&EPA
T,
          Statistical Policy
ASA/EPA Conferences on
Interpretation of
Environmental Data
          I. Current Assessment^
          Combined Toxicant Effects
          May 5-6, 1986

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                                     PREFACE


    This volume is a compendium of the papers and commentaries that were presented at
the first of a series of conferences on interpretation of environmental data conducted by
the American Statistical Association and the  U. S. Environmental Protection Agency's
Statistical Policy Branch of the Office of Standards and Regulations/Office of Policy,
Planning, and Evaluation.

    The purpose of these conferences is to provide a forum in  which professionals from
the academic, private, and public sectors can exchange ideas on statistical problems that
confront EPA in its charge to protect the public and the environment through regulation
of  toxic exposures.  They  provide a unique  opportunity for  Agency  statisticians and
scientists to interact with their counterparts in the private sector.

    The holding  of a research  conference  and preparation of  papers  for  publication
requires the  efforts of many people.  Gratitude is expressed to the ASA Committee on
Statistics and  the  Environment  which was instrumental in developing  this  series of
conferences.   Thanks are also  owed to members of the ASA staff and,  particularly, Ede
Denenberg, who supported the entire effort.  Although there was no provision for a formal
peer  review,  thanks  are also  due  to  the reviewers  who assessed the articles for their
scientific merit and raised  questions  which were submitted  to  the authors  for  their
consideration.

     The views presented in this  conference are  those of the individual writers and should
not be construed as reflecting the official position of any agency or organization.

     Following the first  conference  on "Current  Assessment of  Combined Toxicant
Effects," in  May  1986,  a  second was held in October 1986  on "Statistical Issues  in
 Combining  Environmental Studies,"  from  which  a proceedings  volume  will  also be
published.   The  subject of  the  next conference, scheduled for May  1987,  will be
 "Sampling and Site Selection for Environmental Studies."


                                                   Emanuel Landau, Editor
                                                   American Public Health Association

                                                   Dorothy G. Wellington, Co-Editor
                                                   Environmental Protection Agency
                       U.S. Environmental Protection Agency
                       GLNPO Library Collection (PL-12J)
                       77 West Jackson Boulevard,
                       Chicago, IL  60604-3590

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                          INDEX OF AUTHORS   .


AHDERSOH, Perry  	  30   LITT, Bertram D	  44
BRODERIUS, Steven J	  45   MACHADO, S.G	  22
CHAMOY, Gail	   9   MARGOSCHES,  Elizabeth H	  83
CHEN, Chao  	  28   MOSKA, Carl  	  30
CHEN, J.J	  78   PATIL, G.P	  63
CHRISTENSEH, Erik R	  66   SHELTON, Dennis  	  30
FEDER, Paul 1	  19   TAILLIE, C	  63
HASS, B.S	  78   THORSLUND, Todd W	   9
HEFLICH,  R.H	  78   WEBER, Lavern J	  30
HERTZBERG, Richard C	  75   WYZGA, Ronald E	  84
KODELL,  Ralph L	   1   YINGER, Elizabeth   	  30

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                  TABLE OF CONTENTS

Preface                                                                     iii
Index of Authors                                                              iv
Modeling the Joint Action of Toxicants: Basic Concepts &.
Approaches. RALPH L. KODELL, National Center for lexicological Research           1
Use of the Multistage Model to Predict the Carcinogenic Response
Associated with Time-Dependent Exposures to Multiple Agents. TODD W.
THORSLUND, GAIL CHARNLEY, ICF Clement Associates                            9
Discussion.  PAUL 1. FEDER, Battelle Columbus Labs                               19
Assessment of Interaction in Long-Term Experiments.  S.G. MACHADO,
Science Applications International Corporation                                     22
Discussion.  CHAO W. CHEN, U.S. Environmental Protection Agency                  28
Concentration and Response Addition of Mixtures of Toxicants Using
Lethality, Growth, and Organ System Studies. LAVERN J. WEBER, PERRY
ANDERSON, CARL MUSKA, ELIZABETH YINGER, DENNIS SHELTON, Oregon
State University                                                              30
Discussion.  BERTRAM D. LITT, Office of Pesticides, U.S. Environmental
Protection Agency                                                            44
Joint Aquatic Toxicity of Chemical  Mixtures and Structure-Toxicity
Relationships.  STEVEN J. BRODERIUS, U.S. Environmental Protection
Agency, Environmental Research Laboratory, Duluth                               45
Discussion.  G.P.  PATIL, C. TAILLIE, Center for Statistical Ecology
 and Environmental Statistics, Pennsylvania State University                          63
Development of Models for Combined Toxicant Effects. ERIK R. CHRJSTENSEN,
University of Wisconsin-Milwaukee                                               66
 Discussion. RICHARD C. HERTZBERG, U.S. Environmental Protection Agency         75
 A Response-Additive Model for Assessing the Joint Action of Mixtures. J.J. CHEN,
 B.S. HASS,  R.H. HEFLICH, National Center for Toxicological Research                78
 Discussion. ELIZABETH H. MARGOSCHES, U.S. Environmental Protection Agency      83
 Statistical Directions to Assess Effects of Combined Toxicants. RONALD E. WYZGA,
 Electric Power Research Institute                                                84
 Appendix A: ASA/EPA Conference on Current Assessment of Combined Toxicant
 Effects Program                                                           89, 90
 Appendix B: Conference Participants                                            91

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                MODELING THE JOINT ACTION OF TOXICAHTS:  BASIC CONCEPTS AND APPROACHES

                      Ralph L. Kodell, National Center  for  Toxicological Research
Introduction
   The problem  of  modeling the  joint  action of
drags  and  environmental  toxicant*  ha*  aa*a  a
reanrgence  of   interact  recently,  dna to  a
heightened  awareness of  the  need to protect
haalth and   environment,  and  the  attendent
regulatory  consideration*.    The  assessment of
combined  toxicant effects falls  into the  general
framework of a  mixture problem.  There  is a  body
of  literature  that  deals  with finding  optimal
mixtures  of  various components through  the use of
response  surface  methodology  (Cornell,  1981).
This  approach  has  been  used  successfully, for
example,  to describe  the   effects  of  cancer
chemotherapy treatments (Carter e_t al_,  1984). In
general,  however,  the assessment of mixtures of
agents such as  drugs  and pesticides has tended  to
 follow a more  specialized approach (Kodell  and
 Pounds,   1985).   Most  current   efforts  to  study
 this  type   of  joint  action   are   based  on  the
 seminal  work  of  practitioners such as  Bliss
 (1939),  Gaddura  (1949),  Hewlett  and Plackett
 (1950), Finney  (1952), and Loewe (1953).  In drug
 development, the interest  lies both in enhancing
 efficacious  joint  effects and  in limiting   toxic
 joint  effects.   In pesticide  development, the
 interest  lies  in  enhancing  toxic  effects  to  a
 targeted  population,  while  limiting those   toxic
 effects to untargeted  populations.   This  is
 illustrated in Figure  1.    In addition,  it  is
 important  to  know  of  any inhibitory   effects  of
 one  beneficial drug or pesticide on another.
     Generally  speaking, in  modeling  the   joint
  toxic   action  of   agents   administered  in
  combination,  the  toxic endpoint produced  by
  individual agents  is  known,  and  the objective  is
  to determine whether the joint toxic  action   of
  two or  more agents is in some sense  "additive,*
.  as   opposed   to    being    "synergistlc"   or
  "antagonistic."   In addition  to  basic  research
  and   development   considerations,   this  has
  application in determining  acceptable  levels  of
  exposure  to  environmental   toxicants.   Various
  scientific  disciplines  are   involved,   including
  biostatistics,  pharmacology, toxicology and
  epidemiology.

  Joint Action Nomenclature
      In  looking into  the  problem of investigating
  the Joint action of  toxicants,  one  immediately
  senses  a  lack of  consistency  among investigators
  with  respect  to  the  nomenclature  used  to
  characterize  various types of  joint action.  For
   example,  some authors use  the term  "synergism"
   very  loosely to  describe  any enhanced joint
   effect, while others use a  term "potentiation" to
   describe  certain  types   of  enhancement  and
   synergism to  describe   others.     The  term
   "additivity*  Implies the absence  of  synergism to
   some,  but is  a  special  case  of  synergism to
   others.   Berenbaum (1977)  has  described  the
   inconsistent   terminology  surrounding  synergism
   quite  succinctly,  although a  bit  harshly:
   "Synergy, however, is  a topic on which confusion
   reigns.   The relevant pharmacological  literature
   is often  obscure (some papers,  indeed,  are models
   of incomprehensibility) and is profusely littered
 with technical terms that are  not  always  clearly
 defined.   Several  different  terms  are used  to
 dascrlb* the  came phenomenon  and  the  same  term
 Man* different thing* to different author*."
    Wail*  clearly there is  no consensus  with
 respect to  Joint  action nomenclature, there does
 seem to be  a tendency to classify various types
 of  joint   action  into either  of  two  broad
 categories,    namely,    "interactive"    and
 "noninteractive"  action.    Under  the  latter
• category,  the   concepts   of  "addition"  and
 "independence"  underlie  various null  models  of
 joint  action (Table  1).  To  the  pharmacologist

 TABLE  1. .Concepts a=d  nomenclature associated
           with the broad classifications of  non-
           interactive and interactive joint action.
               Noninteractive Action

     Addition                     Independence
  Concentration     Response      Response
                      Addition      Multiplication
  Similar Action
 Response
   Independence
                 Interactive Action

        Synergism	      	Antagonism—
      Potentiation

      Enhancement

      Supra-Addition•
Inhibition

Attenuation

Infra-Addition
   and   toxicologist,   the  concept  of   addition  or
   "additivity" can imply something about either the
   doses (concentrations) or  the  responses (effects)
   of   toxicants   acting   together.     To   the
   blostatistician, addition  of  doses  is  In  line
   with  the  concept  of  "similar  action,"  whereas
   addition of responses is  related to the notion of
   "independence" of action.   To  the  epidemiologist,
   the  concept of  additivity relates   only  to the
   responses of jointly  acting toxicants, and  stems
   from  the  notion of independence of action.  The
   epidemiologist    includes   the   concept   of
   "multiplication"  of  responses  as  a  form of
   noninteractive joint action, in the  sense  that it
   can  be  interpreted as a type  of  independence of
   action.   Table  2 gives a  cross-classification of
   basic concepts by  scientific disciplines.
      In  the  category  of Interactive   joint  action
   are  included the various departures  from additive
   and  Independent joint action.  These interactions
   are  often  classified as  either  "synergistic"  or
   "antagonistic,"  although  Increased  effects  are
   sometimes described as exhibiting "potentiation"
   or  "enhancement"  rather  then  synergism,  and
   decreased  effects  as  exhibiting  "inhibition"  or

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                           • -v< ' ...,.V f. ,  -',      • '       ,      •   '''     "         .        ' "'      '  ' "V"  '      "  "  ."
                         ^•~-^visu^^
 TABLE 2.  Concepts of noninteractive joint action,
          cf.egorized by scientific disciplines in
          which  they are used.  Cell entries
          represent terms or notions within each
          discipline chat are conmonly used to
          describe the concepts of noninteraction.
          An empty call implies that the discipline
          does not embrace the concept.
"X. Null
X.todel
Disciplines.
Toxicology/
Pharmacology
Epidemiology
Biostatistics
Concentra-
tion
Addition
Addicivity

Simple
Similar
Action
Response
Addition
Summation
Additivity
Uncondi-
tional
Independence
Response
Multipli-
cation

Multipli-
cation
Conditional
Independence
 "attenuation"  rather  than  antagonism  (Table  1).
 Numerous  other terms  have  been used to describe
 interactive  joint action,   including  supra-  and
 infra-addition,  super-  and sub-addition,  hyper-
 and    hypo-addition     and     hyper-     and
 hypo-multiplication.

