistent with d, and d2)

h(d,,d2) = mixture risk from dose d, of chemical 1
and dose dj of chemical 2

[ED10], = dose of chemical 1 that results in a 10%
response, either as a fraction of exposed test animals
that respond, or as a fractional change in a measured
physiological value.

[ED10]2 = dose of chemical 2 that also results in the
same 10% response

Then, designating chemical 1 as the index chemical
in the RPF approach,
    (or equivalently = potj / pot,)

ICED = d, + (RPF2* d2)

h(d1,d2)=f1(ICED) = mixture risk from chemicals 1
and 2 evaluated at the ICED of chemical 1
must judge whether or not the mixture components exhibiting a common toxicologic

outcome also share a common mode of action. At times, the term Common Mechanism
                                           13

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of Action is used.  This term implies a greater degree of understanding of
toxicodynamic events, such that the chemicals in a mixture exhibit a common
toxicologic outcome when tested and that the underlying molecular and cellular
toxicodynamic events leading to this outcome are the same for each chemical, after
they reach the target site. (Toxicodynamic events include the initial interaction of a
toxicant with its molecular or cellular target and subsequent responses to the toxic
insult.) These two terms represent a continuum of toxicodynamic understanding; they
are degrees of scientific resolution. For RPFs, there must be a judgment that chemicals
exhibiting a common mode of action either do or do not share a common mechanism of
action. If judged that they do, then subclasses are not needed. If judged that they do
not, then subclasses should be developed and a second set of assumptions should be
identified and used to combine (or not combine) the toxicities that the subclasses
exhibit.
      The second prerequisite for applying an assumption of dose-addition is that the
chemicals have similarly shaped dose-response functions at least within the region of
exposure of interest for the risk assessment. An evaluation will often be needed of the
expected shapes of the dose-response functions in the low dose region including the
region that may lie below the lowest dose tested in the relevant toxicological bioassay.
In Section 5 of this report, we describe procedures that can be used to evaluate
similarity among the observable regions of dose-response functions.  If there is an
evaluation of shape below the experimental response region, it may include an
assessment of the mechanism/mode of action.
                                      14

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      RPFs are based on comparisons with an index chemical, and the mixture risk is

estimated using the dose-response function of the index chemical. Criteria pertaining to

the inclusion of compounds in an RPF Set apply to the index chemical. The index

chemical should be a well-studied member of the RPF Set; studies on the index

chemical need to provide exposure data for routes of interest and health assessment

data for health endpoints of interest.  To estimate relative potency, toxicity studies of

compounds in the RPF Set need to be comparable to studies conducted on the index

chemical.

3.1.   JUDGMENTS OF COMMON TOXICOLOGIC ACTION

      "Pesticides are determined to have a "common mechanism of toxicity" if they act

the same way in the body; that is, if scientifically reliable data demonstrate that upon

exposure to these chemicals, the same toxic effect occurs in or at the same organ or

tissue by essentially the same sequence of major biochemical events" (U.S. EPA,

2002b). The issue of a common mechanism of toxicity has been addressed by a

working group of experts convened by the International Life Sciences Institute Risk

Science Institute (Mileson et al., 1998).4 The working group presented three criteria to

describe a common mechanism of toxicity: (1) cause the same critical toxic effect; (2)

act on the same molecular target at the same target tissue; and (3) act by the same

biochemical mechanism of action or share a common toxic intermediate. The working

group agreed that all three points are useful to apply to chemicals that may act by a

common mechanism of toxicity, but did not state whether all three points must be met
      Subsequent to the International Life Sciences Institute expert panel, U.S. EPA issued a guidance
document for identifying pesticides with a common mechanism of toxicity (U.S. EPA, 2002b) and a CRA
case study for the organophosphorus pesticides (U.S. EPA, 2001 b).
                                      15

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before a firm common mechanism of toxicity determination can be reached. It is



recognized, however, that precise mechanistic information on animal or human effects



for pesticides and most environmental chemicals is scant. Common mechanism



determinations will therefore be difficult to establish with these three points because



chemicals often exhibit spectra of adverse effects rather than the same critical toxic



effect (criterion 1) (Mileson et al., 1998).



      Knowledge of a chemical's mode of action implies a general understanding of



the key toxicodynamic events that occur at a tissue level, rather than a detailed



description of the cascade of events at the cellular or molecular level such as is



suggested by the term "mechanism of action." For chemical mixtures, the term,



"common mode of action", implies that chemicals exhibit a common toxicologic



outcome in the same tissue when tested. However, the toxicodynamic events that lead



to this common outcome after the chemicals reach the target site are not well



understood; they may be the same (or similar) or not (it is simply not known). A



common mode of action is sufficient justification to consider or employ a dose additive



model. The terms "mode of action" and "mechanism of action" represent degrees of



scientific understanding of toxicodynamic events underlying observed toxic responses



rather than separate categories.



      The distinction between these two terms is discussed here using a hypothetical



cancer assessment to illustrate when dose additive models, such as RPFs, can be used



and when they should not be used. (RPFs are relatively simple mixture risk models



typically developed from empirical bases; as additional detailed toxicodynamic data are



generated for mixture components, these simple models are likely to be replaced by
                                      16

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biologically-based mixture risk models.) Tumors occurring in a specific liver tissue in an
animal bioassay may arise through a number of different modes of action. Consider two
examples.
Example Chemical 1:
      Repeated episodes of chemical-induced liver cell necrosis may result in random
      DMA replication errors as the surviving cells undergo compensatory
      reproduction. These random DMA replication  errors may occur in genes critical to
      control of cell replication (e.g., tumor suppressor genes and  proto-oncogenes)
      and become "fixed" in the genome through replication, ultimately giving rise to
      liver tumors.
Example Chemical 2:
      A mutagen  may interact directly with liver cell  DMA that codes for genes in the
      cell replication cycle and cause a mutation that gets fixed in  the DMA after a
      round of replication. Some of these mutations reduce the cells' ability to properly
      regulate their own replication and this lack of  replicative control  ultimately results
      in tumor development after a series of additional mutations and changes occur in
      the affected cells.
These two chemicals do not share a common mechanism of action because the first
induces carcinogenesis through necrosis and the second induces carcinogenesis
through mutation of the target tissue.
      Now, assume that two chemicals that comprise an environmental mixture both
cause necrosis in the same hepatic tissue when tested individually  in separate animal
bioassays. The ultimate result of the liver tissue necrosis that occurs when each
                                      17

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chemical is tested in a bioassay is the formation of observable liver tumors in the same
tissues (as in Example Chemical 1 in the preceding paragraph, the tumors form when
random errors in DNA replication occur in genes that control the cell cycle get "fixed"
during compensatory liver cell replication). The same bioassay outcome (i.e., liver
tumor formation arising from a specific tissue when each individual chemical is tested)
may occur through a number of different possible necrotic processes that lead to dead
liver cells:
1)    The chemicals may cause liver tissue cell necrosis by the same mechanism of
      toxic action. The chemicals may be shown to cause the same sequence of
      necrotic events in affected liver cells after the chemicals reach the target tissue.
      This is clearly a case of chemicals exhibiting a common mechanism of action.
      Lacking the level of mechanistic detail provided in this description, analysts could
      still logically conclude that the chemicals may share a common mode of action
      based on the occurrence of liver tumors arising in the same hepatic tissue.
2)    The chemicals may cause tissue necrosis in the same liver cells by different
      necrotic mechanisms  (i.e., either different toxicodynamic events or different
      sequences of toxicodynamic events that are observed to lead to cell
      death...ultimately resulting in tumor formation through random errors in
      compensatory replication in remaining living cells). In this case, one could
      reasonably judge that the chemicals still exhibit a common mechanism of action.
      Lacking the mechanistic detail, analysts could still logically judge that the
      chemicals share a common mode of action because of the  occurrence of liver
      tumors arising in the same hepatic tissues.
                                      18

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For the two cases above, RPFs for the chemicals could be developed (given

appropriate data).

3)     If the chemicals cause tumors in different types of cells of the same organ, then,

       based on evidence from animal bioassays, it is concluded that the chemicals do

       not share a common mode of action. Because they are not causing necrosis in

       the same types of cells, it could be concluded that the chemicals cause toxicity

       through different modes of action.This outcome could occur because of

       toxicokinetic differences between the chemicals, toxicodynamic differences

       between the chemicals, or both. In any case, it is not appropriate to use RPFs

       for the assessment of risk posed by this mixture, based on the available

       toxicodynamic information.5
      5ln practice, U.S. EPA (2000) suggests use of the Hazard Index (HI) method as an indicator of
risk when mixture components cause toxicity in the same target organ. In this case dose addition is
loosely defined to accomodate the lack of accessible mechanistic data.
                                       19

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         4. CHOICE OF DOSE METRIC IN CHARACTERIZING MIXTURE
                        TOXICITY BY DOSE ADDITION
      Two separate issues are discussed in this section. F:irst, the potential

significance of using kinetic data in the development of relative potency factors is

described. If kinetic differences exist between test species and humans, relative

potency factors will change when modeling risks on the basis of a administered dose

versus an internal dose. Second, a discussion is presented regarding choice of dose

metric in an RPF-based approach. In modeling human health risks posed by exposure

to a mixture by the RPF method, the type of dose measures employed do not appear to

alter the outcome of the risk estimation procedure.  Two dose measures commonly used

for delivered dose are units of mass (mg/kg) or moles (mmol/kg). The key is to be

consistent in development of an RPF application, using either mass measures or molar

measures.

4.1.   RPF DOSE ISSUES

      Measures of either an administered dose or internal dose may provide the basis

for estimation of relative potency for a chemical group. Administered or applied doses

are the amount of a substance applied to an external body barrier and available for

absorption. Administered doses include those doses applied to external body

membranes such as the gastrointestinal tract, the lungs and the skin. Internal doses

measure or estimate the quantity of a contaminant that is present in an internal tissue

(U.S.  EPA, 1992). The entire administered dose may not cross the barrier. Tissue

concentrations of interest could include those occurring at either toxicologic target

tissues and or tissues not targeted by the chemical.
                                     20

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4.1.1. Administered Dose. Most applications of RPFs in the literature are based on



measures of administered doses. For example, the EPA has developed four sets of



RPFs that estimate the toxicity of a mixture of related compounds based upon



administered dose measures for individual compounds: the dioxins, the polychlorinated



biphenyls, the polycyclic aromatic hydrocarbons, and the organophosphorus pesticides



(OP) (U.S. EPA, 2000, 2001 b). In each case, the risk estimates based on RPFs were



described as interim, pending the emergence of additional chemical mixture-specific



toxicokinetic and toxicodynamic data. The type of dose upon which the RPFs are based



will not alter the interim nature of the risk estimate. Ultimately, biologically-based



mixtures risk models will likely be developed for each case; these models will replace



the simpler RPF models and be based upon the emergence of additional chemical



mixture-specific toxicokinetic and toxicodynamic data.



4.1.2. Internal Dose. Measures or estimates of internal doses may provide an



improved basis both for estimating risks posed by chemical mixtures that occur through



multiple exposure routes and for estimating human health risks for some mixtures by



the same exposure route. To date, RPFs based on internal doses have not been



developed because the ability to predict internal organ or tissue doses through



physiologically-based pharmacokinetic (PBPK) models is relatively new or because,



given the simplistic assumptions of the RPF approach, refined estimates of dose would



provide little resolution to overall uncertainty.



4.1.3. Mixtures Exposures Through Multiple Exposure Routes. In 2002, U.S. EPA



completed a report showing that a multiple exposure route mixtures risk assessment



can be conducted based on internal dose estimates developed in both test animals and
                                      21

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humans for toxicants that do not cause portal of entry effects (U.S. EPA, 2002c). The
document combines exposure modeling results, PBPK modeling results, and the RPF
mixtures risk assessment approach. Human internal doses (e.g., blood, tissue, and
organ concentrations) were estimated using PBPK models, accounting for external
exposures from multiple routes (as dictated by the exposure scenario) and human PK
processes. Hypothetical RPFs were developed for a subset of chemicals based on test
animal data. Although the application of a full PBPK model was recognized as the
preferred approach to estimating rodent internal doses (i.e., blood concentrations), for
the example data used in the report, administered doses were assumed to be 100%
bioavailable to the rat. The rodent response data were assumed to be constant
between internal and external exposures and were used to evaluate the human dose-
response relationship. The use of internal dose measures (i.e., blood concentrations in
both humans and rodents) both for developing the RPFs based on rodent data and as
an indicator of human multi-route exposure provides a necessary and consistent basis
for extrapolating across species. Clearly, these approaches should not be used and are
inappropriate for toxicants that elicit responses at points of contact with the body (e.g.,
skin, intestinal tract, and nasopharyngeal, bronchial and lung epithelium).
4.1.4. Mixtures Exposures Through a Single Exposure Route in Different Species.
For some mixtures, basing RPFs on internal doses may reduce some uncertainty in
applying RPFs for individual exposure routes. From a single route of exposure  to a
given chemical mixture, the animal kinetics and human kinetics that give rise to
respective internal doses of the mixture components may result either in the same
internal doses or different internal doses, when the same amount of chemical is applied
                                      22

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 externally. If the kinetics result in the

 same internal doses or internal doses

 that differ consistently across the

 mixture (i.e., comparisons of the ratios

 of external to internal doses for each

 component between test animal and

 human are constant), then basing

 RPFs on internal dose estimates is not

 necessary, because the relative

 potencies will not change.  When

 kinetic differences between humans

 and test animals lead to non-constant

 differences in internal dose

 concentrations across a chemical class,

 then basing RPFs on internal doses

 provides a more scientifically sound

 basis for applying RPFs (see Text

 Boxes 4-1 and 4-2).