 Null Models  for Noninteractive  Joint Action
    The primary focus of  this  paper will  be  on
 null  models  of  concentration  and   response
 additivlty  as applied  in a pharmacological /
 toxicological  context.  These models and concepts
 win be discussed initially.   Following this, a
.less-detailed discussion  of  the additive  and
 multiplicative models  of relative risk employed
 in epidemiology studies will be given.
    The basic  approach  to modeling  the' joint
 action of two (or more)  toxicants is  founded  on
 tolerance   distribution   theory.    That  is,
 individuals  are presumed to have  varying degrees
 of  tolerance to  a particular  toxicant,  thus
 implying    a   probability   distribution   of
 tolerances.   Dose-response  models are formulated
 without   attempting   to    Identify   specific
 underlying mechanisms of action of the toxicants
 under  study.    Pharmacological  foundations  for
 joint  action  studies are often  attributed  to
 Gaddum (1949) and  Loewe  (1953, 1957),  while
 biostatistical modeling  has  been  developed  by
 Bliss  (1939), Flnney (1952, 1971),  Hewlett  and
 Plackett  (1950,  1959),  Plackett and Hewlett
 (1948, 1967),  Hewlett (1969),  Ashford (1958) and
 Ashford and Smith (1965).    There has  been some
 attempt to formulate more refined models in terms
 of their biological  basis.   For  example, Ashford
 and  Cobby   (1974)  developed  a  class  of  joint
 action models based  on  receptor  theory  and  the
 law  of mass action,  following work  by Plackett
 and  Hewlett (1967) and citing  the early work of
 Gaddum  (1936).   This work was  followed—up  by
 Ashford  (1981).   Although there  has  been some
 application  of  this  theoretical  approach (e.g.
 Chou  and Talalay, 1983;  Svensgaard and Crofton,
 1985), virtually  all  practical investigations  of
joint  toxic action have  followed  the  tolerance
distribution approach.
   As   alluded   to   above,    generally    the
dose—response  models  that have  been*  formulated
for noninteractive joist action are based either
on  concentration addition  or  on  response
addition, or at least they include  theae types of
joint action as special eases.  Among  the author*
who have adopted  the concept  of  concentration
addition in modeling  noninteractive joint action
are  Smyth  vt^  al.  (1969),  Casarett   and   Doull
(1975), Piserchia and Shah (1976),  Berenbaum
(1977), Eby   (1981), and  Unkelbach  and  Wolf
(1984).     Among  those  who   have   modeled
noninteractive  joint action  on  the  basis  of
response  addition  are Webb  (1963),  Holtzman  et
al. (1979), Wahrendorf et al. (1981), Ozanne  and"
Mathieu  (1983) and  Machado  and  Bailey (1985).
Authors  who have  aodeled  on the  basis of  both
concentration addition and  response  addition
include  Broderiua  and Smith  (1979)',  Shelton  and
Weber (1981),  Chou and Talalay  (1983),  Kodell  and
Pounds  (1985), Christenaen and  Chen  (1985),  and
Chen  et al.   (1985).    The  terms  "concentration
addition" and  "response addition" were  introduced
by  Shelton and  Weber (1981).  Their  idea  of
response additivity is slightly more general than
its use in  this  paper.   Loewe  (1953)  used  the
terms  "iso-addition"   and  "hetero-addition"   to
describe  a  broad  concept  of  concentration
addition  and  a  narrow  concept  of  response
addition, respectively.  Steel and  Pecknan (1979)
introduced  the  notion  of  an   "envelope   of
additivity" that is bounded  by  Loewe's  iso-  and
hetero-additivity.

Concentration JVdditivity
   Some  of the  principles  and concepts  that
underlie  concentration  addition will  be   given
prior   to   presenting  a  formal  mathematical
definition.  Under the broad category of similar
action, Bliss  (1939),  Pinney (1971)  and Hewlett
and Plackett  (1959) all  expressed  the  principle
that two toxicants have the same site of primary
action, while  Ashford and Cobby (1974)  expressed
the principle  that both toxicants  act  at all  the
same sites.  Hewlett and Plackett (1959)  regarded
similar action as meaning  that  the physiological
effects leading to the response are additive.   In
this  sense  of additivity,  they allowed for
imperfect  correlation  of  tolerances  to  the  two
toxicants.  In the narrower  sense  of additivity
used   In  pharmacology,   the   tolerances   are
completely positively  correlated,  but apparently
one toxicant is not necessarily a simple  dilution
of the  other  (Hewlett and Plackett,   1959).   In
the  narrowest sense of  additivity  (similar
action)  is   the  concept  of   concentration
additivity (simple similar action)  (Bliss,   1939;
Finney,  1952;  Hewlett and  Plackett,   1959),   in
which one  toxicant is simply a dilution of the
other with  respect to  administrated dose.   This
concentration additivity is also characterized  by
the   perfect  positive   correlation   of   the
individual tolerances  to  the  two  toxicants
(Flnney, 1971;  Hewlett and Plackett, 1959).
   Let  P(dp  denote  the  probability  of  a   toxic
response    to  concentration  dj^  of  toxicant  i
(i-1,2) such that

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                                                                                    S^^
for some monotonic function. T  (I-1.2)'
toxlSt is  a dilution of  the1 oth.r,  then dl -
«d   where p  is the relative potency of  toxicant
P'to  toxicant 1.   The  probability  of a  toxi c
response  to the  combination  of  ^   and dj,
Illuming concentration addition, Is
    The pharmacological  approach  to asses.ing
               addltivity has been through the use
           grams (Hewlett,  1969),  which are plots
         ,  of  dose,  of  the  two  toxicants  that
       y give fixed  levels  of toxic  response.  The
        tit  represents a  given constant response
  alied an isobols  (Figure 2).  Under  the broad
 definition of  additivity, these isoboles are
 straight  lines, but  they  are  not   necessarily
 Parallel.    Under  the   narrow  definition   ot
 'concentration  additivity,  vith perfect  positive
 correlation  of  tolerances,  these  isoboles  are
 parallel  straight  lines  with  slope equal to  the
 neaative  of the relative potency.
     The biostatistical approach  to  assessing  con-
  centration  additivity has involved the fitting of
  dose-response models.  As a  simple illustration,
  cln-ider the  parallel  line  assay  technique
  whereby  a  suitable  linearizing  transformation
  (e.g., probit), Ff)  -  .^ lo« V 1-
  (Flaney,  1971).   Setting  ^  •  32  yields  p   -

  exp[(a2-ai)/B].   Another simple method^is the

  slope  ratio assay technique  whereby  PI       •
     .
   response  to  d.  and  d2 is predicted using  either
   F  or P.  with estimated parameter values,  and  the
   goodnesl-of-fit of the model  is  assessed  (Kodell
   ST Pounds.  1985).    Often  models  °f  greater
   complexity have been  used  (Hewlett  and  Plackett,
   1959; Christensen and Chen, 1985).

   n.anonse Addltjvit?
   -  Afl above, some of  the principles and concepts
    that  underlie  response addition  will  be  given
    orior  to  presenting  a  formal  mathematical
    definition.    Under  the  broad  category  of
    ^indent action, Bliss  (1939)  and  Finney
    (1971) expressed  the  principle that two toxicants
    have different modes  of  action, whereas  Hewlett
    and Plackett (1959)  and Ashford  (1981) pressed
    the principle  that  the toxicants have different
    sites of  action.   Hewlett  and Plackett  (1959)
    .odeled  biological independence without  assuming
    •tati.tical  independence.    That  1.,  their
    definition of  Independent  action  allowed  for
    correlation of tolerances to the  two  toxicants.
    More  narrowly,  some  early  investigators  (e.g.
    Gaddum   1949)  modeled independence  of action in
    the  sense  of  'absence  of  synergism,"  aasuming
    perfect positive  or  negative correlation of
    tolerances.  In the  narrowest  sense of  indepen-
    dence is the concept of simple  independent action
     (Bliss,  1939;  Finney,  1971), which is also called
     response additivity.  This response  additivity is
     characterized by zero correlation  of the Individ-
     ual tolerances to the two toxicants  (Bliss , 1S-J9,
     Finney, 1971; Ashford and Cobby, 1974).
   With  P(d.) as defined above,  the probability
of  a  joint   toxic  response,  assuming  response
additivity, is

         PU^)  -  P(dt)  + [l-P(dl)]P(d2)

                   -  P(d2)  + [!-P(d2)]P(d1)  .

That is, the response to the second  toxicant  over
and  above  that  of  the first is  .imply an added
effect based  on  the proportion  not  responding to
the  first  toxicant, and vice versa.  Note that

    P(d1+d2)       -  P(dx) + P(d2)  -  P(d1)*P(d2)  ,

which  corresponds to  the probability of the union
 of  statistically independent  events.    Although
 response  additivity  doesn't  mean  simply  adding
 response probabilities,  the  last  expression above
 indicates  that  if  these probabilities  are  small,
 then the  product,  P(d,)*P(d0),  will not  greatly
 Influence  the  joint ''response.     However,  some
 authors have just added responses, without  regard
 to their magnitude (Holtrman £t _al_.  , 1979;  Ozanne
 and  Mathieu, 1983).    This  latter approach  Is
 equivalent  to  hypothesizing  independent  action
 with perfect negative correlation of tolerances.
    The  use  of  isobolograms to  identify response
 additivity has  not been  popular, perhaps  because
 of  a  lack  of   agreement  as  to  the   shape and
  location  of  isoboles.  For example,  Webb  (1963)
  and Hewlett  (1969) suggest conflicting  shapes and
  locations of  isoboles  for  response   additivity.
  Indeed,  Christensen  and  Chen (1985) demonstrated
  various  shapes  of  isoboles   under  response
  additivity.
     The  biostatistical  approach  to   assessing
  response additivity  has involved  the   fitting  of
  dose-response  models.    For  example,  a  simple
  procedure  has  been  to  formulate P(d.+d.)  as

  W + *2(d2}  "  Vk^W' for suitably
  chosen  P.  (e.g.,  Kodell  and  Pounds,  1985).

  P(d.+d2) is  predicted from separately estimated

  F  (d.) functions, and  the  goodness-of-fIt of the
  response additivity model is  assessed.   Often,
  more  general models of response additivity have
   been  used  (Hewlett  and  Plackett, 1959;  Shelton
   and Weber,  1981).

   Application  of Concentration  and Response
   Additivity
     The  setting  of  water  quality  standards  for
   multiple  contaminants  is  an  example  of  an
   activity that  requires  either  knowledge  of  or
   assumption, about the  Joint  action of  these
   contaminants.   Citing insufficient information on
   mixtures of environmental contaminants,  the Safe
   Drinking Water Committee  of the-National  Research
   Council (1980) stated that estimates  of  toxicity
   from acute exposures will, out  of  necessity, have
   to  be  based  on  a nonconservative assumption of
   additivity.   The Committee  went  on  to  cite  the
   work  of Smyth et^  al.  (1969),  which   is  based on
    concentration  additivity, as  pertinent.
      With respect to carcinogenic effects from
    chronic exposure,  the  Committee favored  response
    additivity,    stating    that   to    estimate
    quantitatively the cumulative  carcinogenic  risk
    of  several carcinogen.,  the  individual risks

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                                             ^&&£&!f^
might b«  added.   The  Committee stated chat this
approach  assume*  Chat  Interactions are  not
present and  that  the  risks are  small enough so
Chat adjustments  for  joint probabilities are not
needed.
   Let  D, and    D,  denote  exposure  levels  of
toxicants  1  and 2,  respectively,  that correspond
individually to an  acceptable  level of risk, R.
To  insure an acceptable  level of risk,  R, to a
combination, dj+dj, of  toxicants 1  and  2 under
concentration  additivity, then  dj^  and  dj must
satisfy (Finney,  1971)
 Equivalently,
where it1 and IT, are  the respective proportions of

toxicants 1 and 2 in the mixture.  Under response
additivity, if R la an acceptable level of risk.
for  a  conibination,  d^+dj, of  toxicants  1  and 2 ,

then d   cannot pose  an individual risk exceeding

R    and  d-  cannot  pose  an  individual  risk

              where
           -

 exceeding ^
     It should be noted that there  is  a case for
 which  concentration  addition  and   response
 addition  are  indistinguishable  mathematically,
 i.e.,  their  predicted  joint  responses  are
 mathematically identical.   This  is the  case of
 the one-hit model.   Suppose  that
     P(d2) • F2(d2) - 1

 With  a double logarithmic  linearizing  transforma-
 tion,  parallel  lines  with slope- 1 are obtained,
 enabling  estimation of  X  ,  \2  and the relative

 potency,  p  - ^/ \, where d^jp  .   Thus, under
 an assumed  concentration-additive joint response,
                                       d)]
                  1 -
                                - X2d2]  ,
  However,  assuming  a response-additive  joint
  response ,
  P (dj-hlj) • F^) + F2(d2) - F1(d1)*F2(d2)

           - 1 - exp[-X1d1] -I- 1 - exp[-X2d2] - 1
   Thus the assumption of either concentration or
response additivity  leads to  the  same predicted
mathematical joint response function.  Of course,
this  is true  also  for  a  strictly  linear  dose
response model, which Is the limiting form of the
one-hit model aa the dose approaches sero.

Interaction
   As  indicated  earlier,  there  is  no clear
consensus as to what constitutes "interaction" of
drugs or  toxicants.   In  a broad  sense,  several
authors  have   expressed  the   concept   that
interaction  is characterized  by  one  agent's
influencing  the biological  action of  the  other
(Bliss,  1939;  Hewlett   and  Plackett, 1959;
Ashford,  1981).   However,  there  is disagreement
when  this  broad concept  is made  more specific.
Plackett  and   Hewlett   (1967)   pointed   out
differences  between  their concept  of interaction
and  that  of  Ashford and Smith  (1965),  quoting
their  definition  of  interaction fron  an  earlier
paper  (Plackect and  Hewlett,  1952)  as follows:
"[Drugs]  A  and  B are  said to  interact  if  the
presence of  A influences  the amount of B reaching
B's  site  of  action,  or  the changes produced  by B
at  B's  site of action; and/or reversely, with A
and  B interchanged."  Plackett and Hewlett (1967)
contended   that  Ashford  and  Smith's   (1965)
definition  of  "noninteractive"  action  included
only simple  similar action with complete positive
correlation  of  tolerances and  independent action
with zero  correlation  of  tolerances,  whereas
their  own'definition would  include  both  siailar
action  with  incomplete  correlation of  tolerances
and  independent action with  nonzero correlation
of  tolerances as noninteractive.
    The  use   of isobolograms  to  characterize
 "interactive"  departures from  additivity  has
suffered  from  Inconsistent  nomenclature,  as
pointed out  by Hewlett  (1969).   Interestingly,
Hewlett (1969)  reserved   the  term  synergism  to
describe an enhanced effect when only one of two
 agents  is  active individually,  using the  term
potentiation to describe  an enhanced  joint effect
 for two  separately  active agents-   However,  he
 described a decreased joint effect in both cases
 by  the 'term antagonism.   Also,  Hewlett (1969)
 described the joint  action of  two  agents that are
 separately   inactive  but  jointly  active  as
 "coalitive."  Figure 3 illustrates some commonly
 accepted   isobolographic  representations  of
 interactive  joint  action.
    With   respect   to   attempting  to   refine
 characterizations  of joint  interactive effects,
 Loewe  (1957)  seemed critical of  the  role  that
 biostatlatica has played in this effort.   He was
 probably  correct,  to  the  extent  that  he  was
 saying  that tolerance  distribution  models  that
 depend  on  quanta!  response  bioassay data  for
 their  resolution  have  limited ability to define
 basic  biological mechanisms.    Plackett  and
 Hewlett  (1967)  commented on  identiflability
 limitations of tolerance  distribution models.

 The Additive and Multiplicative Models  of
 Relative Risk
    Relative risk is defined as  the  ratio of the
 risk  due to a causal  agent  in  the presence of
 background risk factors  to the risk due simply to
 background  factors.    The  additive model  of
 relative  risk used  in epidemiology  studies

-------

corresponds   to    response   additivity    in
pharmacology/toxicology studies.   It is based on
an  approximation to a  model of "unconditional"
independence of events, wherein causal agents and
background  factors act  independently  of  one
anther (lothman,  1976;  Bogan at  al., 1976).
Howavar,  It corresponds alao  to a  model  of
mutually exclusive (and therefore nonindepandant)
events  (Xodell  and  Gaylor,  1986).   Under  the
additive  modal  of  relative  risk,  the  relative
risk due  to two  agents in combination  is simply
the sum of  their individual relative risks.   More
specifically,  B»u  - RR1  +  K*2 -  1.    All
departures  from  this model  are characterized as
either synergistic or antagonistic.
   The multiplicative model of  relative risk does
not   have   a   corresponding   null  model  in
pharmacology/toxicology studies.   It Is based on
a  model  of  "conditional"  independence   in  a
statistical   sense,   for   an   event   space
appropriately  defined  (Kodell and Gaylor, 1986),
having  arisen  originally  froc  the multiplication
of  attributable  risks  (Walter,  1976;  Walter and
Holford,  1978).   As its  naae implies,  under the
multiplicative  model  of   relative  risk,  the
relative risk due to two agents  in combination is
simply  the product  of  their Individual relative
risks.   That  is, RR^  -  RR1 * RRj.  Departures
from-this  model  are  termed either synergistic or
antagonistic.
    Hamilton (1979)  reviewed  various measures of
synergism that are employed  with two-by-two
tables  of  cohort data  from epidemiology studies.
All have been designed to detect departures  from
the  additive   and  multiplicative  models  of
relative risk.  Investigators  who have discussed
or used   both  the   additive and  multiplicative
models  of relative risk are Kupper  and   Hogan
 (1978),    Koopman    (1981),   Thomas    (1981),
Siemiatycki and Thomas  (1981),  Haailton (1982)
 and Relf (1984), the latter  three  being  concerned
 specifically with  Joint  carcinogenic  risk.
Notably, Hamilton and Hoel (1978)  have  considered
 concentration  additivity,   response additivlty,
 and  response multiplication  all  in  the  same
 context, namely, that  of  joint carcinogenic  risk.
    Siemiatycki  and  Thomas (1981)  formulated
 several    examples   of   the   additive   and
 multiplicative models in  the  context of the
 multistage model of  carcinogenesis.   They  also
 demonstrated a nonidentiflability aspect  of  these
 models, in that data  can be  consistent  with  a
 particular model  even  though the  underlying
 conditions for  that model are not met.   Hamilton
 (1982) also discussed nonldentlfiability aspects
 of  his  postulated  multistage model  for   joint
 carcinogenic!ty.  It  should be noted  that  apart
 from      theoretical     considerations     of
 nonidentiflability,  simple  two-by-two tables of
 epidemiologic  cohort  data,  upon which  many
 studies of interaction of  disease risk factors
 are   (of   necessity)  based,  contain  limited
 Information about  the Joint  action  of  these risk
 factors.

 Discussion
    The  study of  the  Joint  action  of  agents
 administered  in combination is a  very difficult
 undertaking  both  conceptually and  practically.
 Even  though there  is common  ground among
Investigators  of joint  toxic action,  there  It
also   a   great  deal   of  inconsistency   and
disagreement in nomenclature and  concepts.  It is
recommended  that attempts to assess combined
toxicant  affects  be  kept as simple  as  possible,
In  light  of  the crude  data generally  available
for  auch  assessments.   Investigators should  ha
careful  to define their own terms  precisely and
to  fully  understand  the terminology of  othara.
Tents   auch   as  additivity,    independence,
synergism,  and  antagonism should  not  be  used
loosely.   As has been shown, departure  from one
type  of  additivity, say concentration additivity,
might imply another type of  additivity,  say
response additivity, rather than  a synergistic or
antagonistic  form  of interactive  joint  action
(Table 3).

TABLE 3. Illustration of incorrect conclusions
          that can be  reached if only one type of
          "additivity" is considered as a model of
          noninteractive  joint action.  The shape
          of the  underlying dose-response curves
          governs  the  type of error that might be
          made.
Dose-
Response
Convex
Convex
Concave
Concave
True
Situation
Concentration
Addition
Response
Addition
Concentration
Addition
Response
Addition
Null
Hypothesis
Response
Addition
Concentration
Addition
Response
Addition
Concentration
Addition
Incorrect
Conclusion
Synergism
Antagonism
Antagonism
Synergism
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-------
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                    .;f^^Sl^!*s^^
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-------
 i .V^-'-iJiO--  .,   -.  ...
 '-.o-i ^-^VlVMA'-  --' -'.-

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                                         Joint Action Studies
                                                   1
                                              Type of  Agent
                             Drug

                                I
                         Type of Effect
                           Pesticide

                              1
                        Type of Effect
                intended                  Adv"98
                         Nature of Effect
               Adverse                  Intended

                        Nature of Effect
               Therapeutic                  Toxic
                            Objective  	
                Toxic                    Toxie
                           Objective
                 Optimal
                 Mixture
 Safe
Mixture
 Safe
Mixture
Optimal
Mixture
                                                  Safety
                                                Assessment
                                 Concerns
                                 Health
                                 Environment
                   Scientific Disciplines

                           T
                       Pharmacology
                       Toxicology
                       Blostatlstlcs
                       Epidemiology
         Figure 1.   Schematic representation of opposing  objectives in joint action studies, along with
                    concerns that motivate assessment of  combined toxicant effects, and  scientific
                    discipline involved.

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                                          Isoboles for
                                          Simple  Dilution
Figure 2.  Isobologram for  assessing joint action.  An isobole  is a plot of pairs of  doses of
          two toxicants that jointly give a fixed level of toxic response (e.g., 502).  For
          a simple dilution, isoboles for various response levels are parallel straight lines.
                                   Isobole  for
                                   "Antagonism"
                                                Isobole for
                                                "Additivity"
                           Isobole  for
                          "Synergism"
 Figure 3.  Isobologram depicting  some commonly accepted,  but not universally accepted,
           representations of concentration  additivity,  synergism,  and antagonism.
           Isoboles for response  additivity  can lie anywhere within the square, depending
           upon the underlying dose-response curves.

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                         USE OF THE MULTISTAGE MODEL TO PREDICT THE CARCINOGENIC
                                 RESPONSE ASSOCIATED WITH TIME-DEPENDINT
                                       EXPOSURES TO MULTIPLE AGENTS

                     Todd W. Tborsluad and Gail Charaley, ICF  Clement Associates
Introduction
   In a review of multiple agent dose-response
experiments, Filov et al. (1979) notes that the
observed interaction effects are usually highly
dose-dependent.  As a result, such empirical
tests of interaction as proposed by Hamilton and
Hoel (1979), Machado et al. (1983), and Chen and
Kodell (1986) performed at one set of dose levels
may give very little information about interac-
tions at another set of dose levels.
   The "high" dose levels for joint effects are
defined as the exposure values where statisti-
cally siznificant increases in cancer risk are
observed in either epider^iological studies or
cancer bioassays.  Tor the most part, exposure
to complex mixtures of agents in the environment
is at "low" dose levels, i.e., at least three
orders of magnitude below those at which a cancer
response is observable in laboratory tests.  As a
result, empirical tests of interaction observed
in bioassays give little insight into the effects
of complex mixtures at environmental levels of
exposure.  To estimate effects at low dose lev-
els, it is necessary to postulate an underlying
theoretical construct for the carcinogenic pro-
cess that can be  translated into a mathematical
dose-response model.  Such a model will contain
parameters describing various elements of the
process.  The joint effect of exposure to a com-
plex mixture is  determined by the way in which
individual agents affect the parameters describ-
ing various elements of the process.
   The agents in the complex mixture can interact
to affect the process in a variety of ways.
Chemical interaction between agents may create a
different carcinogenic agent.  An example of this
in drinking water is the interaction of chlorine
used as a bactericide with naturally occurring
organic matter  to form trihalomethanes  (Bellar
et al. 1974, Rook 1974).  New compounds may form
within the body  as well.  For example, nitrosa-
tion of certain  compounds in fava beans by en-
dogenous nitrite, when both are present in the
gastric lumina,  leads to the formation of a po-
 tent,  direct-acting mutagenic nitroso compound
 (Vang   et al.  1984).
    Complex mixtures can also act to modify the
exposed individual so that the dose at the site
of  action for one agent is dependent upon the
exposure levels  of the other agents in the mix-
 ture.  Any  event that affects the absorption,
distribution, metabolism, or elimination of a
 compound will  affect the level of that compound
 that is  available to react with DNA or other
 target  species.  For example, simultaneous oral '
 exposure  to  disulfiram  (Antabuse) and inhalation
 exposure  to  ethylene dibromide can greatly in-
 crease the  hepatocarcinogenicity of the latter.
 This increase  is thought to be a result of the
 inhibition of  acetaldehyde dehydrogenase by
 disulfiram,  leading  to  the buildup of toxic me-
 tabolites  of ethylene dibromide in the  liver
 (Wong et  al.  1982).  Another  example  is exposure
to cigarette smoke, which  can induce the  levels
of cytochrome P450 and aryl  hydrocarbon hydro-
xylaae that metabolize polycyclic aromatic  hy-
drocarbons (Conney et al.  1977),  resulting  in
higher intracellular levels  of reactive deriva-
tives capable of  forming  adducts with DNA.
   Another way in which  biological interactions
can enhance initiation is  possible saturation of
the enzyme systems responsible for the  repair of
DNA adducts, allowing some to go  unrepaired and
thus leading to  mutation (Thilly 1983).
   All such chemical-biological interactions are
the result of reactions  at many cellular  sites
with multiple molecules  of the agents.  As  a re-
sult, mathematical models  of the  cancer response
that depend upon such mechanisms  would  be non-
linear at low doses.  For  example, if two chemi-
cals combined to tora a  carcinogenic agent, the
rate of formation would  be proportional to  the
product of the concentrations of  the two  chemi-
cals.  A linear  reduction  in the  concentrations
of the chemicals would thus  result in a quadratic
reduction in the formationof the  carcinogenic
agent.
   The nonlinearity of the typical chemical-
biological interaction strongly suggests  that
mechanisms of carcinogenicity that depenc upon
such interactions are only marginally important
at environmental levels  of exposure.  Even  so,
any information  about chemical interactions or
exposure modification should be used in the
formulation of a model of  the joint effects of
agents, if available, by estimating exposure at
the cellular and molecular levels.  For the math-
ematical model of the carcinogenic response dis-
cussed in the next sections, it will be assumed
that the best available  surrogate measure of dose
at the site of action is used as  the dependent
variable.