       Consider the same 2 component

 mixture  example presented in Text Box

3-1 where chemical 1 again serves as

the index chemical. Rodent data exist
               Text Box 4-1

Mathematical Representations and Formulas for RPF
Based on Internal Doses to Rats (test animal)
Let:
d, = exposure to chemical 1 as a result of its presence
in a mixture (units not specified)

d2 = exposure to chemical 2 as a result of its presence
in a mixture (units not specified; consistent with d,)

I, = internal dose of chemical 1 present in a mixture
(units not specified)

I2 = internal dose of chemical 2 present in a mixture
(units not specified; consistent with I,)

ICED, = index chemical equivalent dose based on
relative potency estimates (units consistent for I,, I2)

f,(*)=dose-response function of the index chemical
for response(s) common to chemicals 1 and 2 (units
consistent with I, and I2; they are based on internal
measures of dose but use the same response
measures as developed in Text Box 3-1)

h(I,,I2) = mixture hazard or risk from joint exposure
of dose d, to chemical 1  and dose d2 to chemical 2;
however, these doses are based on internal measures
I, and I2 rather than administered doses d, and d2.

[ED)0]j, = internal dose of chemical 1 that results in a
10% response, either as a fraction of exposed test
animals that respond,  or as a fractional change in a
measured physiological value.

[ED10]I2 = internal dose of chemical 2 that also
results in the same 10% response
Then, designating chemical 1 as the index chemical
in the internal dose based RPF approach,

RPF2I = [ED10]n / [ED10]I2

ICED, = I, + (RPF2* I2)

h(I1)I2)=f,(ICED,)
                                             23

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such that a RPF2 can be developed based

on the administered doses (RPF2E) or the

internal doses (RPF2I) of chemical 1 and

chemical 2. Response data are available

for the test animals only, and the human

mode of action is considered to be the

same as that in the test animal. The ratio,

RPF2E/RPF2I, in the rodent will either

approximately equal the ratio of human

administered dose to human internal dose

or not. If these ratios are equal, then, when

estimating risk using the RPF approach for

a single exposure route, it does not matter

whether external or internal doses are

used as the basis of the RPF. If these

ratios are not equal, then,  when estimating

risk using the RPF approach for a single

exposure route, it matters  whether external

or internal doses  are used as the basis of

the RPF. The RPFs should be  based on

internal doses because the
               Text Box 4-2
   Potential Use Of Internal Dose Based RPFs

Assume that the toxicodynamics are the same for
humans and rats. Let:
K1R= I,/d, in Rat for chemical 1
K2R= I2/d2,in Rat for chemical 2
K1H= I,/d, in Human for chemical 1
K2H= Vdzjin Human for chemical 2
From Text Box 3-1, the mixture risk in rats is
h(d,,d2)=f1(ICED)
This is based upon ICED = d, + (RPF2* d2), where
RPF2 = [ED10]1/[ED10]2.
An implicit assumption in the Chemical Mixture
Guidance is that RPF2 is the same in rodents and
humans. Thus, the human ICED for d2 is calculated
as the product of the human administered dose and
RPF2.
The risk posed to humans from this mixture is
estimated to be h(d1,d2)=f,(ICED),
where ICED = d, + (RPF2* d2) and_RPF2 = [ED10], /
[ED,0]2. The ratio of [ED10]S is calculated from the
rodent administered dose data.
Proposal:If K1R == K1H and K2R * K2H=» RPF2 is not
a valid estimate of the relative potency of chemical
2 for the human.
Proof:  Let chemical 2 be converted to chemical 1
on a 2 to 1 molar basis in the rat (i.e., 2 moles of
chemical 2 is converted thru some kinetic process
into 1 mole of chemical 1 in the rat). For an RPF
model, chemical 2 would be one-half as toxic
relative to chemical 1 based on the administered
doses  =» RPF2 = 0.5, when chemical 1 is the index
chemical and RPF( = 1.
Let the conversion of chemical 2 to chemical 1
cause toxicity of chemical 2 in the human also and
assume that the toxicodynamics of chemical 1 are
identical for humans and rats.
Because K2R * K2H, the conversion of chemical 2
into chemical 1 will not exhibit a 2 to 1 ratio, the
RPF2 estimated from rodent external data * the
human RPF2. The kinetic differences between
humans and rodents lead to different internal tissue
doses which influence the toxicity of chemical 2
relative to chemical 1.
Further Implication
If K1R/K1H = K2R/K2H => It is valid to apply RPF2
estimated from rat data to human administered dose
data due to kinetic differences. The kinetic
differences between species do not change the
relative potency of Chemical 2 to Chemical 1.
                                             24

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pharmacokinetic differences result in inaccuracies when the RPFs are developed in test



animals and applied to humans.



4.2.   CHOICE OF DOSE MEASURES



      For modeling of a mixture's toxicity or joint action under the assumptions of



relative potency factors, representation of dose as either a molecular (molar)



representation of dose or a representation by chemical mass does not matter in the



conduct of the risk assessment. It does not matter because the molecular weights of



the compounds relative to each other are constant.



      Consider two compounds, C1 and C2, that exhibit a common mechanism of toxic



action. Let the molecular weight of C2 be twice that of C1. Administration of 1 milligram



of C1  elicits the same response in test animals as administration of 2 milligrams of C2.



(Molecules of C1 and C2 are equally potent.) If the experimental evidence for RPFs is



based on single chemical experiments where dose is measured in milligrams, then the



relative potency of C2 to C1 will be 0.5. If the experimental evidence for RPFs is based



on single chemical experiments where dose is measured in moles, then the relative



potency of C2 to C1 will be 1. Because a molecule of C2 has twice the mass of C1, the



conversion of mass doses to molar doses in a  risk assessment will result in an RPF for



C2 of 0.5 (i.e., = Y2). Equivalent human exposures (resulting in the same predicted risk)



result from exposures to 1 mole of each chemical or some mass of C1 and  Yz the same



mass  of C2.  Thus, the chemical potency comparisons when applied to estimate human



risk will be the same regardless of whether the measures are based on moles or



masses.
                                     25

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   5.  BIOSTATISTICAL DOSE-RESPONSE MODELING FOR CUMULATIVE RISK








      Biostatistical modeling results can be integrated with exposures to calculate



cumulative risk estimates depending on expected toxicological action of the mixture



components. Three methods discussed in this section:



1)    Dose Addition: When the chemicals of interest act in accordance with a common



      mode of action, a dose addition approach can be employed.  Dose Addition is a



      chemical mixtures risk assessment method in which doses are summed (after



      scaling for relative potency) across chemicals that have a similar mode of action;



      risk is then estimated using the combined total dose.



2)    Integration of Dose Addition and Response Addition: When mixture components



      can be classified into subgroups within which a common mode of action exists,



      then, by definition, independence of toxic action is expected between subgroups.



      Response addition is a chemical mixtures risk assessment method applied to



      chemicals whose modes of action are independent of each other (i.e., the



      presence of one chemical in the body does not influence the effects caused by



      another chemical); risk of a whole body effect (e.g., non-specific cancer), is then



      estimated by summing the risks (e.g., skin cancer, liver cancer) of the individual



      chemicals. Integrating dose addition and response addition in this case means to



      estimate the subgroup risks and then sum them to estimate cumulative risks



      (U.S. EPA, 2002c).
                                     26

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3)    Joint Dose-Response Model: Finally, a joint dose-response model using scaled



      doses is applied when commonality of toxic mode of action is uncertain. This



      method produces a range of cumulative risk estimates.



EPA-sponsored research on the use of dose-addition in cumulative risk assessment,



focusing on the issue of similarly shaped component dose-response curves, has



resulted in the publication of two papers by Chen et al. (2001, 2003). The information



in this chapter relies heavily on the research presented in the Chen et al. papers, which



are reproduced in their entirety in Appendices A and B.  The first paper (Chen et al.,



2001) demonstrates methods for dichotomous data using the log probit and logistic



dose-response functions. The second paper (Chen et al., 2003) further extends the



statistical methods to continuous endpoints, using cholinesterase inhibition as an



example. To demonstrate use of these models in cumulative risk assessment, without



loss of generality, the discussions in this section are limited to dichotomous data using



the log probit dose-response function.



5.1.  DOSE-RESPONSE MODEL FOR COMBINED EXPOSURES



      To begin discussion of dose-addition as a tool for risk assessment, let F^ and F2



be the dose-response functions for chemical 1 and chemical 2, respectively. Under



dose addition, the response, R, to the combination of doses d, and d2 for chemicals 1



and 2, respectively, is




                      R(d1td2) = Ff(cft + p d2}= F^djp +c/2)                  (5-1)





where p is the relative potency of chemical 2 to chemical 1. When one chemical acts  as



if it is a simple dilution or concentration of the other, then the relative potency between



the two chemicals is constant. In other words, for all response levels, the effective dose
                                      27

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of one chemical is a constant multiple of the effective dose of the other chemical.


Hewlett and Plackett (1959) viewed the concept of dose addition (similar action) in a


slightly broader sense than requiring a constant relative potency between two


chemicals. Mathematically, their characterization can be interpreted as allowing the


relative  potency factor to be different for different response levels. Thus, the biological


bases and mathematical models required to characterize an RPF-based assessment


are different depending on whether or not constant relative potency is assumed.


      Dose addition allows for summing the individual doses into an equivalent dose in


terms of an index chemical and using the index chemical's dose-response function to


estimate the mixture response from the equivalent total mixture dose. A dose-response


function for binary response data, denoted P/of)= F, relates the probability of response


to the dose, d, of chemical i, where F is a probability distribution function. The general


model can be expressed in the logarithm of dose as


                                                                           (5-2)
A commonly used dose-response model, used throughout this disscussion to illustrate


the methods, is the probit function, which is,
                     P(d) = c + (1 - c)r09-=exp(- 1/2 f 2)dt                (5-3)
                                   J-»    V27T    V      '
where the parameter c represents background effect and P(d) is defined to be c when


d = 0. The parameters a and P are the intercept and slope parameters of the

dose-response function under its inverse, F~1(P(d)). For the rest of this discussion, the


log probit function for binary data will be used to demonstrate dose addition methods;
                                       28

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however, other functions and continuous endpoints can also be utilized in these


approaches (see Appendices A and B).


      For an example with two chemicals, if the relative potency of chemical 2 to


chemical 1  is constant, then the dose-response for one chemical can be expressed in


terms of the equivalent dose of the other chemical by using a relative potency factor.  In


this case, p = (di/ cfe) (i.e., Pi(cfi) = P2(di/p) = P^(pd2) = P2(cfe)), where the dose 0(1 of


chemical 1  and 0(2 of chemical 2 are equal effective doses (i.e., they cause the same


magnitude of response). Now, given that Pi(ch) = P^(d^/p}, then



             a, + # log^ = a2 + P2 log(
-------
Hence, because Equations 5-6 and 5-7 are both true, P1 =  (32 , and it can be shown that



two chemicals have a constant relative potency if and only if the slopes of the (log)



dose-response functions are equal. (See Appendix A for a more complete proof.)



5.1.1. Constant Relative Potency. The term, constant relative potency, implies that



for all response levels, the effective dose of one chemical is a constant multiple of the



effective dose of the other chemical. Constant relative potency is a desired condition to



conduct an RPF based risk assessment, at least for the dose ranges pertinent to the



exposure of interest (see Section 2).



      5.1.1.1. Dose Addition — If two chemicals have a constant relative potency



and if the joint response is dose-additive, then the dose-response function from



exposure to di of chemical 1 and cfe of chemical 2, using chemical 1 as the index



chemical  is,




                                                                          (5-8)
For a group of m chemicals in which the relative potency between any two chemicals is



constant, the joint response of the m chemicals can be derived in the same way as



Equation 5-8, using a relative potency factor pt for each component as it is paired with



the index chemical(s).
      F(dl,...dm)=F
as + 0  log ds
(5-9)
where, pt = exp[ (a, - as) / P ] for t * s. In this case, the estimated risk at any set of



doses does not depend on the choice of index chemical (i.e., when constant relative
                                      30

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potency is operational, the risk estimate will be the same regardless of the choice of

index chemical).

5.1.2. Nonconstant Relative Potency. Constant relative potency is a fairly restrictive

assumption that may not hold true for many mixtures. Thus, if the relative potency

between chemical 1 and chemical 2 is different for different response levels, then the

slopes of the dose-response functions for the two chemicals will be different and the

modes of action for the two chemicals may also differ. In this case, at the equal

effective doses of ofi for chemical 1 and cfefor chemical 2 such that Pi(ofi) = P2(of2), it

can be shown that the equivalent dose of chemical 2 in terms of chemical 1 is,
                                                                         (5-10)
and the equivalent dose of chemical 1 in terms of chemical 2 is,




                            n     ,„,                                     (5-11)
                           P2
Under these conditions, the joint response can still be estimated by an index chemical

approach, using doses adjusted by a ratio of the slopes. The joint dose-response from

an exposure to 0(1 of chemical 1 and cfe of chemical 2 in terms of chemical 1 as the

index chemical is,


                                                                         (5-12)
                                      31

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where w = P2 / p^ and p12 = exp[(a2 - a1 ) / PJ. On the other hand, the joint response in


terms of chemical 2 as the index chemical is,



       F(dl,d2)= P2(d2 + d2(dl))=F(a2 + p2 Iog(d2 + p2ld2/w))          (5-13)
where w = P2 / PL and p21 = exp^c^ - a2 ) / P2]. Note that the joint response predicted


from chemical 1 , Pi(ofi +c/i(cfe)), will differ from that predicted from chemical 2, P2(d2 +


c/2(c/i)). For m chemicals, the combined response in terms of chemical s can be derived


as,



                       (              }      (           ( m        }}
      F(dl,...,dm) = P\ ds + £ pstd»«   = F\as + & log I pad?-       (5-14)
The pst = exp[(a, - as) / PJ is a potency ratio of chemical t to the index chemical s, and


wst = Pt / ps, is the slope ratio, where t = 1	m, and t *s.


5.1.3. Constant and  Nonconstant Relative Potencies in the Same Mixture.  In


many cases, a mixture may be comprised of component subsets, where within each


subset a constant relative potency may exist (dose addition for common modes  of


action), but where nonconstant relative potencies occur between subsets (response


addition for independence of action between subsets). In this case, a set of m


chemicals can be clustered into several subclasses of constant relative potency. For


example, the set of six chemicals,


                           {{C1,C2,C3},{C4,C5},{C6}},


represents a set where the chemicals C1, C2, and C3 in the first subclass have


constant relative potency with respect to each other, as do the chemicals C4 and C5 in




                                      32

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the second subclass; the relative potency factor between the last chemical C6 and the



other chemicals is different at different response levels. Two approaches are proposed



here for evaluating a set of chemicals with varying relative potencies.



      5.1.3.1. Integrating Dose Addition and Response Addition — The first



approach is appropriate to apply when data on the toxic modes of action are available



so there is some certainty that the subclasses represent groups of chemicals with a



common mode of action distinctly different from the other subclasses. The toxicity



associated with each subclass is produced independently from the other subclasses.



The statistical method is then to estimate the dose-response function for the chemicals



within each subclass under dose addition, using a different index chemical from within



each subclass, and calculate the joint cumulative risk under response addition as the



sum of the subclass risk estimates.  Hence, the joint dose-response function is



expressed as,





F(dl,...dm) = P\d, + pnd2 + ..]+...+ Pcl[dq + pM+Irf,+1 + ...]+...Pm[
-------
selecting an index chemical from a separate subclass. Based on Equation 5-14

developed above, the joint response for m chemicals can be expressed in terms of a

single index chemical 1 for the entire mixture (i.e., including all subclasses) as,


F(dlt...dm) = P\(d, + p2J2 + ...)+-+(/₯/r/ + PMd£ + ..)+...+ pmd™]      (5-16)


The chemicals in the same subclass will have the same slope ratio w t = (Pt / p.,). Also,

the chemicals within the same subclass will have the same cumulative  risk estimate,

regardless of the choice of index chemical. (The complete derivation for Equation 5-16

can be found in Appendix A.) However, the estimated combined response will depend

on the subclass in which the index chemical is selected, a different subclass will predict

a different risk estimate. Thus a range of risk estimates can be produced, reflecting the

uncertainty in  the mode of action determinations.

5.2.   STATISTICAL ALGORITHMS FOR SUBCLASS GROUPINGS WITHIN A
      MIXTURE

      Two classification algorithms are proposed to cluster mixture components into

subclasses such that the chemicals in the same subclass have a common slope. The

joint response is estimated by fitting the dose-response model of the mixture under

dose addition. Chemicals within subclasses are first combined using simple dose

addition (constant relative potency), and then subclasses of chemicals are combined

using a general form of dose addition (non-constant relative potency).  Thus, the

proposed method allows one to estimate the joint toxic response for chemicals having

different dose-response slopes. (A complete example of the classification algorithms

and subsequent response calculations for six hypothetical pesticides in a mixture are

shown in Section 4 of Appendix B.)
                                      34

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      Since two chemicals have a constant relative potency if and only if the slopes of

the (log) dose-response functions are equal, the clustering algorithm is based on testing

for the equality of the slopes of dose-response functions. Either the likelihood ratio test

or the analysis of variance F test can be used for the comparison. (See Appendix B for

more information on these tests.) The clustering algorithms begin with a fitting of each

                                                      s*  s*.      *>
individual dose-response function for the m chemicals. Let, ft 15/? 2,...fi m denote the


maximum likelihood estimates of the dose-response functions. The estimates of the m

slopes can be arranged in an ascending order:


                              P c< P C2<.< P cm.


That is, the chemical ci has the smallest slope estimate, the chemical 02 has the second

smallest slope estimate, and so on. The classification algorithms are applied to this

ordered set. These iterative (stepwise) processes systematically test the adjacent

chemicals in an ordered set for equal slopes and end up with subclasses of chemicals

that can be characterized as having the same slope. The top-down approach begins

with the assumption that all of the slopes are different and uses an iterative process to

group chemicals with common slopes into subclasses; the bottom-up approach begins

with the assumption that all of the slopes are equal and uses an iterative process to

divide the chemicals into subclasses that have different slopes.

      In classical statistics, when the null hypothesis is rejected, this result does not

imply that the null is then true and can be accepted. For example, in the bottom-up

approach, the procedure keeps dividing the chemicals into RPF subclasses until  the

null hypothesis is not rejected. We complete the procedure when we can accept the
                                      35

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null hypothesis that dose-response slopes among chemicals are the same. In the top-



down approach, the procedure keeps grouping the chemicals until the null model is



rejected. We complete the procedure when we can reject the null hypothesis that dose-



response slopes among chemicals are the same. Thus, the top-down approach may



be a preferred method because there, the procedure is more consistent with traditional



statistics.  For this application, however, useful information is gained using either



approach regarding how different two or more curves might be, offering a quantitative



method to assess similarity in dose-response beyond the more typical visual check



using graphics.  Because we want to "travel up" the dose-response curve of the index



chemical to predict mixture risk, we need some comfort level that the dose-response



curves of the subclass chemicals share a common shape.



5.2.1. Top-Down Approach. In the top-down classification, the procedure begins



using an initial model  in which the slopes of the m chemicals are assumed to all be



different. Figure 5-1 illustrates the iterative procedure followed using the top-down



approach. (See also Table 3 in Appendix B for example calculations.)  The initial



model, MO, of chemicals is denoted by the partition set MO = {{C1}, {C2}, (C3), {C4},



{C5}, {C6}}. Consider the  null and alternative hypotheses, comparing two adjacent



slopes,



                     Hoq: Pc,q = Pc,q+i  versus   Haq:  pc,q * (3c,q+1               (5-17)



for q = 1, 2 ,..., m-1. Under the  null hypothesis, a joint dose-response function can be



fit for the mixture of chemicals Cq and Cq+1, using a constant relative potency model,



based on Equation (5-8) of,
                                      36

-------
                                IT)

                                UJ
                                a:
                                       o
                                       CO

                                       2
                                       0.
                                       Q.
I
Q
                                      -e
                                       CD
                                      JI
                                      O


                                      _o
                                      LL
37

-------
                                                                          (5-18)
The null model Hot? can be represented by the partition set,



      B1 = {{C1},... {Cq = Cq+1},... {Cm}}.                                     (5-19)



The hypothesis of comparing two adjacent slopes, equivalently, can be expressed in



terms of testing the two models, the null model (B1) vs. the initial model (MO):



Hoq: B1 = {{C1},... {Cq = Cq+1},... {Cm}}  versus Haq:  MO = {{C1}, {C2},..., {Cm}}.



Let poq be the p-value associated with the test H^ versus Haq, for q = 1, 2 ,..., m-1; and



let pcr= Max { pc1  pc2  pcm} (i.e., pcr is the largest p value associated with testing for a



common  slope between two adjacent chemicals in the set). When the largest value, pcr,



is less than a pre-specified significance level, say, ak then the procedure stops, we



reject the null model that the chemicals can  be further grouped, and the model MO that



the slopes of the m chemicals are different is concluded. On the other hand, if pcr is



greater than the significance level, then we cannot reject the null model, so the



chemicals Grand Cru are classified into one subclass. That is, a new "initial" model, M1



= {{C1},... {Cq = Cq+1},... {Cm}}, is formed and the procedure continues to the next step.



      Under the model M1, the two chemicals Grand 0-1 can be treated as one




chemical. Let,/?cr  denote the maximum likelihood estimate of the common slope for the





two chemicals Cr and cr*-i.  The m-1 slope estimates are now arranged in ascending



order as:
                                      38

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That is, the two individual slope estimates, /3cr and /?cr+I  are replaced by their common






slope estimate, &„ . The same algorithm is applied by performing m-2 comparisons of





two adjacent estimates. The hypothesis can be expressed as



                          Ho(?:B2   versus    Hag:M1



where



      B2   = {.... {Cq ,Cq+1},..., {Cr,Cr+1}, ...}  if q * (r  - 1)or q * (r + 1)








            = {..., {Cr ,Cr+1, Cr+2}, ...}          ifq = (r+1)



Again, if the largest p-value is less than the significance level, then the procedure stops,



the null hypothesis that the slopes are the same is rejected, and the model M1 is



concluded. If the largest p-value is greater than the significance level, then the null



model is  adopted as a new "initial" model and the procedure continues to the next step.



The procedure keeps grouping the chemicals until the null model is rejected. Note that



in the last step, if the null hypothesis is not rejected, then the model {C1, C2,...,Cm},



that all slopes are equal, is used for the risk assessment.



5.2.2. Bottom-Up Approach. In the bottom-up classification, the procedure starts with



the initial model, MO, where the slopes of the m chemicals are equal, denoted as the



partition set, MO = {C1, C2	Cm}. (The same notation  is  used to illustrate the



parallelism between the two classification schemes.) Figure 5-2 illustrates the iterative



procedure followed using the bottom-up approach. (See also Table 4 in Appendix B for



example  calculations.) We now form a new model B1 = {{C1,... Cq}, {Cq+1,..., Cm}}



constructed by the split of MO into two subclasses. Consider the hypothesis of a






                                       39

-------
                               LO

                               HI


                               =>

                               O
                                     o
                                     CD

                                     2
                                     0.
                                     Q.
Q.
Z)

E
o


CD
                                     -e
                                     CD

                                     O
40

-------
constant relative potency model MO against the alternative model B1 of two subclasses



of constant relative potency factors:



                          Hog:MO    versus   HagiBL



To test every possible combination of two subclasses while holding the order of the



slopes constant, there are (m-1) tests. Let pc, be the p-value associated with the test



Hog versus Hag, for q = 1,..., m-1, and let pcr= Min { pc1 p^  pcm} (i.e., pcr is the smallest



p value associated with testing for a common slope between two adjacent subclasses



of chemicals). If pcr is greater than a pre-specified significance level, say, ab, then the



procedure stops, and the initial model MO where all the slopes are the same is



accepted. On the other hand, if pcr is less than the significance level, then the



corresponding alternative model dividing the chemicals into two RPF groups, M1 =



{{C1,... Cq}, {Cq+1,..., Cm}} is accepted, and the procedure continues to the next step.



The algorithm repeats until a null model is accepted. Note that in the last step, if the null



hypothesis is rejected, then the model that all slopes are different is concluded, {{C1},



{C2}, {C3}, {C4}, {C5}, {C6}}.



      The two clustering schemes described above are tree structure classifications.



The top-down algorithm forms the tree from the top. It assumes that the slopes of the



chemicals are different. In each step, a chemical (or subclass of chemicals) is



combined with another chemicals (or subclass of chemicals) to form a new subclass.



Therefore, the number of subclasses at each step is one less than the previous step.



On the other hand, the bottom-up algorithm forms a tree in a division fashion. It



assumes that the slopes of the chemicals are equal. A new subclass is formed in each



step. These two algorithms may result in different tree structures. In both procedures,
                                      41

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a goodness-of-fit test (a global test) can be performed on the terminal tree against the


two trivial trees {{C1}, {C2}, {C3}, {C4}, {C5}, {C6}} and {{C1, C2, C3, C4, C5, C6}}.


5.3.   CUMULATIVE RISK ASSESSMENT


      The fitted dose-response model for the mixture from multiple chemical


exposures can be used for quantitative risk estimation in terms of the equivalent total


mixture dose of the index chemical. For a group of m chemicals in which the relative


potency factor between any two chemicals is constant, the estimated cumulative risk


from exposure to the specific doses d10,...,dm0, for chemicals 1,	m, respectively, is


derived as,


                        (           ( m      }}
      F(dwt...9dm0)  = F\ as -I- ft log £ pstdt0                            (5-20)
whereds,{l,pst are the maximum likelihood estimates of the model parameters, and




D = (psidlQ + ...,psmdm0J is the equivalent total mixture in terms of the index chemical
s, and pss = 1 .  The cumulative risk can be expressed as a response of the mixture


dose in terms of the dose-response function of the index chemical,


                  P(D) = F(as + ft  logrf)                                (5-21)



Using this equation, either the effective dose (EDpJ for a given response level p% or an


acceptable dose level D* corresponding to a given risk level rcan be computed (i.e.,


P(EDp) = p% or P(D*) = r). In general, when the relative potency factor is not constant,
                                      42

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the estimated cumulative risk will depend on the index chemical. In this case, the


average risk or the maximum risk over all possible index chemicals can be calculated.


5.3.1. Mixtures Reference Dose.  The equations developed using RPFs may be


useful in calculating a mixtures Reference Dose (RfD). The RfD is a "safe" level for


environmental pollutants, which represents a human exposure level below which


deleterious non-cancer effects are not expected to occur (U.S. EPA, 2003b). The RfD


has traditionally been derived by dividing an  experimental dose level, a No-Observed-


Adverse-Effect Level (NOAEL) or a Lowest-Observed-Adverse-Effect Level (LOAEL)


from an animal toxicity study by several uncertainty factors (UFs), and a modifying


factor (MF):


                         NOAEL or LOAEL
                  RfD =	                             (5-22)
                    J         UFsXMF                                V     '





An  alternative method is to replace the NOAEL or LOAEL by a modeled benchmark


dose (e.g., the lower 95% confidence limit on an ED10, that is, an effective dose that


produces a 10% response).  These UFs are  used to specifically account for uncertainty


in the RfD estimate due to extrapolations across species (UFA), within species (UFH),


across durations of exposure (UFS), between experimental dose levels (UFL) and from


weak to strong databases (MF). In the absence of statistical treatment, the default


value of these UFs has typically been set equal to 10. For a single chemical, a


benchmark dose (e.g., ED10) often serves as the point-of departure for low-dose


extrapolation in order to minimize model dependency at low dose levels.
                                     43

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      Using the mixture dose response models in this section, a mixtures reference



dose (RfDJ can be developed. For a mixture of components with the same mode of



action, Equation 5-21 can be used to calculate the point-of-departure for the mixture.



The RfDm in terms of an index chemical is defined as RfDm = £Dp/UFm; where EDp is



the mixture dose corresponding to a risk level of p% and UFm is the uncertainty factor



for the mixture. The UFm would need to consider all of the same UFs shown above for



the single chemicals RfD development. For given exposure doses, risks above the



RfDm can be calculated using an appropriate mixture dose response model (Wilkinson



et al., 2000). (A complete example of the this procedure is shown in Appendix A.)
                                     44

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                              6. CONCLUSIONS





      An important question in mixtures risk assessment research is how to assess a



mixture containing some chemicals that share a common toxic mode of action and



other chemicals that do not. Current additivity methods have evolved to handle either



the former (dose addition) or the latter (response addition). Alternatively, the risk



assessor may choose to do the assessment based on whole mixture data. The



biostatistical methods developed in this report provide alternative methods to evaluate a



mixture under three scenarios. The simple case occurs when there is certainty that a



common toxic mode of action is operating, so a dose addition approach can be applied.