Multistage Model

   The most utilized quantitative model of  the
carcinogenic process is  the  simple multistage
model described  by Armitage  and Doll (1954).
This multistage  model provides a  satisfactory
explanation of the power law for  the age  inci-
dence of many forms of epithelial carcinoma.  It
also explains the time-dependent  effects  of vari-
able exposures,  including  cigarette smoking
(Armitage 1985). The multistage  model  is based
upon the assumption that the carcinogenic process
is a series of ordered,  irreversible transforma-
tions in a single cell.  After going through a
fixed number of  transformations, a cell  is con-
sidered to be a  tumor that will grow and  be ob-
served some time in the  future.
   If these transformations  occur at the molec-
ular level, it is reasonable to assume  that a
single molecule  of an agent, if it enters the
critical reaction, can cause the  transformation
from one stage to the next.   Under this assump-
tion, the probability of a transformation is

-------
linearly related Co  the degree of exposure  at the
molecular  level.
   Tor constant  exposure  to a single agent,  the
transformation rate  from  stage i to stage i+1 may
b* expressed as
                                                   more and Keller (1978) , we define  the  following
                                                   variables :

                                                        P. (t) - probability that a cell is in the
                                                                 t\\
                                                                i   stage at time t and
where
     o.  • background transformation rate,

     0  - transformation rate per unit  of  expo-
           sure ,  and

     x  • a constant that is directly proportional
           to the best surrogate measure of expo-
           sure at the site of action.

Assuming that there are  a total of k stages and a
fixed  time w fron the appearance of a cell in the
kth stage to deatr. by a  tunor,  the age-specific,
agent-induced  cancer death rate [h(x,t)J  is ex-
pressible as
         k-1
h(x,t) •  H  (a.+B.x)
         i-0
                                    /(k-1)!,   (2)
 where
          age attained.
 The probability of death from a tumor by age t in
 the absence of competing mortality is simply
P(x,t)
               1-exp -/wh(x,v)dv -
                                         (3)
      1-exp -  n  (a.+B.x) (t-w)*Yk!
               i-0
 The derivation of these results is presented
 clearly in the recent Armitage (1985) paper.
 Generalization of Multistage Model to Account for
 Variable Exposure to Multiple Agents

    The multistage model has previously been gen-
 eralized to account for either exposure to mul-
 tiple agents or variable exposure over time.
 Whittemore and Keller (1978) describe the complex
 equations that can be used to obtain estimates of
 risk under variable exposure conditions using the
 multistage model.  Day and Brown  (1980) give re-
 sults for the case where observation continues
 after exposure ends.  Crump and Howe (1984) de-
 rive an expression for the special case where one
 or  two specified stages of the multistage process
 are assumed to be exposure-dependent, and expo-
 sure is taken to be a time-dependent step func-
 tion.  The multistage model has also been modi-
 fied by Siemiatycki and Thomas (1981) , Hamilton
  (1982) , and Reif  (1984) to account for exposure
 to  multiple carcinogenic agents under constant
 exposure  conditions.
     To generalize the multistage model to account
 simultaneously  for variable exposure and multiple
 agents, we  start with a time-dependent exposure
 model.  Following the approach taken by Whitte-
                                                        X.(t) • transition rate from the  i to the 1+1
                                                                stage at tine t.

                                                   The probability that a cell is in the  1C   stage at
                                                   time  t (given that it is in the initial  untrans-
                                                   formed state at t • 0) can be described by the
                                                   following set of simple differential equations:

                                                        dPQ(e)/dt — xQ(t) pQ(t)   PQ(0) - 1

                                                        d?1(t)/dt — \1(t)P1(t)    Pi(0) - 0
     dP, (t)/dt
       K.
                                                                    X.  ,(t)P.  , (t)     P, (0)  »  0  U)
                                                                     lC~ —    K." 'i.         K.
                                                            To account for exposure to multiple  carcino-
                                                         genic agents, we define the transition  rate  to be
                                                              Xt(t)
                                                                                                  (5)
                                                         where
                                                                    " background transition  for  i.  '  stage,
                                                                number of agents,

                                                                1 if the jth
                                                                                                      .th
                          agent affects  the  i
     "ii                                     stage
       J     0 if otherwise,

     6. .   • unit exposure transition rate for

             j   agent on i   stage, and

     x  (t) » exposure to j   agent at time t.


   This formulation assumes that each of  the mol-
ecules or produced radicals from all of  the
agents are acting independently of each  other
with regard to their probability of causing  a
cell transformation.  This is reasonable  when
cell transformation probabilities for a  single
cell are very small, as would be the case when
some individuals in the exposed population are
free of the tumor in question.
   Since the probability that a single specified
cell will be transformed is very small,  it fol-
lows that to a close approximation, PQ(O •  !•
Using this assumption, Whittemore and Keller
(1978)  showed that the preceding set of differen-
tial equations has the following approximate
iterative solution:
     PQ(t) -


     Pi(t) '
                                                                   Xi-i(v)Pi-l(v)dv
(6)
                                                    In addition, we assume that

                                                      o The time required for a cell  in its  kth
                                                        transformed state to grow  into  a death-

-------
                                     '  '..  • ,.' •''."'.,' * "•'•':"''.  (^'•,f
                 •&&$^^\&\*ii.^i fe if* r«^-^aS&S^iV^t<2^ii-::•.

     causing  tumor is approximately constant and
     equal  to the value w;

   o The probability that a  given  cell will cause
     • tumor  death is very small;

   o An organ contains H cells of  a specified
     type,  each one of which is capable of caus-
     ing a  tumor death;

   o H is very  large;

   o Each of  the cells acts  independently with
     regard to  undergoing  transformations and
     causing  a  tumor.

Then, the age-specific death rate  associated with
a specific  type of  tumor  in  a given organ may be
expressed,  to a close approximation,  as

     h(t) • N[dP, (t-w)/dt]  « N'X.  . (t-u)P,  (t-w),
                K               tC~*      K."~i
                                              (7)

and the probability of death from  that  tumor by
age t in the absence of  competing  risk  is
     P(t) - 1 - exp - /Q h(v)dv .-

     1 - exp - /* N[dPk(v-w)/dv]dv.
     (8)
   To illustrate how equations 5 through 8 can be
used to estimate the risk associated with mul-
tiple-agent, time-dependent exposures, several
simple examples will be presented in the follow-
ing sections.

Example of Interaction Effects for' Multiple
Agents with Continuous Exposures at Constant
Levels

    For continuous, constant exposures, the trans-
 ition rates are constants (over time) that are
obtained  from equation 5 by substituting x-t for
x^(t).  Using this notation,  the transition rates
have the  form

      V ' xi' "i * Ji au Vj'         (9)
 and the possibility of a death from a tumor by
 time  t  is
      PCxj^....,*„.*£_•  1 -  exp -

                       n   X.(t-w)k/kl
                      i-0   1
 At low environmental  levels of  exposure,

      P(Xl,x2,...,Vt)  :  [JQ Xt]
    (10)
t!,  (11)
 where

 k-i     k-i     k-i  t k-l
 irXi M iroi + 2 U^aijAif  2 ^ *«*«• <12)
 i«o     i-o     l-o  I l-o          ' •
since all higher-order exposure terms sre ap-
proximately equal to zero.
   A number of important implications follow
from these results.  When exposure to multiple
carcinogenic agents occurs, each agent may af-
fect one or more of the transition rates in one
or s»re cell types.  If two agents affect dif-
ferent cell types, their effect on the produc-
tion of tuaors will be Independent if the appro-
priate mortality adjustment is made.
   The probability of a tumor in this case is
one minus the product of the probabilities that
each agent does not cause a tumor.  If the prob-
ability that each agent will cause a tumor is
low, the probability that the joint exposure will
produce a tumor is, to a very close approxima-
tion, equal to the sum of the probabilities that
each agent causes a tumor.  Where two agents act
only on the same single stage of a cell type,
the probability tr.at joint exposure will produce
a tunor is equal tc the SUE of the probabilities
for each exposure.  When tne agents act on dif-
ferent stages of :r.e sane cell type, there is a
multiplicative exzisure effect tarn as well as
the additive teras.
   At high doses, the multiplicative ex-posure
effect term can dominate the carcinogenic joint
response, and the Joint effect can be much
greater than the sum of the individual effects.
However, if both exposures are reduced by sev-
eral orders of magnitude, the joint effect would
be, to a very close approximation, equal Co the
sum of the individual effects.  The same results
hold  when hundreds of compounds are combined.
If each one  is reduced three or more orders of
magnitude, the deviation from additivity is not
an appreciable relative amount.  As a result,
the multistage model predicts additivity at en-
vironmental exposure levels for almost all situ-
ations that would be routinely encountered.
   The main  exception to this rule is when one
of  the agents  remains at a high level.  In these
•cases,  the  incremental risk associated with ex-
posure to  low  levels of an agent can be dominated
by  its multiplicative interaction with exposure
to  high  levels of another agent.  As a result,
particular concern must be paid to agents that
affect  the  same  cell type as cigarettes, since
cigarettes  are  the single deliberately uncon-
 trolled  carcinogen to which we are exposed at a
high level  in  our environment.
    To  demonstrate the general premise that under
multistage  theory, an observed extensive syner-
glstic  effect  in a multiple-agent bioassay does
not imply a major departure from low-dose addi-
 tivity,  the  following numerical example is given.

 Simplest Multistage Model that Results in a
 Synergistic  Effect

    The simplest  multistage model that results  in
 a greater than additive effect arises from the
 assumption that  each of two agents affects the
 transition rates of different single stages  in
 the multistage process.
    Thus, for two agents  (m «  2), if the first
 agent affects the  ic^ and  the second, the jtn
 stage and no other  transition rates are affected,
 it follows that

-------

      u
                                                  Example of Interaction Effects when Multiple-
                                                  Agent Exposures are not Continuous and
                                                  Concurrent over Time
Substituted Into equation 5, this gives the
result
     Xi -
j
               6
                j2
                                                     Variable and noncontiguous  exposure patterns
                                                  •ay be accounted for by treating  the  time-
                                                  dependent exposures, xj(t), aa specific step
                                                  functions that allow equation  4 to  be solved  in
                                                  a closed form.  The following  simple  example  il-
                                                  lustrates this general approach.
                                                     Consider the case where exposure is to two
                                                  agents (m - 2) with the following exposure
                                                  patterns:
   Assuming that competing mortality  from causes
other than the tumor under Investigation is min-
imal at the termination of the experiment, the
probability that a tumor will be  observed may be
expressed as
                                                       Vc) - o
                                                       x2(t)
elsewhere
                                                                       elsewhere
                                                                                                (15)
                                              (13)
where

                              11
                                      B2  '
                                                       s, » starting  cizie  of  exposure  to first
                                                            agent,

                                                       f. • stopping  tine  of  exposure  to first
                                                            agent,
    Consider  a model of the form  of  equation  13
 that  has  the following properties:
                                                       s. - starting  time  of  exposure to second
                                                            agent,  and
    o  One  agent  is  twice as  potent  as  the  other,

    o  0.1  of  one agent and 0.2  of  the  other  gives
      about a 9% response in a.  bioassay if each
      agent is given by itself,

    o  Responses  at  exposures of 1-10   of  the
      single  agent  values give  a risk  of 1-10-5
      for  each agent singularly and 4-10-5 for
      joint exposure to both agents, and

    o  The  background risk is about 5-10~6.

    A numerical model  that  meets these conditions
                                                       f^ "  stopping  tiae  of  exposure to second
                                                             agent.

                                                  It is assumed  that  xi(t)  affects the first  stage
                                                  only and that  X2(t)  affects the last or kth
                                                  stage only.  Under  this  assumption, the transi-
                                                  tion rates have  the following time-dependent
                                                  form:
                                                                °0B01X1       91  *  t  «  fl
 P(xrx2)-l-exp (-0.000005) (1+189,
 or

 P(x , xJ-l-ex?-(0. 000005+0. 948640x +0.474320x2

 +89,991.8x^2).