The second case occurs when the mixtures can be divided into independent mode of



action subclasses; dose addition and response addition can be integrated to make the



assessment. The third case occurs when mode of action is uncertain, so a joint



dose-response modeling procedure is used to create a range of risk estimates.  Thus,



these approaches enrich the available library of mixture risk assessment methods



beyond what is currently published by the U.S. EPA (1986, 2000). Further, these



approaches are available if needed for the evaluation of additional pesticide mixtures



under FQPA. Finally, the results presented here are generalizable to assessments of



other environmental mixtures; the risk assessments that support environmental



regulations of important environmental mixtures such as dioxins, polychlorinated



biphenyls, and OPs are based on concepts of additivity (U.S. EPA, 1989b, 2000,



2001 b).



      The research results in this report can be applied to reduce uncertainties in



RPF-based risk assessments of chemical mixtures. These results also show how
                                     45

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mixtures risk assessments can be conducted using additivity concepts. Various sources

of uncertainty exist in most mixtures risk assessments, including uncertainties

addressed in this report regarding several factors:

      Common mode of action across mixture components (Sections 2, 3)

•     Similarly shaped dose-response curves across mixture components (Sections
      2,5)

      Value of internal vs. external dose estimates for developing RPFs (Section 4)

      Choice of dose metric (moles vs. mass) to use in a cumulative risk assessment
      (Section 4)

      Cross-species extrapolation of relative potency factors (Section 4)

      Estimating risks for a mixture with two or more common mode of action
      subclasses (Section 5).

      Biostatistical modeling in this report presents ways to combine dose-response

information, partitioning the mixtures into common mode of action subclasses. These

models can then be used to estimate risks for specific exposure scenarios or used to

develop toxicity values, such as a reference dose for the mixture.  Three RPF-based

methods are discussed, reflecting what is known or uncertain about the mixture

toxicology. These approaches can be applied using internal or external doses.

Development of approaches based on internal doses may reduce some toxicokinetic

uncertainties associated with RPFs based on administered doses. In the Chen et al.

papers (2001, 2003) in Appendices A and B, external doses were used to develop

statistical methods for grouping components into common mode of action subclasses.

The next step in this process is to use RPFs based on internal doses and compare

subclass groupings and modeling results with those developed using external doses.

Recommended future RPF research on pesticide mixtures is to:
                                      46

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1)    Develop kinetic models for pesticide mixtures in rodents.

2)    Using experimental cholinesterase inhibition measures, determine RPFs based
      on both external and internal dose estimates for the rodent.

3)    Determine if the RPFs based on internal dose estimates significantly differ from
      RPFs developed from external doses for the rodent.

4)    Apply the biostatistical methods for grouping by common dose-response curves
      using RPFs based on internal and external doses and compare the groupings
      that result.

5)    Develop kinetic models for pesticide mixtures in humans.

6)    Estimate human risks using rodent cholinesterase inhibition responses, RPFs
      based on rodent internal doses, and human  internal dose estimates using the
      three approaches presented in Chen et al. (2001, 2003), as appropriate.

7)    Compare subclass groupings and human risk estimates for all scenarios of
      internal and external RPFs.

8)    Evaluate the toxicity of different human exposure scenarios with the RPF models
      developed.

      This  research was undertaken to continue exploring and developing cumulative

risk assessment strategies based on dose addition  concepts beyond current

applications and is intended to improve future applications of RPF based risk

assessments.
                                      47

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                              7.  REFERENCES
Chen, J.J., Y.J. Chen, G. Rice et al. 2001.  Using dose addition to estimate cumulative
risks from exposures to multiple chemicals. Reg. Toxicol. Pharmol. 34(1):35-41.

Chen, J.J., Y.J. Chen, L.K. Teuschler et al.  2003. Cumulative risk assessment for
quantitative response data. Environmetrics. 14:339-353. (DOI:10.1002/env587)

Finney, D.J.  1971.  Probit Analysis, 3rd ed. Cambridge University Press, Cambridge,
England, p. 269-282.

Hewlett, P.S. and R.L. Plackett. 1959. A unified theory for quantal responses to
mixtures of drugs: Non-interactive action. Biometrics. 15:591-610.

Mileson, B.E., J.E. Chambers, W.L. Chen et al.  1998.  Common mechanism of toxicity:
A case study of organophosphorus pesticides. Toxicol. Sci. 41(1):8-10.

NRC (National Research Council). 1988. Complex mixtures:  Methods for in vivo
toxicity testing. National Academy Press, Washington, DC.

U.S. EPA. 1980. Comprehensive Environmental Response, Compensation, and
Liability Act (CERCLA or Superfund). 42 U.S.C. s/s 9601 et seq.

U.S. EPA. 1986. Guidelines for Health Risk Assessment of Chemical Mixtures.
Federal Register. 51(185):34014-34025.

U.S. EPA. 1989a. Risk Assessment Guidance for Superfund. Vol. 1. Human Health
Evaluation Manual (Part A). EPA/540/1-89/002.

U.S. EPA. 19896. Interim Procedures for Estimating Risks Associated with Exposures
to Mixtures of Chlorinated Dibenzo-p-dioxins and -dibenzofurans (CDDs and CDFs) and
1989 Update. Risk Assessment Forum.  EPA/625/3-89/016.

U.S. EPA. 1992.  Guidelines for Exposure Assessment. Federal Register
57(104):22888-22938.

U.S. EPA. 1996. Safe Drinking Water Act Amendments. National Drinking Water
Clearinghouse. Online, http://www.epa.gov/safewater/sdwa/text.html

U.S. EPA. 1997.  The Federal Insecticide, Fungicide, and Rodenticide Act (FIFRA) and
Federal Food, Drug, and Cosmetic Act (FFDCA) As Amended by the Food Quality
Protection Act (FQPA) of August 3,1996; U.S. Environmental Protection Agency, Office
of Pesticide Programs, document # 730L97001, March, 1997. Food Quality Protection
Act (FQPA) Public Law 104-170. Online, http://www.epa.gov/oppfead1/fgpa/
                                     48

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U.S. EPA. 2000. Supplementary Guidance for Conducting Health Risk Assessment of
Chemical Mixtures.  Office of Research and Development, Washington, DC.
EPA/630/R-00/002. Available in PDF format at: www.epa.gov/NCEA/raf/chem  mix.htm

U.S. EPA. 2001 a. National-Scale Air Toxics Assessment for 1996. Science Advisory
Board Preliminary Draft. EPA-453/R-01-003 Office of Air Quality Standards and
Planning, RTP, NC. Online, http://www.epa.qov/ttn/atw/nata/natsaov.html

U.S. EPA. 2001 b.  Preliminary Cumulative Risk Assessment for the Organophosphorus
Pesticides.  Office of Pesticide Programs, Washington, DC.

U.S. EPA. 2002a.  Framework for Cumulative Risk Assessment. EPA/630/P-02/001A.
Risk Assessment Forum, Washington, DC.

U.S. EPA. 2002b.  Guidance on Cumulative Risk Assessment of Pesticide Chemicals
That Have a Common Mechanism of Toxicity. OPP, Washington, DC. Online.
http://www.epa.gov/oppfead1/trac/science/cumulative guidance.pdf

U.S. EPA. 2002c.  The Feasibility of Performing Cumulative Risk Assessments for
Mixtures of Disinfection By-Products in Drinking Water.  EPA/600/R-03/051  Final Draft.

U.S. EPA. 2003a.  Regional/ORD Workshop on Cumulative Risk Assessment, Dallas,
TX, Nov. 4-8, 2002. Washington, DC.

U.S. EPA. 2003b.  Integrated Risk Information System. Office of Research and
Development, National Center for Environmental Assessment, Washington, DC.
Online, http://www.epa.gov/iris

Wilkinson, C.F., G.R. Christoph,  E. Julien et al. 2000.  Assessing the risks of
exposures to multiple chemicals with a common mechanism of toxicity: How to
cumulate? Reg. Toxicol. Pharmol. 31:30-43.
                                     49

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                     APPENDIX A

                  CHEN ET AL., 2001
           USING DOSE ADDITION TO ESTIMATE
CUMULATIVE RISKS FROM EXPOSURES TO MULTIPLE CHEMICALS
                        A-1

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 Using Dose  Addition to Estimate Cumulative Risks from
                 Exposures  to  Multiple Chemicals


         James J. Chen1, Yi-Ju Chen1, Glenn Rice2, Linda K. Teuschler2,
            Karen Hamernik3, Alberto Protzel3, and Ralph L. Kodell1

                  1 Division of Biometry and Risk Assessment
                   National Center for Toxicological Research
                      U.S. Food and Drug Administration
                           Jefferson, Arkansas 72079

                2  National Center for Environmental Assessment
                     U.S. Environmental Protection Agency
                            Cincinnati, Ohio 45268

                           3 Health Effects Division
                     U.S. Environmental Protection Agency
                            Washington, DC 20460
Send correspondence to:
*Dr. James J. Chen
Division of Biometry and Risk Assessment
NCTR/FDA/HFT-20
Jefferson, AR 72079
Tel:(870)-543-7007; Fax:(870)-543-7662; E-mail: jchen@nctr.fda.gov

* The views presented in this paper are those of the authors and do not necessarily represent those
  of the U.S. Food and Drug Administration or U.S. Environmental Protection Agency

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                                     SUMMARY

   The Food Quality Protection Act (FQPA) of 1996 requires the EPA to consider the cumulative
risk from  exposure to multiple chemicals that have a common mechanism of toxicity. Three
methods, hazard index (HI), point of departure index (PODI), and toxicity equivalence factor
(TEF), have commonly been considered to estimate the cumulative risk.  These methods are
based on estimates of EDio (point of departure) and reference doses from the dose response
functions of individual chemicals. They do not incorporate the actual dose response function of
the mixture from multiple chemical exposures. Dose addition is considered to be an appropriate
approach to cumulative risk assessment because it  assumes that the chemicals of interest act
in accordance with a common mode of action (a similar action). This paper proposes a formal
statistical procedure  to estimate the cumulative risk by fitting the dose response model of the
mixture under dose addition. The relative potency between two chemicals is estimated directly
from the joint dose response model of the mixture. An example data set of four drugs representing
four chemicals is used to illustrate the proposed procedure and compare it to the HI, PODI, and
TEF methods.

Key  Words: Chemical mixture; Low-dose extrapolation; Relative potency factor (RPF); Similar
action; Toxicity equivalence factor (TEF);

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                                    1. Introduction

   Regulatory agencies use risk assessment to derive acceptable levels of exposure to chemicals
that may exist as contaminants in food, drinking water, air, or the environment.  Risk assessment
for toxic agents  is usually conducted to evaluate the potential risks from exposure to a single
toxic agent through a single route of exposure. Although it is important to establish safe levels
of exposure for humans for each toxic agent,  people frequently are exposed to  many chemicals
simultaneously or in sequence by different routes.  The exposures to multiple  chemicals could
cause unexpected cumulative potential effects through  various media. The risks may combine
additively, multiplicatively or in some other fashion. The combined risk may be greater, or less
than what would be predicted from data on individual chemicals. Concerns about the problems
of multiple chemical exposure have been an important issue. The risk associated with exposure
to more than one toxic chemical by different routes may be characterized by cumulative exposure
and risk assessments.

   Assessing the cumulative toxicological effects of multiple chemicals has been addressed from
time to time (NRC, 1988; EPA, 1986, 1999a).  Methods and data, which can be used to estimate
the risk of exposures to multiple chemicals, have been developed over the years.  But there is no
consensus on appropriate statistical methods for cumulative risk assessments (CRA). The Food
Quality Protection Act (FQPA) of 1996 requires that, in future risk assessments, the EPA must
consider not only the risk of a single pesticide  chemical residue, but also the risk of exposures to
other pesticide residues and substances that have a common mechanism of toxicity. The FQPA
specifically focuses on available information concerning  the potential cumulative effects of such
exposures.

   The issue of a common mechanism of toxicity has recently been addressed by  a working group
of experts convened by the ILSI Risk Science Institute (RSI) (Mileson et al., 1998).  The working
group presented  three criteria to describe a common mechanism of toxicity:  1)  cause the same
critical toxic effect; 2) act on the same molecular target  at  the same target tissue; and 3) act by

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the same biochemical mechanism of action, or share a common toxic intermediate.  The working
group agreed that all three points are useful to apply to chemicals that may act by a common
mechanism of toxicity, but did not state whether all three points must be met before a firm com-
mon mechanism of toxicity determination can be reached.  It is recognized, however, that precise
mechanistic information on animal  or human effects of pesticide chemicals is scant.  Common
mechanism determinations will therefore be difficult to establish with these three points because
chemicals often exhibit  a different spectrum  of adverse effects in different organs and tissues
(Mileson et al. 1998).

   Wilkinson, et al., (2000) evaluated three methods of assessment of cumulative risk from ex-
posures to multiple chemicals: hazard index (HI), point of departure index (PODI), and toxicity
equivalence factor (TEF). They also considered two the other methods of assessment: the margin
of exposure  (MOE) and cumulative risk index (CRI) that are the reciprocals of the PODI and
HI approaches, respectively.  The approach of these methods is based on estimates of reference
doses or point-of-departure doses (e.g., EDio) from the fitted individual dose response functions.
There is no attempt to  incorporate the dose response function of the mixture from combined
exposures to multiple chemicals. In this paper, we propose a quantitative approach to estimating
the cumulative risk by directly fitting the dose-response function of the mixture through the dose
addition model.

   Under the assumption of a common mode of action (chemicals are non-interactive and act on
similar biological systems in eliciting a common response) for multiple chemicals, the chemicals
are commonly assumed  to act as if one is a simple dilution of the other.  The joint action  of
the chemicals, then, can be described by "dose addition" (Finney,  1971).  The assumption  of
addition of individual exposures (dose addition) to predict a cumulative toxic effect is reasonable
(Wilkinson,  et al., 2000). Furthermore, dose additivity is consonant with EPA policy that "pes-
ticide chemicals that cause related  pharmacological effects will be regarded, in the absence  of
evidence to the contrary, as having  an additive deleterious actions"  (CFR, 1998); also the EPA

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(1986) recommended no-interaction approaches of dose addition for risk assessment of mixtures.

   Let FI and F2 be the dose response functions for chemical 1 and  chemical 2, respectively.
Under dose-addition, the response to the combination of d\ and d2 for  chemical 1 and chemical
2, respectively, is
                               R(d1,d2)  =  Fl(dl+pd2)
where p is the relative potency of chemical 2 to chemical 1. When one chemical acts as if it is a
simple dilution of the other, then the relative potency between the two chemicals is constant. In
other words, for all response levels, the effective dose of one chemical is a constant multiple of
the effective dose of the other chemical.  Hewlett and Plackett (1959) viewed the concept of dose
addition  (similar action) in a slightly broader sense than requiring a constant relative potency
between  two chemicals.  Mathematically,  their characterization can be interpreted as allowing
the relative potency factor to be different  for different response levels.

   Dose  addition allows for summing the individual doses into an equivalent dose in terms of an
index chemical, and using the index chemical's dose-response function to estimate the response
from the equivalent total mixture dose.  Dose addition is considered to be an appropriate ap-
proach to cumulative risk assessment because it assumes that the  chemicals of interest act in
accordance with a common mechanism of toxicity. The main purpose  of this paper  is to pro-
pose an approach to calculating cumulative risk under the broader definition of dose addition in
which the relative potency is not constant (Hewlett and Plackett, 1959). The approach involves
estimating the relative potencies between chemicals from the joint dose response function of the
mixture through addition of the doses of individual compounds.

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                 2. Dose Response Model for Combined Exposures

   A dose response function for binary response data, denoted P(d) = F, relates the probability
of response to the dose, d, where F is a probability distribution function. The general model can
be expressed in the logarithm of dose as
                                                     d>0,
or in the un-transformed dose as
Two commonly used dose response models are the probit model and the logistic model. The
log-probit model is
                      P(d] = c+ (1 - c) r+ Qg  -^=exp(-l/2t2)dt
                                       J—oo     v2?r
and the log-logistic model is
                                 „ , /-,   ^ exp(a +/31ogd)
                                    v     'l + exp(a + £logd)'
where the parameter c represents background effect and P(d) is defined to be c when d = 0.
The parameters a and /3 are the intercept and slope of the dose response models under F-1(P(d)).

   Consider only two chemicals and denote the dose response functions for chemical 1  and
chemical 2 as
and
If the relative potency p of chemical 2 to chemical 1 is constant, then the dose response for one
chemical can be expressed in terms of the equivalent  dose of the other chemical, i.e., Pi(di) =
Pz(di/p) — P\(pd
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The above equality holds for all rfj. In particular, letting di = 1, the equation becomes cti =
a2 — falogp.  This implies logp = (a2 — ai)/fa. Similarly, If P2(d2) — Pi(pd2), then
                         a2 + $2logd2 = ai + /?ilogd2p, for all  d2.
It implies analogously that logp =  (a2 — a\)/P\.  Hence, fl\ = fa. Conversely, assume that the
slopes of the dose response functions are equal (f3\ = fa = /?). If Pi(di) = ^2(^2) then
                                ai + /31ogdi = a2 + /31ogd2.
The relative potency of chemical 2 to chemical 1 is logp = («2 — ai)//3. Thus, the relative potency
p between the two chemicals is constant for all dose (response)  levels. We have shown that two
chemicals have a constant relative potency if and only if the slopes of the (log) dose response
functions are equal.

   If the dose-response functions are modeled in terms of un-transformed doses instead of log
doses, then the  relative potency is constant if and only if the intercepts of the dose-response
functions are equal, where the relative potency is the  ratio of the slopes.  The remainder of this
paper will address only log-dose models.

2.1 Constant Relative Potency

   If two chemicals have a constant relative potency  and if the joint response is dose-additive,
then the dose-response function from exposure to d\ of chemical 1 and d2 of chemical 2 is
                          F(dl,d2)  =   Pl(dl+pd2}
For a group of m chemicals in which the relative potency between any two chemicals is constant,
the joint response of the m chemicals can be derived as
                                                 m
                       F(d1,---,dm]   =  PiCdi+
                                                t=2
                                                          i=2

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where  pt = exp[(at — Q.\)/f3\  is the relative potency of chemical t  to the index chemical  1,
t = 2, • • • , m.  The joint response can also be expressed in terms of  any  other chemical as an
index chemical s,
                                     = F(as + (3log(ds
where p't = exp[(at — as)//3}. It can be seen that p't = pt/ps for t — 1, -,m, where p\ = 1. The
two models are equivalent, i.e., the estimated risk  at any set of doses does not depend on the
choice of index chemical.

2.2 Non-Constant  Relative Potency

   If the relative potency factor  between chemical 1 and chemical 2 is different for different
response levels, then the slopes of the dose response functions for the two chemicals are different.
At the equal effective doses of d\ for chemical 1 and d2 for chemical 2 such that Pi(di) = P2(d2),
it can be shown that the equivalent dose of chemical 2 in terms of chemical 1 is
                                               i
and the equivalent dose of chemical 1 in terms of chemical 2 is
                               j fj \       fa\— a2\,/3
                               d2(di) = exp( — - - K
                                              P2
Under dose- addition, the joint response from an exposure to d'i of chemical 1 and d2 of chemical
2 in terms of chemical 1 is
where w = fal&\, and p12 = exp[(a2 — a\}/P\[. On the other hand, the joint response in terms
of chemical 2 is
                         F(di,d2)  = P2(d2(d!) + d2)

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where £21 = exp[(cci — 02) /Az]- Note that the joint response predicted from chemical 1, P\(d\ +
^1(^2)), wm differ from that predicted from chemical 2, P2(<^2(^i) + ^2))-
   For m chemicals, the combined response in terms of chemical s can be derived as
                                                         t+s
The pst = exp[(at — as)//3s]  is a potency ratio of chemical t to the index chemical 5,  and
           is the slope ratio, t = 1, • • • , m, and t ^ s.

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2.3 General Cases

   For a set of m chemicals, the chemicals can be clustered into several subclasses of constant
relative potency. For example, the set {{Cl, C2, ..}, {Cl, C(l+l)..}, .. {Cm}} represents that the
chemicals 1, 2, ..  in the first subclass have constant relative potency with respect to each other
as do the chemicals Cl, C(l+l) ..  in the second subclass; the relative potency factor between
the last chemical Cm and the other chemicals is different at different response levels. For this
example, the joint response in terms of chemical 1 is

      F(dlj • • • , dm)   =  PiUdj + P2d + ...) + ... + (pldT + pi-Kfili + -) + - + PmC")-

The chemicals in the same subclass will have the same slope ratio wt (= 0tlPi)- Also, the chemi-
cals in the same subclass will have the same cumulative risk estimate, regardless of which is used
as the index chemical.
                            3.  Cumulative Risk Estimation

   The fitted dose response model for the mixture from multiple chemical exposures can be used
for quantitative risk estimation in terms of the equivalent total mixture dose of the index chemical.
For a group of m chemicals in which the relative potency factor between any two chemicals is
constant,  the  estimated  cumulative  risk from exposure  to the specific doses dio,---,dmO for
chemicals 1, . . ., m, respectively, is
                                                      m
                         F(di0, • • • , dm0) = F(as + /31og(]T) (5stdto)),
                                                      t=i
where as, /3, pst are the maximum likelihood estimates of the model parameters, and D = psidiG+
• • ' iPsmdmo) is the equivalent total mixture in terms of the index chemical s, and pss = 1.  The
cumulative risk can be expressed as a response of the mixture dose in terms of the dose response
function of the index chemical
                                           10

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Using this equation, either the EDP for a given response level p% or an acceptable dose level D*
corresponding to a given risk level r can be computed, i.e., P(EDP) = p% or P(D*} = r.

   In general, when the relative potency factor is not constant, the estimated cumulative risk
will depend on the index chemical. In this case, the average risk or the maximum risk over all
possible index chemicals can be used.

   For a single chemical, a benchmark dose (e.g., ED10)  often serves as the so-called  point-of-
departure for low-dose extrapolation in order to minimize model dependency at low dose levels.
The above  equation can be used to calculate the point-of-departure. The reference dose for the
mixture in  terms of an index chemical is defined as
                                    Ref = EDP/GUF,

where EDP is the mixture dose corresponding to a risk level of p% and GUP is the group uncer-
tainty factor. For given exposure doses, the estimated risk unit with respective to the risk at the
reference dose can be calculated (Wilkinson, et al. 2000).


                   4. An Example for  Cumulative Risk Estimation

   A data set of four analgesics given by Finney (1971, Chapter 6, p 104) is used as an example to
illustrate the proposed procedure. These represent typical toxicological data obtained from dose
response experiments. The four analgesics can be regarded as four chemicals having a common
mode of toxicity. The logistic dose response function is used in the analysis,
                           r>(j\      (i    \  exp(a + fllogd)
                                            1 + exp(a + (3logd}.

Table 1 contains the  maximum likelihood estimates with standard  error estimates in  paren-
theses and  the maximum value of the log-likelihood  (LL) of the fitted logistic  dose  response
                                           11

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function for the four chemicals.  The EDio, ED.io, and specific exposure  doses with corre-
sponding predicted risk of the four  chemicals are also listed in Table 1,  The EDio  and ED.10
are used later to describe the three cumulative risk assessment methods  presented by Wilkin-
son et al.  (2000).  We are interested in estimating the cumulative risk at the exposure doses
d10 = .005, d20 = .010, d30 = .005, d40 = .010. The sum of the four individual risks is 6.67 x 10~5.


The likelihood ratio (LR) test is used to test for the equality of the slopes. The LL value under
a common  slope model is -729.225. The LR %2 statistic under the null hypothesis is 2(729.225-
(209.358+157.447+139.797+221.716)] = 1.814. The x2 value shows no evidence of any differences
among the  four slopes.

   The data set of the four chemicals is fitted to the model of constant relative potency given
by
               0*2, 0*3,04) = C+ (1 -C)-
                                        exp(as
where psi is the relative potency factor of chemical t to the index chemical s.  Table 2 contains
the maximum likelihood estimates with standard error estimates of the coefficients of the dose
response function, the equivalent exposure dose D  with the predicted cumulative risk, and the
EDio and ED.i0 using four different index chemicals (s — 1,2. 3,4).  Note that  pa,pf>, and pc are
the estimates of the relative potency factors between chemicals relative to the index chemical. For
example, when s = 1, then pa = p12, Pb = pis, pc = Pu-  The maximum likelihood estimates of
the model parameters are c = .056,6; = —2.605,4 — 1-90, pi2 = 1.26, pis = 3.61, and pu = 0.34.
The total mixture dose is D = .005 + 1.26 x .010 + 3.61 x  .005 + 0.34 x .010 =  0.0391. The
predicted cumulative risk is  1.47 x 10~4.  The predicted risk can be computed using a different
index chemical. Table 2 shows that risk estimate is the same regardless of which chemical is se-
lected as the index chemical.  For a convex dose response function, the estimated (low dose) risk
based on simply summing the individual risks (6.67x 10~5 shown in Table 1) will underestimate
the cumulative risk through dose addition (1.47  x  10~4 shown in Table 2) under  a model of a
                                           12

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common mode of action.

   Alternatively, using the ED\Q as the point-of-departure, the reference dose for the mixture
in terms of the index chemical 1 can be calculated by

                                   Ref= 1.2820/GUF.

If GUF = 50, then Ref = 1.2820/50 - 0.026. This value is smaller than the mixture dose 0.039.
Similarly, the reference doses for the mixture in terms of other index chemicals 2, 3,  and 4 are
0.020, 0.007, and 0.076 respectively.  These values are  smaller than their corresponding mixture
doses shown in Table 2.
   For illustration purposes, assume that the  relative potency factors between chemicals 1, 3,
and 4 (with each other) are constant, and the relative potency factors between chemical 2 and
chemicals 1, 3, and 4 are different. The four chemicals  are grouped into two subclasses { {1,3,4},
{2} }.  If chemical 1 is used as the index chemical  (to represent the subset {1,3,4}), then  the
joint dose response function is
          p/ ,   ,  ,   , x     ,(,    ,   exp(ai
         P(di,d2, d3, d4) = c+(l- c) — - -.
                                    1 + exp(a
If chemical 2 is used as the index chemical, then dose  response function becomes
              ,   ,  ,x      ,       ^  exp(a2  + /?2log(d2
Table 3 contains the maximum likelihood estimates of the model parameters.  Table 3 shows
that chemicals 1, 3, and 4 give the same predicted risk (1.39  x 10~4). But the cumulative risk
predicted by chemical 2 is  1.75 x  10~4.  The estimated slope ratio between the chemical 2 to
chemical 1  (or 3, 4) is w =  1.12 = 1/.89.

                                     5. Discussion

   Wilkinson et al.  (2000) described the three methods, HI, PODI, and TEF, of cumulative
risk assessment based on the estimates of the EDio and reference doses of individual chemicals.
                                           13

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For given exposure doses, the risk unit estimate can be obtained by multiplying an uncertainty
factor (UF) for the chemical.  In the present context, using the EDi0 as the POD, the risk unit
for the three methods is
         HI   =  UF x (0.005/0.9110 + 0.010/1.4214 + 0.005/0.3397 + 0.010/2.9082)
              =  UF x 0.031 (= PODI = TEF),
where UF is the common uncertainty factor for the four chemicals. The risk unit estimate is 0.31
when UF=40,  and it is 3.10 when UF = 100. The ED.io can also be used as POD to calculate
the HI, PODI, and TEF but with UF = 1, the risk unit is HI (0.005/0.0462 + 0.010/0.1855 +
0.005/0.0311 + 0.010/0.1733) = 0.380 (= PODI = TEF).
   The ED.io can also be used as the reference dose in the TEF method as in the context of
the Wilkinson et al.  (2000) examples. When the EDio is used as POD and ED.jo as reference
dose, applying the TEF method to estimate the risk unit will depend on the choice of the index
chemicals. For example, the risk unit estimate for TEF method in terms of the index chemical
1 is
     0.9110 x (0.005/0.9110 + 0.010/1.4214 + 0.005/0.3397 + 0.010/2.9082)             4
                                  00462                                6.05x10  .
In the  same way, the calculated risk units are 5.15 x 10~4, 3.34  x 10~4, and 2.35  x  10~4 for
chemicals 2, 3, and 4 as the index chemical. The risk predicted from the proposed dose-addition
model  given in Table 2 is 1.47  x  10~4 irrespective of which chemical is  selected as the index
chemical.