    This model implies that to achieve a meaning-
 ful (i.e., doubling) joint exposure effect at low
 environmental doses, the joint experimental syn-
 ergistic effect would have to be very large.
    Two agents given together at levels of about
 5% of the single-agent doses would produce about
 a 992 response, while the single-level doses
 given by themselves would yield about a 9% re-
 sponse.  An interaction of this magnitude is un-
 precedented.  This hypothetical situation is
 depicted in Table 1.
     In the next section, implications concerning
 the ordering of exposure will be investigated.
                                                              X (t)  - a   s=l,2,...,k-2  0 < c < »    (16)
                                                               S       S
                                                                                              t < s.
                                                                    (1)
                                                        \-lW  '   X     "  Vl+Bk-1.2x2   Vt
-------
where

         k-2
     A-   H   a/(k-l)!.
         J-0   J

for w  - 0, Che age-specific rat* defined in
equation 7 is

     h(t)  - NX.  ,(t) P. _.(t).
    Since Pfc-iCt)  ia  functionally dependent upon
BI  and f i and \t_i(t)  is functionally dependent
upon  82 and f2,  it  follows that h(t) , as defined
in  equation 16,  is dependent upon the ordering
over  time of sj,,  f]., 32, and f2-  For example,
if si
         32
                                that
 of  \fc_i (t)  and ?k-l(t)  have  the  time-dependent
 representation

                       X  ?           t     f^t  i

 where the X    and P    are  defined  in equations
 16 and 17, respectively.   A  schematic  representa-
 tion of equation 19 that  illustrates how the
 structural form of the age-specific  rate.is time-
 dependent is shown in  Figure 1.   Other structural
 relationships can be derived in  the  same manner
 for alternative orderings of the exposures.
    To explore the effect  of  the  timing of  expo-
 sure on the interaction or synergism of the
 agents, we will estimate "relative risks"  for the
 following situation.  It is  assumed  that agent
 exposures were selected so that  each of the
 transition rates is increased by a relative fac-
 tor of A during the exposure interval.  This  im-
 plies that
       301X1
               3k-1.2*2

                 Vl
                          i.
                equation 21 and used to obtain the relative  risks
                also depicted in Table 2.
                   The results obtained using this approach  con-
                form to one's intuitive sense of reasonableness.
                Ho synergism (i.e., effect greater than one)  ex-
                ists if the exposure that affects the last stage
                •ads before the exposure Chat affects Che first
                stage begins.  Also, Che greatest synergism  ex-
                ists when exposure that affects Che first stage
     (18)       ends before that which affects the last stage
                starts.  In this situation, the relative risk
                rises slightly from I + A/3 to 1 + a/2, as the
                number of stages increases from k - 2 to k » «.
                In contrast, the relative risk decreases rapidly
                from 1 + A/4 to 1 as the number of stages in-
                creases from k » 2 to k » «, for the situation  in
products        which both exposures are given during the same
                half of the time period.  It is possible to  de-
                rive comparable results for any set of assump-
                tions about the stages affected and any step
                functions of exposure.  However, the algebraic
                fora aay be very cc-plex.
                   In the final section, the most important  prac-
                tical problem concerning joint exposure will  be
                investigated — nanely, how to cope with the
                potential interaction of cigarette staoke and
                other carcinogenic agents.

                Joint Effect of Cigarette Smoke and Other
                Agents on Respiratory Cancer

                   As a first step in attempting to estimate  how
                siEoking cigarettes tsocifies the quantitative  ef-
                fect of other agents on cancer rates, it is  nec-
                essary to develop a model for the effects of
                cigarettes alone.  Ideally, we would use the
                combined data from as many sources as possible
                in such an endeavor.  Unfortunately, the only
                data currently available in the open literature
                in a form amenable for fitting with a multistage
                model are found in the Doll and Peto (1978)
                paper; they are reproduced here as Table 3.   It
                is recognized that a number of problems exist in
                using these data.  Among the more important  are
                the following:

                   o British cigarettes and/or smoking patterns
      (20)            are different from those in the United
                     States.
 Under  this assumption, the relative augmented
 risk for  the  two exposures given together, as
 compared  to the sum of the two given separately,
 can be derived.  For most situations at environ-
 mental levels of exposure, this relative risk in
 the absence of competing risk may be expressed as


      R*(0,t)  - /C  {h[x.(v),x,(v)]-h(0,0)} dv
                 0

               * /C  lh[x.(v),0]+h[0,x,(v)]
                 0
                              -2h(0,0)}  dv.   (21)

     The age-specific cancer rates for the four
 combinations  of exposure depicted in Table 2 are
 derived in a  Banner analogous to the preceding
 example.   These rates are then substituted into
                                                           o The data are in a form that results  in  a
                                                             loss of information, since they are  com-
                                                             bined into various groupings rather  than
                                                             being presented for each individual.

                                                           o No information is given on rates after  the
                                                             cessation of smoking.

                                                           It is hoped that the availability of addi-
                                                         tional U.S. data and more complete data for  the
                                                         Doll and Peto  (1978) cohort will eliminate these
                                                         problems in the future.  However, for the present
                                                         we will fit the data given in Table 3 to  various
                                                         forms of the multistage model to illustrate  the
                                                         general approach for predicting the modifying
                                                         effect of cigarettes on the cancer potency
                                                         of other agents.
                                                           It is assumed that the 1   and k   stage  of a
                                                         k-stage model are affected by cigarettes.  In

-------
                                                           ;^
          we assume that each individual in the
             siting at age 91 and continued un-
 Oor
til the end of the observation period.  Also, we
I^a constant lag or weighting tt»e of length
? which will be estimated fro. the data.  Under
thJUi S^Ptlona, the equations for the transi-
tion rates «y »• «*P""«d •»
                     6oixi(0
                             1 - 1,2,.... k-2   (22)
 where the number of cigarettes  smoked per  day has
 the functional form
      x,(t)
sl - c
elsewhere.
 „
    To  laconorac. a const.ut las f-ra of
                        »ith «-•  1° ••M
  tory cancer may  be  written  as
                A(t-w)
                      k-1
                                    w  <  t  <  S.-
  h(x,t) »
                                               (23)
               A(t-w)
                     k-1
     If k-4,  then 13.9 <  w < 23.3

     If k-5,  then  4.8 <  w < 22.2

     If k-6,  then  0   <  w < 14.2

However, l£k<3ork>7, the fit la statisti-
cally rejected at the 0.05 level for all values
of w.  If the data on risks after the cessation
of smoking were available, it is likely that
only a k of 5 or 6 would fit the data; we  would
expect a short lag because risks fall quite
quickly after smoking is stopped.
   For  the purpose of illustration, we will use
our  best-fitting model to predict  the  effect of
smoking on the augmented risk associated  with
another agent.   If  it is assumed  that  a  second
agent  affects  the  1st  stage  of  the multistage
process,  the  transition  rate for  the  1st  stage
 is expressed  as
                                 v° - v
                            For the case where x?(t) » X2 for all  t,  the  age-
                            specific rate under this added assumption has  the
                            form
                                        A(t-w)k"1[l-t-B02x2]  w <  t  <  SL +  w

                            h(x,,x,,t)  -                                  (25)
                                                                                              ^k-l
                                           ^oiV^'V'
                                                          where
   where
          k-1
      A •  n  a
          3-0  3
      The parameters A, BQ1, Bk_j. i,  and w can  be
   estimated from the data in Table 3 using the max-
   Srtikelihood method.  To do so, it is assumed
   that the observed number of respiratory cancer
   detths in each cell has a Poisson distribution
   ££ mean hU.OFl, where PT is the total number
   of  oerson-years observed for each cell.  The
   °plrPameter estates that maximize the likelihood
   are shown in Table  A.  The goodness of fit of the
   Sel  ^illustrated  in  Table 5.  It is assumed
   ^hat for each cell, ^ equals 19.2,  the average
    age at which people started  smoking  for the  en-

    tirThe0parameters k and  w are highly negatively
    correlated, so that other estimates  give  almost
    as good a fit.  Models that contain  values  for
    k  and w that fall in  the range  shown below  do
    not give a  statistically significant worse  fit
    at the  0.05 level, as measured by the log likeli-
    hood  criteria, than the best fit shown in
    Table 4.
                                  B02 * VV
                                To obtain information about the parameter
                             it is assumed that an animal bioassay is avail-
                             able.  In terms of our previous parameters, the
                             probability of response for the animal may be
                             expressed as
       P(x2,t)
                                                           BQ2x2
                                                                                             (t-w)*]
(26)

k
                              Of  course,  only  the  whole term,  (A/k)Bn_(t-w)
                              can be estimated from the quantal  animal  data
                              alone; however,  in conjunction with the human
                              data, 892 may be estimated separately.
                                 The augmented risk associated with continuous
                              exposure to 12 while smoking x; cigarettes per
                              day from age S]^ to t under the assumption of no
                              competing risk may be expressed as
                                                                            (27)
   P(x2/x1,t)-[l-exp-;oh(x1,x2,v)dv]-

     tl-exp-/Qh(x1>o,v)dv] .

   For a low background rate, the augmented risk is,
   to a close approximation,
                                                  .k
                                               t-w)

             s+w  <  t                             (28)

-------



   To illustrate the general approach, w« shall
assume that the bioassay gave a linear tern esti-
mate of
     f
t-w)* • 0.2.
Substituting this value and v - 13.642, t « 70,
• • 19.2, and BV-! 1 " 0.31044 into equation 28
gives the numerical result       _    ^     t  -,>

P(x2/x1,t-70)-0.2x2
            D.31044x
             •0.2X-+0.0618X.X-.

   Let us  assume that without cigarette use the
predicted  risk  is I'lO   based only on the animal
bioassay.   This implies that x_ equals 5'10~5.
"sine equation  29, we calculate the augmented
risk associated with the second agent alone in
the presence of cigarette smoke for individuals
who started sacking at age 19.2 and continued
until death or  age 70.  These results are de-
picted  in  Table 6.
   Thus, a person who smokes tuo  packs a day  in-
creases  his or  her augmented risk by more than
one order  of magnitude.  The interesting philo-
sophical  public  health question arises:  Does  so-
ciety have the  responsibility for protecting  an
 individual from a second agent that increases the
 involuntary risk by about 1-10-^  if the individ-
 ual  smokes two. packs a day, when  the voluntary
 risk  he  or she  assumes for smoking  is about
 1-10~1  or  three orders of magnitude higher.
    Ic summary,  the approach suggested here to
 adjust  for cigarette use employs

    o  Human data to estimate the effects of ciga-
      rettes,

    o  A""1""*1 data  to estimate the  effects asso-
      ciated with  the second agent,  and

    o Multistage theory to predict the joint  ef-
      fects of cigarettes and the  second agent in
      the  absence  of  any  actual  joint  exposure
      data.
                   REFERENCES

 ABM1TAGE, P.  1985.  Multistage models of car-
     cinogenesis.  Environ. Health Perspect.
     63:195-201
 ARMITAGE, P., and DOLL, R.  1954.  The age-
     distribution of cancer and a multistage theory
     of  carcinogenesis.  Br. J. Cancer 8:1-12
 BELLAR,  T.A., KICHTENBERG, J.J., and KRONER, R.C.
     1974. The occurrence of organohalides in
     chlorinated drinking water.  J. Am. Water
     Works Assoc. 66:703-706
 CHEN,  J.J.,  and KODELL, R.L.  1986.  Analyses of
     two-way  chronic studies  (Submitted for publi-
     cation)
CONNEY, A.H., PANTUCK, E.J.,  HSIAO,  K.C., KUNTZ-
   MAN, R., ALVARES, A.P.,  and KAPPAS, A.  1977.
   Regulation of drug metabolism in  man by envi-
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CRUMP, K.S., and HOWE, R.B.   1984.  The Multi-
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DAY, N.E., and BROWN, C.C.   1980.  Multistage
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   JNCI 64:977-989

DOLL, R., and PETO, R.   1978.   Cigarette sacking
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   non-smokers.  J. Epideniol.  Communitv Health
   32:303-312

FILOV, V.A., GOLV3EV, A.A.,  LIl'ELINA, E.I., inc
   TOLOKO;,TSIV, N.A., eds.   1979.   Quantitative
   Toxicology.  John Wiley  &  Sons, New York

HAMILTON, M.A.  1982.  Detection of  interactive
   effects in carcinogenesis.   Blomed. J.
   24:483-491

HAMILTON, M.A., and HOEL, D.G.  1979.  Detection
   of  synergistic effects  in  carcinogenesis.
  • Unpublished manuscript

MACHADO,  S.G., SHELLABARGER,  C.J., and LAND,  C.E.
   1983.  Analysis of interactive effects between
   carcinogenic treatments  in long-term aniral
   experiments.  Unpublished  manuscript
 REIF,  A.E.   1984.
    JNCI  73:25-39
                                                                Synergism in carcinogenesis.
 ROOK,  J.J.   1974.   Formation of haloforms during
    chlorination  of  natural waters.   Water Treat-
    ment  Exam.  23:234-243

 SIEMIATYCKI, J.,  and  THOMAS, D.C.  1981.  Bio-
    logical  models and statistical interactions:
    An  example  from  multistage carcinogenesis.
    Int.  J.  Epidemiol. 10:383-387

 THILLY,  W.G.   1983.   Analysis of chemically in-
    duced mutation in  single cell populations.
    In  C.W.  Lawrence,  ed.   Induced Mutagenesis:
    Molecular Mechanisms and their Implications
    for Environmental  Protection.  Plenum Press,
    New York, pp.  337-378.

 WHITTEMORE, A.,  and KELLER, J.B.  1978.  Quanti-
    tative theories  of carcinogenesis.  SLAM Rev.
    20:1-30
 WONG,  L.C.K.,  WINSTON,  J.M., HONG,  C.B. and
    PLOTNICK, H.   1982.  Carcinogenicity and tox-
    icity of 1,2-dibromoethane in the rat.
    Toxicol. Appl. Pharmacol. 63:155-165
 YANG,  P., TANNENBAUM, S.R., BUCHI,  G. ,  and LEE,
    G.C.M.  1984.  4-Chloro-6-methoxyindole is  the
    precursor of  a potent  mutagen (4-:hloro-6-
    methoxy-2-hydroxy-l-nitroso-indolin-3-one
    oxime) that forms  during nitrosation of the
    fava bean (Vicia fava).  Carcinogenesis
    5:1219-1224

-------
^I2^^
                               FIGURE 1

               SCHEMATIC REPRESENTATION OF THE TIME-DEPENDENT,
                ACE-SPECIFIC DEATH RATE SHOWN IN EQUATION 19
   WEI.
                                  P(U
                  sl        32            f2
          NOTE:  X   is defined in equation 16 and P   , in equation 17.
                                 TABLE  1

                             BIOASSAY DESIGN
                                           Xl
    x2                  0    1.05414-10"5   4.8743-10"3      l-lfl"1

    0                    0        1-10"5          —        9.0503-10"2

    2.10828«10~5       1-10"5    4-10"5

    9.7486'10~3        —          —        9.852-10"1

    2.10"1        9.0503'10"2

    NOTE:  This is the design  required to estimate an interaction
           tern large enough to  double the risk over that predicted
           by additivity at environmental levels of exposure.
           Underlining indicates a test group in the hypothetical
           bioassay.

-------
>'.: ..:It&'':^:&&^
                                                TABLE  2


                 RELATIVE  AUGMENTED RISK  {R*)  OF JOINT EXPOSURE COMPARED WITH
                          SUM OF RISK  ASSOCIATED WITH SINGLE EXPOSURES
                                                               0] +  h(0,x2(v)l -  2h(0,0)|dv
                     Tine Interval
           0  to t/2                 t/2 to  t
          ..                     _____      Functional
 No.     x.(t)       X2(t)      x^t)     '  Xj(t)         Fora  of R«         k«2       k«5


 10           0          x.          x-         1 + A/2k          I * •J     1 +  A    1
 2        0           x2         xj_          0          1                  111


 3        Xj^          0          0           x2         1+  A(2 fc


 4        x,          x,         0           0          1 +  A/2*
                                                   TAOLZ 3


                                 • tSPHUTOBt CANCER DATA rROM DOLL  AND PETO  (UH)
                                                      Elpoiuc* (Ci9*c«ttta p«r o>yl — «
H«0i«n A««
42. S y««t« old
•No. of eanetcs ob«*rv*d
No. e( p*r>on-ytar« ob>tfr«d
47. S y*>c« old
No. ot p«CBon-y««c» ooa«EV«d
S2.S y»«n old
No. o( eaneto ob»i**d
Ho. 0( p«rion-y«»r» obl«tv*d
51. S y««n old
No. o( ptcion-ytao oburvtd
t2.S year* old
Ho. of eancara ooa«rv*d
No. of patcon-yoaca obaocmd
»7.5 yvaca old
Ho. of camera obatrvod
No. of paraon-yoara obaocvod
72. S ycaca old
No. of cancats obatrvad
Ho. of p*raon-y*ara obmved
0.0

0.0
17,I4(.J
0.0
15, 132.5

1.0
12, 224.0
2. 0
1,105.5

0.0
«. 241.0

0.0
4,111.0

1.0
2,723.5
2.7

0.0
1,214.0
0. 0
1,000,5

0.0
153.5
1 0
1* u
425.0

1.0
505.5

0.0
3*2.5

1.0
242.0
4.4

0.0
2,041.5
0.0
1,745.0

0.0
1,542.5
0.0
1,355^0

1.0
1.0(1.0

1.0
143. J

2.0
494.5
11.)

1.0
3,795.5
1.0
3.205.0

2.0
2.727.0
"1.0
2,2(8.0

1.0
1.714.0

1.0
1,214.0

4.0
142.0
14.

0.
4.124.
|a
3,995.

4.
3,271.
Q^
2, 4<4l

2.
1,»29.

2.
1.217.

4.
413.
0

0
0
Q
0

0
J
g
S

0
:

0
0

0
S
20.

1.
7,044.
^
4,440.

4.
S.5I).
•
4.157.

11.
2.0(1.

12.
1.1)0.

10.
1,055.
4

0
0
Q
5

0
0
g
S

0
5

0
0

0
0
25.

0.
2,523.
7
2.545.

3.
2, (20.
C
2.101.

4.
l.SOt.

S.
974.

7.
527.
4

0
0
g
5

0
0
g
5

0
5

0
I

0
0
30.2

1.0
1.715.5
2 0
2.123.0

3.0
2,224.5
( 0
1, 923^0

11.0
1.1(2.0

9.0
7(1.5

2.0
317.5
31.0

0.0
192.5
0 0
1.150.0

3.0
1,211.0
4 Q
1.0(3.0

7.0
124.0

9.0
515.0

5.0
213.0
   77.5 y*ara old

     Ho. Of canctra  obittv«d           2.0      0.0      0.0     4.0      5.0      7.0     4.0     2.0      2.0
     HO. of paiaon-ycaca oba«c««d   1,772.0    201.5    517.5    547.0    370.5    912.0    209.5   130.0     11.5


   NOTE:   HO. of person-years obatrved cetera to the total nuaber oC peraon-yeara obaerved  in that aqe group at
          that expoaure level.

-------

                                    TABLE  4

            RESPIRATORY CANCER  AND  CIGARETTE  SMOKING  DATA:
                     MAXIMUM LIKELIHOOD ESTIMATES  OF
                    PARAMETERS IN THE MULTISTAGE MODEL
Coefficient                                Maximum  Likelihood  Estimate
A                                                    0.283404971489-10"10

BQ1                                                 0.575320316865

Bk.1(]>                                              0.310436883121

Lag tine  (w)                                     13.6420002494


NOTE:    This  is a  five-stage  model;  stages  1  and  5  are  affected
          The age at the  beginning  of  exposure  is  19.2  years.
                                      TABU 1

             uuiurui u»cii AW ciuirm wuiw OATAI  cooiwoi or rir or woci TO ouiiivu O»TA
                                                 14.0    10.1    31.4    18.3
    41. 1 y.«f A old
      •O. o* CAAC.lt oe«.l*»4   0.0    a. 9


    41.1 r*A«A 014
      •o. Of CAMdl <*MC~<   I.I    I.I    1. 1     1.1     '.»     >-°    !•>    1-1    °-°
      ••. t»i CAAgiH p r«<* •'»
      •». 01 CA^c.n MunM   1.1    1.1    I.I     1.0     I.I     •• I    >••    I.I    4.1
      ••. *f «i«n«i» fCMlct^  1. 13311  1. 13111  I.SI7S4   1.7041   1.113?   7.37M   4.1137  1.7344)  4.1431

    ta. » r*A«« *14
      •>. «l UAC.r. e>A.cra<   1.0    1.1    1.0     1.1     1.1    II. I    '.I    It. I    7.0
      ••. 4>f c«MC«(a r(*^kctM  1.3IM  1.11114  0.7li*l   3.3412   3. 1113   I.7f4l   1.4111  7. MSI  4.7731

    47. S y«Ail »U
      ••. 01 CAAC.CI obMC<^4   I.I    1. 1    1.1     3.1     3.0    13.1    S.O    1.0    f.O
      •a. of CAMMA pfMictAd  1.1371  1.31717  I. Ills   3.7)11   4.&3I4  10.147   7.23IS  7.3141  7.4411

    73.1 y«A(A A14
                                   1.0     4.0     4.9   11.0    7.0    3.9     1.0
     77.1 TAAIA old

      MO. Of CAOCAtA OOAAIVA4
     NTTtl  CAllA i«r. e*llAOAA« AO th«t tM *t««ict«« VAlwA 1A ACAACAC tbAA 4M ««I«A I to 3.S. If POAAlAtA. OAOCAAA of
                                     TABLE  6

               AUGMENTED RISK ASSOCIATED WITH SECOND AGENT
                         FOR VARIOUS SMOKING LEVELS
              Cigarettes Smoked             Augmented  Risk
              pec  Day on                     Associated with
              Average                          Second Agent


              0                                   1.00-10*1
              10                                  4.09'10~|
              20                                  7.18-10~=
              40                                13.36-10"5

-------
                                         ;..,.,     •..«--,-.••     _  . ,f-\ •  •  ;.- ;   ...  .  ,•••-:., ,,,>r.'   -. ,,.,,'Ji-ij-'.. ,-• r--  -
                          ^j!&'^L4:::^^^^^>i%^a^s-i^&:^
                   DISCUSSION

Paul I. Feder, Battelle Columbus Division

   I enjoyed reading the Thorslund and Charnley
paper.  It 1s well written and presents good
•ethodology and useful applications.
   The main theme of the paper Is the description
and estimation of health risks associated  with
low dose environmental exposures to «ult1ple
agent mixtures.  Determination of the presence,
absence, nature, and extent of Interactions among
mixture components at low environmental exposure
levels 1s of considerable Importance.  A key Idea
of the paper Is that the presence or absence of
empirically determined, high dose Interactions
observed 1n laboratory bloassays 1s Irrelevant to
Inferring the presence or absence of interactions
among mixture components at low environmental
levels of exposure.
   In order to make definitive statements  about
the presence and nature of interactions at low,
environmental exposure levels, 1t is necessary to
understand the biological and cheir.ical mechanisms
by which the mixture components interact with one
another and with the body.  There may be chemical
interactions among mixture components;
differential behavior among components with
respect to environmental transformation and fate;
saturation of various Internal enzymatic
processes of metabolism, detoxification, or
genetic repair by some mixture comconents,
thereoy altering the effects of otr.ers.  Certain
mixture components may modify the pharmacokinetic
characteristics of other components, thereby
altering their concentrations at the site  of
action.  Individual mixture components may not be
carcinogenic, Just combinations as with
Initiators and promoters.  Any mechanistic
Information concerning the modes of action of the
mixture components and their Interrelations
should be Incorporated Into the dose-response
models that extrapolate the observed high  dose
laboratory effects to predict health effects at
the low environmental exposure levels.  Thorslund
and Charnley assume away many of these
mechanistic and pharmacokinetic considerations
when they state "...For the mathematical model of
the carcinogenic response...1t will be assumed
that the best available surrogate measure  of dose
at the site of action 1s used as the Independent
variable...".  This 1s easier to assume than to
verify.  In all fairness though, the biological
mechanisms of action are often not very well
understood.
   Thorslund and Charnley generalize the
multistage model to account for wltlple agents
and variable exposure.  Their models are a class
of empirical dose response models that predict
health effects due to Joint exposure, based on
Individual component data.  The models are
motivated by the mechanistic considerations
underlying the multistage model and provide a
plausible explanation of many high dose
Interactions that are observed 1n laboratory
data.   In the absence of specific Information
about  the nature and extent of the biological
mechanisms and interactions, this class of models
offers a workmanlike approach to describing the
low dose behavior of mixtures and the low dose
Interactions that are operative, among the
mixture components.   It  provides  an empirical
extension of  component add1t1v1ty to Incorporate
linear by linear Interaction  terms Into the
predictions.
   Thorslund  and Charnley  state  "... At high
doses the Multiplicative exposure effect tern  can
dominate... and the Joint effect can be much
greater than  the sun  of  the Individual effects.
However, 1f both exposures are reduced by several
orders of magnitude,  the joint effect would be,
to a very close  approximation, equal  to the sum
of the Individual effects... the multistage model
predicts add1t1v1ty at environmental  exposure
levels for almost all situations  that would be
routinely encountered".
   While 1n principle the  Thorslund and Charnley
model Implies low dose component  addltivity, the
viewpoint above  is somewhat of an  overstatement.
It discounts  pharmacokinetic  interactions such as
saturation of elimination  or  repair processes  anc
it ignores the question  of *nat constitutes a
"low" dose.   Several  examples will  be presented
below in which the Thorslunc  and  Charnley mocel
1s predictive of joint toxlcity of a two
component mixture at  environmental  levels of
exposure, but yet where  component  additlvity does
not hold.  Thus,  the  Thorslund and Charnley model
1s not synonymous with component  additlvity of
r1slcs~at Jow, environmental exposure levels.
 Implications of the Thorslund and Charnley *
   The simplest example "of  tne  Thorslunc  ana
Charnley model corresponds  to the  case  of a  two
component mixture and two stages.   Let  U,  and  V
denote the concentrations of components 1 and  2;
assume that  each component  affects  a different
stage.  Equation (13) expresses the risk  of  a
tumor for this special case as
P(U,V) * 1 - exp[-A(l+BU)(l+CV)]
(13)
At low environmental exposure  levels,  P(U,V) can
be approximated as

P(U,V) s A + ABU + ACV + ABUCV
       = POO + P10 + P01 + Pll.