   The HI, PODI, and TEF methods all assume that the dose response functions for the chem-
icals considered have a similar slope.  The relative potency factors among  chemicals are often
based on a particular effective dose  EDP (e.g., ED10) of individual dose response functions. In
this approach, the relative potency estimate will depend on the choice of the particular effective
dose if the slopes are not estimated to be equal. For example, the relative potency between
chemical 3 and chemical  1 is  0.9110/0.3397 =  2.68 based on ED10, and  it is 0.0462/0.0311  =
                                          14

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1.49 based on ED.io (Table  1). Because the proposed approach takes into account a common
slope in fitting the joint dose response function, the estimate of the relative potency for a subset
of chemicals that have the same slope is invariant to the choice of effective doses or the index
chemicals. In Table 2, for example, the estimated relative potency of chemical 3 to chemical 1 is
3.61. This value can be also  computed from the ratio of the ED10 or ED.io of the chemical 1 to
chemical 3.

   The proposed approach of fitting a single joint dose-response function to the dose response
data (from all chemicals) is  consistent with the current approach  to a single chemical risk as-
sessment. The fitted dose-respopnse function can be used to estimate the cumulative risk for a
given set of exposure doses or  to derive a reference mixture dose from a benchmark dose from
the index chemical.  The proposed approach is similar to the TEF method.  But, unlike the
TEF method, the proposed method will give the same predicted risk regardless of the choice of
the index chemical under the constant relative potency model. Perhaps  most  importantly, the
proposed approach can be used when the relative potency factor differs for different subclasses
of chemicals.  This flexibility, which is based on a broader than usual concept of dose addition,
makes the procedure broadly applicable for estimating cumulative risk.


                                      References

CFR (1998).  Code of Federal Regulations, 40,  180.3, July 1, 1998.
EPA (1986).  Guidance for Health  Risk from Exposure to Chemical Mixtures.  U.S.  Envi-
     ronmental Protection Agency. Fed. Reg. 51, 34014.
EPA (1999a).   Guidance for Conducting Health Risk  Assessment of Chemical Mixtures.
     U.S. Environmental Protection Agency, Washington, D.C., April 1999. Unpublished
     draft document.
                                           15

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EPA  (1999b).  Proposed Guidance on Cumulative Risk Assessment of Pesticide  Chemi-
      cals That Have a Common Mechanism of Toxicity.  U.S. Environmental Protection
      Agency, Washington, D.C., August 1999.  Unpublished draft document.
Finney  (1971). Probit Analysis, Third Edition. Cambridge University Press, Cambridge.
Hewlett, P. S. and Plackett, R. L. (1959).  A unified theory for quantal responses to mix-
      tures of drugs: non-interactive action. Biometrics, 15, 591-610.
Mileson, B. E., Chambers,  J. E., Chen,  W. L., Dettbarn, W., Ehrich, M., Eldefrawi, A.
      T., Gaylor,  D. W., Hamernik, K., Hodgson,  E., Karc2mar, A., Padilla, S., Pope,
      C., Richardson, R. J., Saunders, D. R., Sheets, L. P., Sultatos, L. G., and Wallace,
      K. B. (1998).  Common mechanism of toxicity: A case study of organophosphorus
      pesticides. Toxicol.  Sci. 41, 8-20.
National Research Council (NRC) (1988). Complex Mixtures: Methods for in Vivo Toxicity
      Testing.  Natl. Acad. Press, Washington, D. C.
Wilkinson, C. F., Christoph, G. R., Julien, E., Kelley, J.  M.. Kronenberg, J., McCarthy,
      J., and Reiss, R. (2000).  Assessing the risks of exposures to multiple chemicals with
      a common mechanism of toxicity: How to cumulate? Reg.  Toxicol.  and Pharmacol.
      31, 30-43.
                                           16

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Table 1.  The maximum likelihood estimates (standard errors) of the coefficients of the logistic
dose-response model, and the estimated ED10, ED.io and maximized log-likelihood value for the
four chemicals.
Chemical c
I

2

3

4

Sum
0.00
(0.27)
0.12
(0.15)
0.00
(0.59)
0.00
(0.33)

a.
-2.05
(1.54)
-2.87
(1.31)
-0.07
(1.41)
-3.98
(2.59)

13 LL
1.58 -209.358
(0.71)
2.32 -157.447
(0.71)
1.97 -139.797
(0.88)
1.67 -221.716
(0.77)

ED 10 ED. 10 Exposure Pred.
0.9110 0.0462 0.005 2.98xlO~5

1.4214 0.1855 0.010 0.11 xlO~5

0.3397 0.0311 0.005 2.73xlQ-5

2.9082 0.1733 0.010 0.85xHT5

6.67xKT5
                                           17

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Table 2.  The maximum likelihood estimates (standard errors) of the coefficients of the joint dose
response function from multiple exposures of chemicals having constant relative potency factors.
              a      P      pa      pb      pc      D       Pred.        EDi0   ED
.10
 1   0.056  -2.605   1.90    1.26     3.61    0.34   0.0391   1.47x 10~4     1.2820  0.1071
     (0.09)  (0.67)  (0.30)   (0.39)   (0.03)   (0.13)
 2   0.056  -2.165   1.90    0.79     2.86    0.27   0.0310   1.47x 10~4     1.0170  0.0850
     (0.09)  (0.60)  (0.30)   (0.08)   (0.32)   (0.03)
 3   0.056  -0.167   1.90    0.28     0.35    0.09   0.0108   1.47x 10~4     0.3553  0.0297
     (0.09)  (0.32)  (0.30)   (0.03)   (0.01)   (0.04)
 4   0.056  -4.674   1.90    2.97     3.75    10.72  0.1160   ].47x 10~4     3.8090  0.3183
     (0.09)  (0.97)  (0.30)   (0.29)   (1.14)   (0.38)
                                             18

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Table 3.  The maximum likelihood estimates (standard errors)  of the coefficients of the joint
dose response function from multiple exposures of chemicals that do not have constant relative
potency factors.


 s     c      a      P      pa      Pb     PC       w       D        Pred.     ED10    ED
.10
 1   0.054  -2.535   1.84    1.08    3.64    0.34     1.12    0.0328  1.39x 10~4   1.2426  0.0958
     (0.10)  (0.67)  (0.31)  (0.41)   (0.03)   (0.28)   (0.20)
 3   0.054  -0.153   1.84    0.27    0.30    0.09     1.12    0.0090  1.39x 10~4   0.3405  0.0262
     (0.10)  (0.32)  (0.31)  (0.03)   (0.01)   (0.08)   (0.20)
 4   0.054  -4.541   1.84    2.97    3.21   10.80    1.12    0.0973  1.39x 10~4   3.6966  0.2849
     (0.10)  (0.98)  (0.31)  (0.29)   (1.18)   (0.84)   (0.20)

 2   0.054  -2.389   2.08    0.93    2.93    0.36     0.89    0.0505  1.75x 10~4   1.1297  0.1170
     (0.10)  (0.72)  (0.43)  (0.23)   (0.32)   (0.14)   (0.16)
                                             19

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Data set of four chemicals from Finney (1971)
Chemical
1
1
1
2
2
2
3
3
3
4
4
4
4
4
Dose
1.50
3.00
6.00
1.50
3.00
6.00
0.75
1.50
3.00
5.00
7.50
10.00
15.00
20.00
Response
19
53
83
14
54
81
31
54
80
13
27
32
55
44
Total
103
120
123
60
110
100
90
80
90
60
85
60
90
60
                      20

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                      APPENDIX B

                    CHEN ET AL, 2003
CUMULATIVE RISK ASSESSMENT FOR QUANTITATIVE RESPONSE DATA
                          B-1

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                  Cumulative Risk Assessment for
                     Quantitative Response  Data


         James J. Chen1, Yi-Ju Chen1, Linda K. Teuschler2, Glenn Rice2,
            Karen Hamernik3, Alberto Protzel3, and Ralph L. Kodell1

                  1 Division of Biometry and Risk Assessment
                   National Center for Toxicological Research
                      U.S. Food and Drug Administration
                           Jefferson, Arkansas 72079

                2 National Center for Environmental Assessment
                        Office of Research Development
                     U.S. Environmental Protection Agency
                            Cincinnati, Ohio 45268

                            3 Health Effects Division
                          Office of Pesticide Programs
                     U.S. Environmental Protection Agency
                             Washington, DC 20460
Send correspondence to:
*Dr. James J. Chen
Division of Biometry and Risk Assessment
NCTR/FDA/HFT-20
Jefferson, AR 72079
Tel:(870)-543-7007; Fax:(870)-543-7662; E-mail: jchen@nctr.fda.gov

* The views presented in this paper are those of the authors and do not necessarily represent those
  of the U.S. Food and Drug Administration or U.S. Environmental Protection Agency

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                                     SUMMARY

   The Relative Potency Factor approach (RPF) is used to normalize and combine different toxic
potencies among a group of chemicals selected for cumulative risk assessment. The RPF method
assumes that the slopes of the dose response  functions are all equal; but this method depends
on the choice of the index chemical, i.e., different index  chemicals will give different predicted
mean estimates.  This paper is part of an approach  to explore and develop cumulative risk
assessment strategies.  As part of this approach this paper proposes a procedure for cumulative
risk assessment from exposure to multiple chemicals that have a common mechanism of toxicity.
We propose two classification algorithms to cluster the chemicals into subclasses such that the
chemicals in the same subclass have a common slope. The joint response is estimated by fitting
the dose response model of the mixture under  dose addition. The proposed method will give the
same predicted mean response regardless of the selection of the index chemical for the chemicals
in the same subclass.  The proposed method  also allows  one to estimate the joint response for
chemicals having different slopes. An example data set of six hypothetical pesticide chemicals is
used to illustrate the proposed procedure.

Key Words: Chemical mixture; Classification  tree; Point  of departure (POD); Relative potency
factor (RPF); Similar  action.

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                                    1. Introduction

   Quantitative risk assessment is used to derive acceptable exposure levels or to estimate the
risks from exposure to chemicals that may exist as contaminants in food, drinking water, air, or
the environment. Estimation of potential risks for toxic agents is usually conducted on a single
toxin by a single route of exposure. However, people frequently are exposed to many chemicals
simultaneously or in sequence by different routes from different sources.  Exposures to multiple
chemicals could cause unexpected potential adverse effects through a variety of toxicological in-
teractions. Various chemical components may induce similar or dissimilar effects over time. The
Food Quality  Protection Act (FQPA)  of 1996 requires the Environmental Protection Agency
(EPA) to consider not only the risk of a single pesticide chemical residue but also the risk of
exposures to other pesticide residues and substances that have a common mechanism of toxic-
ity. The FQPA specifically  states the available information concerning the potential cumulative
effects of such exposures. The process of risk assessment of concurrent exposure by all relevant
routes for a  group of compounds that cause a common toxic effect by a common mechanism is
designated as cumulative risk assessment.

   The issue of determining a common mechanism of toxicity has been addressed by a working
group of experts convened by the International Life Sciences Institute (ILSI) Risk Science Insti-
tute (RSI) (Milesonet al., 1998). Subsequently, the EPA (EPA, 1999; http:/www.epa.gov/oppfeadl/
trac/science/)  has issued a guidance document for identifying pesticide chemical that have a com-
mon mechanism of toxicity.  Recently, the EPA issued the results of the revised cumulative risk
assessment for organophosphorus pesticides [http;//www.epa.gov/pesticides/cumulative/]. The
current paper is part of an approach to continue exploring and developing cumulative risk assess-
ment strategies. In this paper, we assume that common mechanism groups can be satisfactorily
determined.  In this context, a common mechanism group is defined as a group of pesticides
determined to  cause a common toxic effect by a common mechanism of toxicity. Such chemicals
are said to occupy the same "risk cup"  (EPA, 1999).

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   One important issue in cumulative risk assessment is how to incorporate the probability model
for estimating cumulative risk. Methods and data, which can be used to conduct risk assessment
of exposures to multiple chemical mixtures, have been developed over the years (NRC, 1988;
EPA, 1986,  1989, 1999). Because of complexity of evaluating multiple chemicals there are no
statistical methods for assessing risks from  multiple chemicals that can be routinely applied to
all chemical mixtures.  Methods for risk assessment of chemical mixtures fall into two general
approaches:  1) whole mixture of concern, and 2) component-based.  The whole mixture approach
involves either direct evaluation of the mixture of concern or an assessment of the mixture of
concern using data available on a sufficiently similar mixture. The component-based approach
considers the additive or interactive actions among the mixture components. The existing toxico-
logical database for pesticides contains data generated primarily to evaluate the hazard potential
of individual chemicals.  The most widely used component-based methods are dose addition and
response addition. Dose addition assumes  that the chemicals act on the same biological site,
similar biological systems and behave similarly in terms of the primary physiologic processes
(absorption, metabolism, distribution, elimination), and elicit a common response (EPA, 2000a).
Response addition assumes that the chemicals behave independently of one another, so that the
body's response to the first chemical is the same whether or not the second chemical is present;
in simplest terms, a response addition model is described by statistical independence. Given that
cumulative risk assessment will be based on the chemicals sharing a common toxic effect that
arises by a common mechanism of toxicity, dose addition is considered to be the most  appropriate
model to use for estimating cumulative risk.