   In this expression POO represents the
background risk, P10 and P01 represent the
additional risks due to each component
separately, and Pll represents a linear by linear
Interaction term between components 1  and 2.
When Pll 1s small  relative to P10 and  P01, the
component additional risks are essentially
additive.  Thus, to determine  when component
additlvity 1s a reasonable assumption it 1s
necessary to determine conditions under which Pll
1s small relative to P10, P01.  The expressions
for POO.PlO.POl.Pll, Imply that

  Pll « P10P01/POO - (P10/POO)P01
      - (P01/POO)P10.

Thus Pll 1s small  relative to P10.P01 1f

   P10/POO « 1 and P01/POO « 1.

   Define a relative risk as the ratio of the
absolute risk to the background risk.   That 1s,
let

-------

R(U,V) • P(U,V)/POO, RIO - P10/POO,
R01 • PO/POO, and Rll « Pll/POO.  Then
R(U,V) • 1 + RIO + R01 + R10R01.

The product ter» 1s small If Rll x R10R01 «1.
   Therefore, what constitutes a "low" dose  In
the Thorslund and Charnley Model depends on  the
level of background risk.  To have additive
componentwise risks, the additional risks
associated with each component must be small
compared to the background risk, or equlvalently
the relative risks must be small compared to 1.
If the additional risks for each component are
large relative to background, the product term
will dominate; the component effects will appear
to Interact.
   Re1f  (1984) presents a number of
epldemiological examples that show the
relationship between joint effects and individual
component effects at environmental levels of
exposure.  We Illustrate the predictiveness  or
lack thereof of the Thorslund and Charnley model
for these examples.

   1. Lung cancer associated with smoking
       (component 1) and uranium mining (component
      2).
    POO  *  .57  x  10"4
    POO  >  P10  »  5.87  x 10
    POO  +  P01  «  2.27  x 10
                        -4
                        -4
        P10/POO
        P01/POO
9.30
2.98.
 The  additional  component risks are  large  compared
 to background.   The joint risk estimated  by  the
 Thorslund and Charnley model 1s
104 P
       (U,V)  * .57  +  (5.87 -  .57) +  (2.27  -  .57)
             + (5.87  -  .57)(2.27 - .57)/.57
             » .57  +  5.3 + 1.7 + 15.81  * 23.38.
 The observed value  1s 22.7.  In this example  the
 component additional risks are large relative to
 background,  the product term dominates,  and the
 components appear to be Interactive.   The
 Thorslund and Charnley model is predictive at the
 environmental level of exposure but componentwise
 additivlty does not hold there.

    2. Lung cancer associated with  smoking
       (component 1) and asbestos work  (component
       2).
   POO
   POO + P10
   POO + P01
          1.13 x 10

                12.3  x  10
                5.84  x  10
-4
-4
P10/POO « 9.88
P01/POO - 4.17.
 The additional  component risks are large  relative
 to background.   The Joint risk estimated  by the
 Thorslund and Charnley model 1s

 104P(U,V) • 1.13 +  (12.3 - 1.13) +(5.84 - 1.13)
           + (12.3 - 1.13)(5.84 - 1.13)/1.13
           - 1.13 + 11.17 + 4.71 + 46.56
           « 63.57.

 The observed value 1s 60.2.  In this example, as
 In the first, the component additional risks are
 large relative  to background, the product term
 dominates, and  the components appear to be
 Interactive. The Thorslund and Charnley  model Is
 predictive at the environmental level of  exposure
 but componentwise additivlty does not hold there.
                                                          3. Lung cancer associated with  smoking
                                                             (component 1) and asbestos mining
                                                             (component 2)

                                                       ROO - 1
                                                       ROO + RIO • 12
                                                       ROO + Rol • 1.6.

                                                       The additional relative risk for smoking 1s large
                                                       whereas that for asbestos mining 1s  not.

                                                          R(U,V) • 1 + 11 + 0.6 + 6.6  • 19.2.

                                                       The observed value 1s 19.0.  In this example the
                                                       component 1 additional relative risk is large,
                                                       the product term 1s large relative to the
                                                       additional effect for component 2, and the
                                                       components appear to be interactive.  The
                                                       Thorslund and Charnley model is predictive at the
                                                       environmental level of exposure but  componentwise
                                                       additivlty does not hold there.

                                                          4. Abnormal sputum cytology  associated with
                                                             smoking (component 1) and uranium mining
                                                             (component 2)
                           POO *  .04
                           POO +  P10  *  .11
                           POO +  P01  -  .08
                        P(U,V) =  .04  +  .07
                               -  .22.
+ .04
 P10/P11 * 1.75
 P01/P11 « 1.0.
(.07)(.04)/.04
                                The observed  value  is  .22.   In this examole the
                                component  additional  risks  are comparable or
                                moderately large  relative to background.  The
                                Thorslund  and Charnley model is predictive at the
                                environmental  level  of exposure but componentwise
                                additivlty does not  hold there.

                                   5.  Oral  cancer associated with smoking
                                      (component  1)  and alcohol use (component 2)
                                                          ROO - 1
                                                          ROO + RIO
                                                          ROO + R01
                                               1.53
                                               1.23,
             The additional componentwise  relative risks  are
             small relative to background.

                R(U,V) - 1 + .53 +  .23  +  (.53)(.23)  =  1.88.

             The observed value 1s  5.71.   In  this example the
             Thorslund and Charnley model  predicts essentially
             componentwise additivlty;  1t  1s  not predictive at
             the environmental level of exposure.

                6. Renal cancer associated with  smoking
                   (component 1) and exposure to cadmium
                   (component 2)

                ROO * 1
                ROO + RIO = 1
                ROO * R01 = .8.

             The additional component relative risks-are
             essentially zero.  Thus, R(U,V)  1s  at most 1.
             The observed joint relative risk 1s 4.4.  In this
             example the Thorslund  and  Charnley  model  predicts

-------
                                                 :':^^ns^
componentwise addHlvHy; 1t 1s not predictive at
the environmental level of exposure.

   The performance of the Thorslund and Charnley
model with Relf's examples has a number of
Implications.

   1. The Model  predicts some observed component
      Interactions at environmental levels.

   2. The model  does not predict all observed
      component  Interactions at environmental
      levels.

    3.  Environmental  exposure levels In  a  number
      of the examples  were sufficiently high  for
       the product term 1n  the  model's  expression
       for risk to dominate.  Thus,  the  Thorslund
       and Charnley model  1s  not  synonymous  with
       componentwise addltivity of  risks at
       environmental'exposure  levels.

    4. What constitutes "low"  levels of exposure
       and "high" levels of exposure for the
       purposes  of the model  1s based on risk
       levels relative to background.  Exposure
       levels that might be quite low on an
       absolute  basis could still be "high" with
       respect to componentwise addltivity 1n the
       Thorslund and Charnley model.

     5.  Irrespective  of whether or not the model
        predicts  component  addltivity, Inferences
        concerning the  Joint effects of  multiple
        components can  be based on  Individual
        component data  alone.   Component data  are
        the most readily  available  for  risk
        assessment purposes.

                     Conclusions

     The USEPA Guidelines for the Health Risk
   Assessment of  Chemical  Mixtures (1985),  page 12
   state  "...When little or no quantitative
   Information 1s available on the potential
   interactions among the components, additive
   models are  recommended for systemic
   toxicants...". This paper carries the above
   recommendation a step further; the model accounts
   for linear by  linear Interactions empirically,
   based  on  component data.  This provides a good,
   empirical  modeling approach 1n the absence of
   specialized mechanistic  Information.  The model
   does  not always predict  componentwise addltivity
   at low, environmental  levels of exposure.   It
   predicts some, but not all, observed
   environmental  Interactions among  mixture
   components.
      What level  of subdivision of  the mixture into
   components should be used when carrying out the
   risk calculations?  If the composite, tested  as a
whole, 1s not carcinogenic at the laboratory dose
levels, can testing be stopped without
considering componentwise tests?  I believe that
the answer Is no!  The USEPA Mixtures Guidelines
(1985), page 11 state '...Even If a risk
assessment can be made using data on the nlxture
of concern or a reasonably similar Mixture, 11
may be desirable to conduct a risk assessment
based on toxlclty data on the components In the
mixture...1n a chronic (high dose) study of such
a mixture  (containing carcinogens and  toxicants),
the presence of the (acute) toxicant could mask
the activity of the carcinogen...the toxicant
could  induce mortality (at high  doses)  so  that at
the maximum tolerated dose of the mixture, no
carcinogenic effect could be observed...".
However,  at  low,  environmental  levels  of exposure
the acute toxicant might  have no effect and  so
the carcinogenic  comoonent might be  active.
 "...The  mixture  approach  should  be  modified  to
 allow the risk assessor  to evaluate  the potential
 for masking,  of  one  effect by another,  on  a case-
 by-case  basis".
    Thus, a sensible  empirical approach to
 carrying out risk assessments on mixtures in the
 absence  of specific  mechanistic information
 concerning componentwise interactions, would be
 to carry out dose response estimation and risk
 calculations based on componentwise testing at a
 number of different levels of decomposition of
 the mixture, ranging from the entire composite to
 very homogeneous constituents.   At each level of
 decomposition the componentwise risks would be
 combined  based on the -Thorslund and Charnley
 model to  obtain  composite risk  estimates.  Large
 discrepancies 1n the composite  risk estimates at
 differing levels of decomposition would indicate
 the  presence  of  synergistic or  antagonistic
 component Interactions.
    In conclusion, I found this  paper to be very
 interesting,  thought provoking, and well  written.
 It raises as  many or more questions about
 methodology  for  risk assessment on mixtures as it
 resolves.

                     References

  (1)   Relf,  A.E., Synerglsm 1n  Carclnogenesis,
       Journal of the National Cancer  Institute,
       Vol.  73,  No.  1, 1984, pp  25-39.

  (2)   Thorslund, T.W. and Charnley, G., Use of
       the Multistage Model  to  Predict  the
       Carcinogenic  Response Associated With the
       Time-Dependent  Exposures  to Multiple
       Agents, Proceedings of the ASA/EPA
        Conference on Current Assessment of
        Combined Toxicant Effects.  1986.

  (3)    USEPA, Guidelines for the Health Risk
        Assessment of Chemical Mixtures.  Final
        Report ECAO-CIN-434,  1985.