   Dose-addition models presented in the literature are often in terms of a probability measure
(e.g., Finney, 1971; EPA, 1986). Let  FI  and F2 be the dose-response functions for chemical 1
and  chemical 2,  respectively. Under dose addition, the response to the combination of di and
d-2 for chemical 1 and chemical 2 is Fi(di + pd2) = F2(di/p + da),  where  p is the  relative po-
tency of chemical 2 to chemical 1.  FI and F2 are the probability of occurrence of a toxic  effect
for chemical 1 and chemical 2, respectively. The commonly used  models are the probit,  logis-

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tic, and multistage models. The data for a probability model are quantal responses measured
by the presence or absence of a toxic endpoint such as death.  Recently, Chen et al.  (2001)
applied the dose-addition model approach to estimating cumulative risk for quantal effects by
directly fitting the combined dose-response function for a set of chemicals in the same "risk cup".

   The toxic responses from exposures to pesticide residues often are measured by a continuous
quantitative value, such as altered blood concentration  or altered neurological function. In the
context of the FQPA, EPA has recently concluded that  the organophosphorus  pesticides act by
a common  mechanism of toxicity, which is manifested through inhibition of acetylcholinesterase
(Mileson et al., 1998). The common endpoints measured in cholinesterase bioassays are plasma,
red blood cell, and brain cholinesterase activity levels. In this paper, we develop a dose-addition
model for quantitative response data to estimate cumulative risk.

   Risk is customarily defined as the statistical probability of the occurrence of an adverse effect
at a  level of exposure. Dose-response models for adverse quantal response data are well defined
since an adverse effect is self-evident, that is, the occurrence of an  adverse effect is observed on
individual subjects empirically. By contrast, a clear-cut adverse effect for continuous quantitative
responses is difficult both to define and to observe unequivocally.  The characterization of risk
for continuous quantitative responses in terms of probability of occurrence does not naturally
follow. Methods for risk estimation of continuous quantitative response data for a single toxin
have been proposed  by many  authors (e.g., Crump,  1984;  Gaylor and Slikker, 1990; Chen and
Gaylor, 1992; Kodell and West,  1993; Chen et al.,  1996). Dose response  modeling of continuous
quantitative data for cumulative risk assessment has not been developed.  The main purpose of
this paper is to propose an approach to estimating the cumulative response and cumulative risk
of an adverse continuous quantitative effect for an individual concurrently exposed to pesticides
in a common mechanism group.
                  2. Dose Response Model for Combined Exposure

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   Let y(d] be a control-adjusted response variable (after a proper scaling or transformation,
if necessary)  of an individual exposed to a chemical at dose d.  The control-adjusted response
y(d) is calculated either by subtracting the  control mean from responses in the treatment groups
(difference scale) or by dividing the responses by the control mean  (ratio scale).  Assume that
y(d) has a normal distribution with mean E(y(d)) = /u(d) and variance a.2 (Note that we assume
a constant variance across dose groups of a chemical.) The mean response is often expressed as
a linear function of the natural logarithm of dose,
                                    fj,(d)  = a + (3 log d,
where a is the response for d = 1.  The parameters a and J3 are the intercept and slope of the
log-dose response function, respectively.

   Without loss of generality, suppose c is a critical value for an abnormally low level of response,
a level below which a response is considered to be atypical. For example,  c may be a certain
threshold  such as a 3 standard deviation reduction (difference)  from the control mean or 20%
reduction relative to the control mean. Under the difference scale, c can alternatively be expressed
as c — —ka, where k is appropriately chosen to yield  a specific low percentage point of the
distribution of unexposed individuals.  For exposure to a given dose d, the proportion of the
individuals with response y(d) below the critical value c = —ka is given by
                                 P(d)  =  P[y(d)
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value c. By expressing c in terms of k and a the probability of an adverse effect can be calculated.

2.1 Dose Addition  Model

   Consider  only  two chemicals and denote the dose response functions for chemical 1 and
chemical 2, respectively, as

and
If n\(di) = ^2(^2), the ratio of the equally effective doses pi2 = d\/d2 is called the relative
potency of chemical 2 to chemical 1.  Chen et  al.  (2001) showed that two chemicals have a
constant relative potency if and only if the slopes of the (log)  dose response functions are equal,
i.e., /?! = /32.  The combined mean response can be derived through addition of doses of chemical
1 and chemical 2  based on the relative potency factor.  Briefly,  under dose addition, if two
chemicals have a constant relative potency, then  the dose-response function from exposure  to di
of chemical 1 and d2 of chemical 2 is
If the relative potency factor between chemical 1 and chemical 2 is different for different response
levels, the joint  response from exposure to di of chemical 1 and d2 of chemical 2 in terms of
chemical 1 is
where ty12 = /?2/A> and /o12  = exp[(a2 - ai)//?i].  The cumulative response from exposure to
chemical 1 and chemical 2 can also be expressed in terms of chemical 2. However, if the relative
potency is not constant,  then the response predicted based on chemical 1 will differ from that

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predicted based on chemical 2.

   For a set of m chemicals, the chemicals can be clustered into several subclasses of constant
relative potency {81,82,..., Sk}, where each Si consists of chemicals having constant relative po-
tency. For example, the set {{ci, c2,..}, {cg, c9+i-.}, ..{c™}} represents that the chemicals GI, c2, ..
in the first subclass have constant relative potency with respect to each other as do the chemicals
cq, Cg+i,.. in the second subclass; the relative potency factor between the last chemical c™ and the
other chemicals is different at different response levels. For notation simplification, let c, = i. The
joint dose response function from exposure to the set of m chemicals {{1, 2,..}, {q, q + 1..}, --{m}}
in terms of chemical 1 (called  the index chemical) is
   /z(di, • • •, dm)   =  0i [fa + pl2di + ...) + ... + (piqd?1' + Pi(,+i)
-------
number of subjects in dose group i from chemical /, and g\ is the number of dose groups from
chemical /. Suppose y^\ is normally distributed with mean ni(du) and variance of. Estimation of
the mean and variance parameters for an individual chemical can be obtained by the maximum
likelihood method. The log-likelihood function for the chemical / is
where /J,i(du) = ai + fy log dn. The log- likelihood function for the m chemicals in terms of the
chemical s (the index chemical) is
                            " 1=1 i=l j=l

where ns(Du) = as + /3S log Dit and Dit = dis + YT^s Pstd™tst.

   Denote the maximum likelihood estimate (MLE) of as, /3S, pst, wst, and of as ds, (3S, p~st, wst,
and of, respectively.  If the  control-adjusted response y^i is measured on the difference scale,
then the estimated cumulative risk from exposure to the m chemicals in terms of the chemical s
can be derived from
                                  •  =,[-*-
If y^i is measured on the ratio scale, then the risk estimate is given by
In both cases, the estimated probability -P(-D) will depend on the standard deviation of a selected
index chemical.
                          3. Tree Classification Algorithms
   In this section, we propose two classification algorithms to cluster a group of chemicals into
subclasses of constant relative potency factors.  Since two chemicals have a constant relative
                                           9

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potency if and only if the slopes of the (log) dose response functions are equal, the clustering
algorithm is based on  testing for the equality of the slopes of dose response functions.  Either the
likelihood ratio test (LR) or the analysis of variance F test can be used for the comparison.

   The procedure begins with a fitting of each individual dose response function  for the m
chemicals. Let /3j, /32,  • • •, An denote the MLEs of the slopes of the dose response functions. The
estimates of the m slopes can be arranged in an ascending order:
                                                 — MCm-
That is, the chemical c\ has the smallest slope estimate, the chemical c2 has the second smallest
slope estimate, and so on.  Two tree classification algorithms, top-down and bottom-up, are
proposed.

   In the top-down classification, the procedure starts with the model that the slopes of the m
chemicals are all  different, denoted as MO = {{!}, {2}, • • • , {m}}.  Consider the hypothesis of
comparing two adjacent slopes,

                         H0g : &, = (3Cg+1   versus  Hag : (3Cq ^ /3Cq+1
for q = 1, 2, • • • , m — 1. Under the null hypothesis Hog, the dose addition model for the mixture
of chemical cq and chemical cq+\ is
The null model H0g can be represented by the partition set Bl= {{ci}, ...{cg,cg+1 }, ...{cm}}.
The hypothesis of comparing two adjacent slopes, equivalently, can be expressed in terms of
testing the two models:
   H0g :  Bl = {{d}, ..., {c9,c9+i},..., {cm}}  versus   Ha(? : MO = {{ci}, {c2}, • • • , {cm}}.
Let pCq  be the p- value associated with the test Hog versus Ha?, for q = 1, 2, • • • , m — 1; and let pCr
= Max  (pcnPc2i ' ' ' iPcm_i}- If Pcr is IGSS than a pre-specified significance level, say, at, then the
                                            10

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procedure stops, and the model MO that the slopes of the m chemicals are different is concluded.
On the other hand, if pCr is greater than  the significance level, then the chemicals  cr and cr+\
can be  classified into the one subclass. That is, the model Ml= {{ci},  ...,{ CT,CT+\ }, ...}  is
accepted, and the procedure continues to the next step.

   Under the model Ml, the two chemicals CT and cr+1 can be treated as one chemical.  Let J3'Cr
be the MLE of the common slope for the two chemicals CT and cr+\. The (m — 1) slope estimates
listed in the ascending order become
That is,  the two individual slope estimates /?Cr and /3Cr+1 are replaced by their common slope
estimate 0'^. The same algorithm is applied by performing m — 1 comparisons of two adjacent
estimates. The hypothesis can be expressed as

                              H0g : B2   versus  Hag :  Ml

where
           B2= {...,{cq, cg+1},...,{cr, cr+l}, ...},    if g + (r - 1) or q ± (r + 1)
              = {-,{CT-I, CT, cr+1},...},            if q  = (r - 1)
              = {...,{cr,cr+i,cr+2},  ...},             if q = (r + 1).
Again, if the largest p- value is less than the significance  level, then the procedure stops, and the
model Ml is concluded. If the largest p-value is greater  than the significance level, then the null
model is accepted and the procedure continues to the next step. Note that in the last step, if
the null hypothesis is not rejected, then the model {{ ci, c2, • • • , cm}} that all slopes are equal is
accepted.

   In the bottom-up classification,  the procedure starts with the model that  the slopes of the
m chemicals are equal, denoted as MO  = {{ci,c2, • • • ,cm}}. (We  use the same notations to
illustrate the parallelism between the  two classification schemes.)  Consider the model Bl  =

                                           11

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{{GI, ...cg}, {cg+1, • • •, Cm}} constructed by the split of MO into two subclasses, q = 1, 2, • • •, m—1.
Consider the hypothesis of a constant relative potency model MO against the alternative model
Bl of two subclasses of constant relative potency factors:

                             H0g : MO  versus   Haq : Bl.

There are  (m-1) tests.   Let pCq be the p-value  associated with the test  H09 versus  Hag,  for
q = 1, • • • ,m — 1, and let pCr = Min  {pcl,pC2, • • • ,Pcm_i}- If P^ [S greater  than a pre-specified
significance level, say,  a^ then the procedure  stops and the model MO is accepted.  On the
other hand, if p^ is less than the significance level, then the corresponding alternative model
Ml= {{CI...CT}, {cr+i, • • •, Cm}} is accepted, and the procedure continues to the next step. The
algorithm repeats until a null model is accepted. Note that in the last step, if the null hypothesis
is rejected, then the model that all slopes are different is concluded, {{ci}, {02}, • • •, {cm}}.

    The two clustering schemes described above are tree structure classifications. The top-down
algorithm forms the tree from the top. It assumes that the slopes of the chemicals are different.
In each step, a chemical (or subclass of chemicals) is combined with another chemicals (or sub-
class of chemicals) to  form a new subclass. Therefore, the number of subclasses at each step is
one less than the previous step. On the other hand, the bottom-up algorithm forms a tree in a
division fashion. It assumes  that the slopes of the chemicals are equal. A new subclass is formed
in each step. These two algorithms may result in different tree structures. In both procedures, a
goodness-of-fit test (a global test) can be performed on the terminal tree against the two trivial
trees {l,2,...,m}  and {{l,2,...,m}}.
                                     4.  An Example

   A data set consisting of a group of six chemicals was constructed for the example. The data
are the measures of different cholinesterase activity levels. These data represent typical endpoints
measured in a cholinesterase bioassay. Table  1 shows the sample size (n), mean response, and

                                            12

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standard error (S.E.) for each of the five dose groups of the six chemicals. The control means for
the six chemicals are 340, 345, 334, 304, 359, and 450. The Bartlett test indicates that a constant
variance model among dose groups is rejected for every chemical. Therefore, a natural logarithmic
transformation of the response is applied to achieve a constant variance. The constant variance
model for a given chemical appears to be adequate for the transformed data (Bartlett test).  The
transformed data are then adjusted by subtracting their respective control means.  The linear
dose-response function using the natural logarithm of the dose,
                                   /i(d) = a + /3 log(d).
is fit to the control-adjusted data for each chemical.  Table 2 contains the maximum likelihood
estimates (MLEs) with standard error estimates in parentheses and the log-likelihood  (LL) of
the fitted dose response function for each of the six chemicals.  The six slope estimates listed in
ascending order are given  as
      /?3(-0.289) < /34(-0.260) < /?5(-0.232) < ^(-0.221) < &(-0.212) < #$(-0.169).
The likelihood ratio test is used in the analysis. The significance level for the top-down approach
is set  to be at=0.25, and for the bottom-up approach is a.b = 0.05.

   In the first step of the top-down classification the different relative potency model M0=
{{3},{4},{5},{2},{1},{6}} is compared with each of the five models:  Blj ={{3,4},{5},{2},{1},
{6}},  B12={{3},{4,5},{2},{1},{6}}, B13={{3},{4},{5,2},{1},{6}}, B14={{3},{4},{5},{2,1},
{6}},  and B15={{3},{4},{5},{2},{1,6}}.  The model B13 gives the largest p-value 0.6048 (>
0.25). Therefore, the model Bis is used as the null model in the  next step,  and  the procedure
continues. Table 3  provides the details of the analysis in each step.  This procedure concludes
that the six chemicals are classified into three subclasses as {{3,4},{5,2,1},{6}}.

   In the first step  of the bottom-up classification, the constant relative potency model MO =
{{3,4,5,2,1,6}} is compared with each of the five models: Bli ={{3},{4,5,2,1,6}}, B12={{3,4},{5,
2,1,6}}, B13={{3,4,5},{2,1,6}}, B14={{3,4,5,2},{1,6}}, B15={{3,4,5,2,1},{6}}.  The details of
                                           13

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analysis in given in Table 4. The bottom-up classification comes to the same three subclasses of
constant relative potency model, {{3,4},{5,2,1},{6}}. In this example, the top-down algorithm
requires four steps, while the bottom-up algorithm requires three steps.

   The goodness-of-fit test can be performed using the likelihood ratio test to compare the termi-
nal tree {{3,4},{5, 2,1},{6}} against the trivial trees {{3},{4},{5},{2},{1},{6}} and {{3,4,5,2,1,6}}
for the top-town and bottom-up  procedures, respectively.  In the top-down  procedure,  the p-
value associated with the test of comparing the null model {{3,4},{5,2,1},6} and the alternative
model {{3},{4},{5},{2},{1},{6}} is  greater than the significance  level 0.05  (p-value=0.4984).
Therefore,  we conclude that  the model with the three subclasses  of constant relative potency
{{3,4},{5,2,1},{6}} is adequate. Similarly, the goodness-of-fit test indicates  a significant fit of
the same model with the p-value 0.00246 (< 0.05) by testing the null model  {{3,4,5,2,1,6}}
against the alternative model {{3,4},{5,2,1},{6}} for the bottom-up classification.

   Both classification algorithms  indicate that the six chemicals can be grouped into the three
subclasses  {3,4},{5,2,1},{6} of constant relative potency factors. The  data set of six chemicals
can be fitted based on  the three subclasses.  For example, if chemical 1 is  used as the index
chemical (to represent the subset  {5,2,1}), then the joint dose-response function is


Using chemical 2 or chemical  5 as an  index chemical, it will have a similar dose response function
and  give the same prediction at  given exposure levels.  If chemical 3 is  chosen as the index
chemical (to represent the subclass {3,4}), the joint dose-response function becomes
Finally if chemical 6 is used to be the index chemical, then the joint dose-response function is
given by
           , • • • , d6)  -  «6 + /36log(d6
                                            14

-------
   Table 5 (columns 1-11) contains the maximum likelihood estimates with standard errors of
the model parameters for the different index chemicals. The notations pa, pb, pc, pd, pe, wa, and
Wb denote the relative potency factors and slope ratios between chemicals relative to the index
chemical, and a is the standard deviation of the index chemical. Suppose we are interested in a
cumulative risk assessment at the exposure doses d\ = 0.030, d? = 0.035, d3 = 0.020, d± = 0.020,
d5 = 0.030,  and de = 0.002. The equivalent exposure dose D can be estimated using an index
chemical. For instance, when the index  chemical s — 1, the maximum likelihood estimates of
the coefficients are p\2 = 1-032, p\$ = 1.600, p\3 = 0.437, p\4 = 0.273, p\6 = 4.718, w13 = 1.215,
w\6 = 0.762, and al = 0.171. The total mixture dose is D = 0.030+1.032x0.035+1.600x0.020+
0.437 x (0.020)1-215 + 0.273 x  (0.030)1'215 + 4.718 x (0.002)0-762 = 0.2560.  The predicted mean
response is -0.4585 or, taking anti-logarithm, a 36.8% reduction of activity of cholinesterase. The
total mixture dose and predicted responses are shown in the last two columns of Table 5. It can
be seen that the chemicals 1,  2, and 5 give the same predicted mean response of  -0.4585 as do
the chemicals 3 and 4 (-0.5247).

   Alternatively, the combined response may be computed for each subclasses of chemicals with
the joint dose-response function being the sum of the three dose-response functions for the three
subclasses
    i , • • • , de)  =  [«! + /?ilog(d! + pi2d2 + pi5d5)] + [a3 + /%log(d3 + p^d^} + [a6 + /%logde)].
The mixture dose for the subclass {1,2,5} is D = 0.030 + 1.032 x 0.035 + 1.600 x 0.020 = 0.1141
with chemical 1 as index chemical. The predictive response is -0.2796. Similarly, the mixture
dose for the subclass {3,4}  is D = 0.3360 (chemical 3  as index chemical) with the predictive
response -0.2844, and the predictive response for chemical 6 is -0.0548. The estimated cumula-
tive response becomes (-0.2796)+(-0.2833)+(-0.0548)=-0.6177. This alternative  approach uses a
response addition to combine results from the dose-additive subclasses.
                                            15

-------
   The estimate of adverse probability based on the joint dose-response function from exposure
to the six chemicals can be calculated in terms  of the critical  value c — — k
-------
generally defined as a point estimate of the dose or exposure level that is used to depart from
the observed range of empirical response (or incidence) data for the purpose of extrapolation
(EPA, 2000b). In the case of a cumulative risk assessment, POD is a dose reflecting a uniform
response for the common toxic effect for each chemical.  The RPF is defined as the ratio of
the POD  of the index chemical to  that of each other chemical in the group.  The exposure
dose to each chemical is multiplied by the RPF to express all exposures  in terms of the index
chemical.  The summation of these values provides a total combined exposure dose expressed in
terms of the index chemical for prediction. In the present context, suppose that the estimated
dose corresponding to the predicted mean response of -0.05 is  defined to  be the POD, such
that a + ft log(POD)  = —0.05. For the given exposure doses considered in this example, Table 7
contains the estimate of POD, the total combined (equivalent) exposure dose, predicted response,
and  the cumulative risk  estimate for each index chemical.   For example,  the total exposure
normalized to the chemical 1 is

   dRpp  =  0.030(0.0327/0.0327) + 0.035(0.0327/0.0387) + 0.020(0.0327/0.1782)
          +  0.020(0.0327/0.1894) + 0.030(0.0327/0.0315) + 0.002(0.0327/0.0019)  - 0.1965.

The predicted mean response associated with the total combined exposure &RPF can be estimated
by using the individual dose response model of the index chemical 1
                            =  -0.774 + (-0.212) x log(0.1965) = -0.4290.

The cumulative risk estimates using the RPF method is calculated as

                         P(cW) = $ [-3 - ~"14?°1 = 0.3169.
Tables 6 and 7 show that the risk estimates obtained from the RPF method are,  on average,
slightly smaller than those obtained from the proposed model. On the other hand, the estimated
risk based on simply summing the individual probabilities (0.0155, second column of Table 6)
will heavily understate the risk estimated either from RPF method or from the proposed method,
                                           17

-------
as does the predicted mean response (-0.3001, shown in Table 2).  The alternative version of
the proposed approach, which combines dose addition and response addition, gives substantially
higher risk estimates than the version that employs dose addition only.

   The proposed dose addition model is similar to the RPF method. Although the RPF method
assumes that the dose response functions for all chemicals have a similar slope (a constant relative
potency), different index chemicals will give different predicted mean estimates. The proposed
method does incorporate the actual dose response function of the mixture from multiple chemical
exposures.  The method  allows one to estimate the joint response for the chemicals in a common
mechanism group but having  different relative  potency factors. ]t will give the same predicted
mean response  regardless of the selection of the index chemical for  the chemicals in the same
subclass, but risk estimates will depend on the  variance of the index  chemical.
                                           18

-------
                                      References

CFR  (1998). Code of Federal Regulations, 40, 180.3, July 1, 1998.
Chen, J. J., Chen, Y-J, Rice, G., Teuschler, L. K, Hamernik, K., Protzel, A., and Kodell,
     R. L. (2001). Using Dose Addition to Estimate Cumulative Risks from Exposures to
     Multiple Chemicals. Reg. Toxicol. and Pharmacoi, 34, 35-41.
Chen, J. J. and Gaylor, D. W. (1992).  Dose response modeling of quantitative response
     data for risk assessment. Commun. Stat. Theory Methods, 21, 2367-2381.
Chen, J. J., Kodell, R. L., and Gaylor, D. W. (1996). Risk assessment for nonquantal toxic
     effects.  Toxicology and Risk Assessment, 503-513.
Crump, K. S. (1984).  A new method for determining allowable daily intake.  Fundam.
     Appl. Toxicol., 4, 854-871.
Environmental Protection Agency (1986).  Guidance for Health Risk from Exposure to
     Chemical Mixtures.  Fed. Reg., 51, 34014.
Environmental Protection Agency (1999).  Guidance for identifying Pesticide  Chemicals
     and Other Substances that have a Common Mechanism of Toxicity. Fed.  Reg. 64,
     5795-5796.
Environmental Protection Agency (2000a). Supplementary Guidance for Conducting Health
                                                                                 *
     Risk Assessment of Chemical Mixtures. U.S. Environmental Protection Agency, Risk
     Assessment Forum.  EPA/630/R-00/002.
Environmental Protection Agency (2000b).  Proposed Guidance on Cumulative Risk As-
     sessment of Pesticide Chemicals That Have a Common Mechanism of Toxicity. U.S.
     Environmental Protection Agency, Washington, D.C., June 2000. Unpublished draft
     document.
Finney (1971).  Probit Analysis,  Third Edition. Cambridge University Press, Cambridge.
                                          19

-------
Gaylor. D. W. and W. L. Slikker (1990).  Risk assessment for neurotoxic effects.  Neuro-
     toxicology, 11, 211-218.
Hewlett, P. S. and Plackett, R. L.  (1959). A unified theory for quantal responses to mix-
     tures of drugs: non-interactive action. Biometrics, 15, 591-610.
Kodell, R. L. and R. W. West (1993). Upper confidence limits on excess risk for quantita-
     tive responses. Risk Anal, 13, 177-182.
Mileson, B. E., Chambers, J.  E., Chen, W. L., Dettbarn, W., Elhrich, M., Eldefrawi, A.
     T., Gaylor, D. W., Hamernik, K., Hodgson, E., Karczmar,  A.,  Padilla,  S., Pope,
     C., Richardson, R. J., Saunders, D. R., Sheets, L. P., Sulta,tos, L. G., and Wallace,
     K. B.  (1998). Common  mechanism of toxicity:  A case study of organophosphorus
     pesticides. Toxicol.  Sci., 41, 8-20.
National Research Council (NRC) (1988). Complex Mixtures: Methods for in Vivo Toxicity
     Testing.  Natl. Acad.  Press, Washington, D. C.
Wilkinson, C. F., Christoph, G. R., Julien, E., Kelley, J. M., Kronenberg, J., McCarthy,
     J., and Reiss, R. (2000).  Assessing the risks of exposures to multiple chemicals with
     a common mechanism of toxicity: How to cumulate? Reg. Toxicol. and  Pharmacol.,
     31, 30-43.
                                            20

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                Table 1: A hypothetical group of six chemicals*
Chemical Dose
1 control
0.02
2.3
22.5
213
2 control
0.017
1.7
17.0
177
3 control
0.05
2.0
19.0
205
n
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
Mean*
340.0
323.8
167.0
80.0
45.6
345.0
360.8
187.4
74.4
50.2
334.0
360.8
268.8
68.6
38.0
S.E.
6.97
15.19
11.63
3.41
1.63
15.20
26.58
12.92
1.50
3.80
7.52
8.05
8.71
1.99
2.77
Chemical Dose
4 control
0.03
1.1
15.0
168
5 control
0.019
1.3
13.8
189
6 control
0.01
0.1
10.8
250
n
10
10
10
10
10
5
5
5
5
5
5
5
5
5
5
Mean
304.0
382.3
266.4
87.6
44.4
359.0
299.8
220.0
92.6
35.0
450.0
301.6
264.2
104.6
59.6
S.E.
5.65
4.93
6.27
1.87
1.51
9.75
18.57
13.49
1.86
2.55
13.03
24.24
21.23
6.09
8.89
* Data represent hypothetical events for inhibition of the activity of the enzyme of
  cholinesterase in laboratory animals treated with increasing doses of six different
  chemicals.
f Mean activity of cholinesterase after dosing.
                                       21

-------
Table 2: The Maximum Likelihood Estimates (Standard Errors) of the coefficients of the
    individual dose response model, and Log-likelihood values for the six chemicals
Chemical
1

2

3

4

5

6

sum
a
-0.774
(0.041)
-0.768
(0.047)
-0.548
(0.080)
-0.483
(0.039)
-0.853
(0.072)
-1.104
(0.056)

0
-0.212
(0.011)
-0.221
(0.013)
-0.289
(0.023)
-0.260
(0.012)
-0.232
(0.020)
-0.169
(0.014)

a LL d
0.170 7.0439 0.030
(0.027)
0.201 3.6799 0.035
(0.032)
0.322 -5.6999 0.020
(0.051)
0.233 1.5159 0.020
(0.026)
.0.307 -4.7730 0.030
(0.049)
0.248 -0.4775 0.002
(0.039)
1.2891 0.1202
Pred(d)*
-0.0316

-0.0278

-0.0832

-0.0642

-0.0385

-0.0548

-0.3001
          * Pred(d) is the natural logarithm of the predicted response.
                                        22

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-------
Table 6: The estimated cumulative risks from the individual model and
   the two proposed dose addition approaches for the six chemicals
Individual model Dose- addition model
Chemical
1
2
5
3
4
6
sum
P(d)
0.0024
0.0021
0.0020
0.0031
0.0032
0.0027
0.0155
P(D]
0.3748
0.2396
0.0666
0.0961
0.2544
0.0645

Pa(D)
0.7368
0.5353
0.1616
0.1397
0.3679
0.3052


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Table 7: The estimates of the POD, total exposure dose
predicted response (Pred(d#pp)), and cumulative risk
         for the six chemicals from the RPF method
Chemical
1
2
3
4
5
6
POD
0.0327
0.0387
0.1782
0.1894
0.0315
0.0019
dRPF
0.1965
0.2324
1.0701
1.1370
0.1892
0.0114
Pred(d/jpF)
-0.4290
-0.4455
-0.5676
-0.5164
-0.4668
-0.3480
P(dHPF)
0.3169
0.2166
0.1079
0.2166
0.0695
0.0552

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