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                                             ^j&s
                                 *
                            ASSESSMENT OF INTERACTION IN LONG-TERM  EXPERIMENTS

                      S. G. Machado, Science Applications International Corporation
1.  IKTRDDUCTIOH
  This paper will address th« problem of uses-
sing interaction between carcinogens or toxic
substances in long-tern factorlally designed
animal experiments.  The general context. Is that
of long-term screening tests for carcinogens, for
which the analysis is based on Cox regression
methods, See, for instance, Peto et al. (1980).
The designs considered are 2x2 and 2x2x2.
The method easily generalizes to other factorial
designs.  The problem came to the author's atten-
tion via a request from Dr. C.J. Shellabarger
of Brookhaven National Laboratory who had com-
pleted a 2 x 2 x 2 experiment to examine inter-
actions between radiation and chemical carcino-
gens in the induction of manraary tumors in rats,
and was not sure how to analyze his data.  He
had previously conducted 2x2 experiments and
to add a third treatment seemed a natural next
step.
  In the statistical and epidemiological litera-
ture,  there has been a lot of discussion in
recent years about what is meant by interaction.
Distinctions have been made between statistical
and biological interaction, and interaction  in
the public health sense.  As statisticians are
well aware, presence of interaction in a linear
model  depends on the scale of measurement being
used.  For instance, for a two-way layout with
one observation per cell, interaction  can be  "got
rid of" by a suitable power transformation of the
data.  To  reduce confusion, in an area which is
complicated enough, it is important for the
statistician to define what is meant by inter-
action at  the outset of a study, and what  is
meant  by  "synergism" and "antagonism," since
 these  terms do not mean the same things to all
 scientists.
   In  this  paper, the kind of  Interaction
 considered, between agents A  and B,  is  that
which occurs if  the effect of A  and B  taken
 together  is unexpectedly larger  or  smaller  than
 that  of  the sum  of the effects of A  and B  taken
 separately.  Synergism Is said to  occur  if  the
 joint effect  is  larger  than expected,  and,  con-
 versely,  antagonism occurs  if  the joint  effect
 is smaller than  expected.  The situation  in
 which only A  produces  the effect  of  interest,
 but the presence of B  modifies  the  effect  of A,
 is not considered.  A  and B  are  presumed  to
 have the same  site of  action.
   The underlying model  for  no interaction
 considered here  is Finney's  definition of  simple
 independent action of  different  agents and the
 background (1971).   This  is  equivalent to
 Hewlett and Plackett's model  of  "dissimilar
 nonlnteractive action."
   For long-term  experiments,  under the propor-
 tional hazards assumption,  the model results in
 a linear relationship  between the hazard  func-
 tions rather than  the  multiplicative one  commonly
 assumed for interaction (see  Wahrendorf  et al.,
 1981).  Other researchers  have looked at additive
 as well as multiplicative models for no  inter-
 action, for instance,  Thomas (1981)  in the
 context of general relative risk models-,  and
Prentice et al. (1983) for the analysis  of  an
extensive cohort study.
  The work for this paper was done with  the
assistance of Dr. Kent Bailey, of the National
Heart, Lung and Blood Institute, and is  essen-
tially a continuation of that of Wahrendorf et
al.  The contribution of Korn and Liu (1983)  who
took a non-parametric approach, i.e., without
making the proportional hazards assumption, will
be briefly mentioned.

2.  2x2 FACTORIAL EXPERIMENTS

  The hypothesis to be tested Is that of inde-
pendence of action, i.e., of tumor inducing
potential, between carcinogens ?iven in  combina-
tion.  Suppose, ir. a 2 x 2 experiment with
treatments A and 3, that HQO animals receive  no
treatment and n^Q, HQ^, r\n animals receive,
respectively, doses d^ of A, dg of B and
(d^ •*• d-g).  The animals are observed throughout
the experiment for occurrence of tumors  of
interest; times of tumor appearance or of deaths
from unrelated causes are noted.  Let qij(t),
i,J-0,l, be the probability that an animal  in
the group (i,j) remains tumor-free up to time t.
Let QIJ » qij(T) where T is any time after  the
last event.  Let a^j be the number of aniaais
with tumors in grouo (1,1) at time T.  Finney's
hypothesis of simple independent action  of  A and
B is:

              Ho:  QiiQoo " QioQoi

Synergism corresponds to the left hand side
being much less than the right hand side of this
expression; conversely for antagonism.
  Various methods have been proposed for test-
Ing this hypothesis, see Wahrendorf et al., Korn
and Liu, or Hogan et al. (1978).  Perhaps the
simplest conceptually is the likelihood  ratio
test.  The likelihood is proportional to:
                      mij   nij ~™i j
          Hijd - Qij)   QIJ

  The log-likelihood is first maximized  with
^Qlj) as four independent parameters, i.e.,
Qij • (nij-mij)/nij , and then with the {Qij }
constrained by the null hypothesis.
  Taking time into account, Finney's hypothesis
of independent action of A and B becomes:

  HQ:  log ([qii(t)qoo(«)]/lqiO
-------

Synerglsm corresponds co the expression on the
left of (1) being greater than zero  and,  con-
versely, antagonism corresponds  to the expression
being less than  zero.
  If che proportional hazards assumption  is
•«4e. aiMly  that Aij(t) may be  expressed as
A00(t)fij where  fj.j does not depend  on t,
then the null hypothesis becomes, without loss
of generality:

  HQ:   1 + fil - «10 - f01  • °  .

the arbitrary scale factor having been absorbed
into the arbitrary function AQO^i'   T^6
alternative hypothesis allows for all three
parameters to be free.  Let f -  (foo if 10 >f01t
fll)'.  For  testing HQ it is convenient  to
express f  in  terms of parameters (6) .   Let
              Si     SV4
Tv^CS)   •  (l,e  ...,e  )'  for arbitrary H>-1
and  let W  be  any 1x3 aatrix of  constants  with
colucr.s orthogonal to the interaction contrast
vector  (1,-1,-1,1)' • VAB, say.  There are many
choices for W;  a natural  choice  is  to define W
as  the  first  three columns of the design matrix
 Under HQ, f is modelled as  f  - WT2(S), and
 under the alternative hypothesis, f » 13(8).
 The null hypothesis HQ is  tested by the likeli-
 hood ratio statistic, X -  2(Lg-Lc), where LS,  LC
 are, respectively,  the maximized log-likelihood
 functions under the saturated and constrained
 models; X has approximately Xi  distribution
 under the null hypothesis.   Technical details  for
 the likelihood maximization are described in
 Machado and Bailey (1985).

 3.  EXAMPLE 1

   The data for this example is from Table 2 of
 Kom and Liu (1983).  Female rats were treated
 with two chemical carcinogens, labelled NTA and
 MNNG.  The endpoint of interest was death from
 any cause.  All rats alive at the end of the
 experiment, as well as 4 accidently killed early
 on, are considered censored observations.  The
 results of the analysis were:
Saturated
  Model

Lc - -291.481
                                Constrained
                                   Model
/\
                                  - -294.588

                                  -  /0.25
                                     11.56
      f  -  -5.31

vAB'f -  0
    The likelihood ratio test, X - 6.21 (a -
  0.013),  indicating significant departure from the
  null hypothesis; thetinteraction Is antagonistic,
  fro* the sign of vAB*f.  This result corresponds
                          to  those of Korn and Liu.
                           For comparison, the time-Independent  likeli-
                          hood  ratio test was 2(Ls - LC) " 4.32;  this
                          statistic has approximately Xj.  distribution,
                          thus   -nis group may be excluded
                          from  the analysis and f_ - ( f ^Q ,IQ^ , f^)  iay
                          be  modelled by VJT(p) with
                                            W  -
                          This represents the additive  model  subject  to
                          f(30 * 0.  If the parameters are  estimated as
                          In Machado and Bailey,  the log-likelihood is
                          the log-likelihood for  the additive model (1)
                          for all four groups.  The saturated model is
                          fitted similarly by omitting  the control  group
                          and modelling (f 1Q ,fQ1 ,f u)   by  T2( P).

                          5.  EXAMPLE 2.

                            This example was described  in  Machado and
                          Bailey.  The data  are from a  nine-month study
                          to investigate possible interaction between
                          the known carcinogens dlethylstilbestrol  (DES)
                          and dimethylbenzanthracene (DMBA)  in the
                          Induction of mammary adenocarcinomas In female
                          AC1 rats (Shellabarger  et al.,  1980).  The
                          results of the test for interaction are:
Saturated
  Model

LS - -152.46

0  »  |-0.52\
      \ 1.471
A.
                                                            Constrained
                                                              Model

                                                         LC - -156.848

                                                         •  -   -0.68
                                  0.60 I
                                 \4.33/
                                                                                         f  -
                                      o.5
                                        .5
  The test statistic X - 8.77  (a- 0.004),
indicating very significant departure  from
the null^hypothesls.  Since the  contrast,
fll-?10-foii ls greater than 0,  the  Inter-
action is synergistlc.

  The likelihood ratio test from the time-
independent test is 6.61 (*-  0.01), again
indicating significant interaction.

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                                                                                    y^^
6.   PROPORTIONAL HAZARDS ASSUMPTION.

  The proportional hazards assumption should
b« chtcked since gross departures from propor-
tionality may well affect the behavior of
the interaction teats in an adverse way.
talbfleisch and Prentice (I960, ch. 4) recommend
using "log-minus-log" plots:  plots for all the
treatment groups of log(-log(q(t)) versus log(t),
where q(t) is any estimate of qj.j(t), will show
constant separation over time if there is pro-
portionality.  For small to moderate sized
samples, it may be difficult to discern from the
plots whether there actually is constant separa-
tion over time.  In this case the uniformity of
the fit of the saturated and constrained models
and of the behavior of the interaction statistics
over time may be checked by estimating sets of
parameters for different partitions of the time
axis.  Most likely, divisions into early versus
late, or early, middle and late tiae periods will
be sufficient.
  Suppose T is any time beyond the time to the
last event and let a  time  t be chosen so that
the time axis is partitioned into (0,t], (t,T].
Let Ls(l), Ls(2) be the maximized log-likelihoods
under the saturated model, respectively, for the
time periods  (0,t) (events after t considered
censored) and  (t,T] (individuals with either
events or censoring times  before t excluded from
the analysis); let Lc(l),  Lc(2) be similarly
defined maximized  log-likelihoods for the con-
strained model.  Then a  test for the uniformity
of  the saturated model over time is 2(Ls(D +
1.5(2) - Lg) which  has approximate chi-square
distribution  with  degrees  of freedom 3  for the
2x2  case.   A test  for  the uniformity of the  fit
of  the constrained model  over time is 2(L<;(1) +
L(;(2) - LC) which  has approximate chi-square
distribution  with  degrees  of freedom the number
of  parameters in the  model, e.g., 2 in the 2x2
case.   If  there  appears  to be no lack of unifor-
mity  of  fit  of the saturated or constrained
models and  no evidence of  any interaction, then
 one would  be  comfortable in accepting  the null
 hypothesis.   If  there seems to be evidence of
 interaction and  the  Interaction is of  the same
 type  in each time  period,  and also if  there is
 uniformity of fit  of  the saturated model, then
 the overall test of  interaction under  the pro-
 portional hazards  assumption  can be used.  If
 there is evidence  of  nonproportionality,  the  two
 time periods  could be considered separately with
 respect to the presence or absence of  inter-
 action.   It is worth checking  the  consistency  of
 the conclusions  of such an analysis when  different
 values of t are chosen.
   If the proportional hazards  assumption  does  not
 appear to hold,  with different  partitions of  the
 time axis, then the non-parametric methods  of
 Korn and Liu (1983)  may be more appropriate.

 7.  EXAMPLES

   Figure 1 shows the "log-minus-log"  plot for  the
 data of Korn and Liu discussed in  Example 1.
 There is no reason to suspect  departure from  pro-
 protionality of the hazard function.
  Figure 2 shows the "log-minus-log"  plot for
the data of Example 2.  The plot  indicates some
departure from proportionality since  the curve
for the group receiving both DES  and  DKBA seems
steeper than those of the groups  receiving a
•ingle treatment.  The time axis  was  partition-
ed into two periods:  (0.136) and (137,266),
day 136 being approximately the half-way point
in time and in numbers of events.  The maxi-
mized log-likelihoods and estimated parameters
for the saturated model were, for the early
time period:  LSU) - -80.63, Lc0.10) indicating no strong
evidence for lack of uniformity;  the  estimated
3 coefficients are not very similar but are far
from significantly different.  The xj  tests
for Interaction  for the early and late tine
periods are 5.45 (a - 0.020) and  3.56
(a " 0.059), respectively; moreover,  the
interaction contrasts are 2.35 and 3.32 and
thus there is significant synergism for both
time periods.  Similar results were obtained
for various t between 129 and 190, Thus it
appears that there was a synergistlc  inter-
action between DES and DMBA In this experiment.

8.  KORN AND LIU'S 2 STATISTIC

  Korn and Liu proposed a statistic for
continuous time  data which does not rely on the
proportional hazards assumption.   They made the
reasonable suggestion that the model  of inde-
pendent action with no further assumptions is a
good starting place for an analysis.
  Restricting attention to the 2x2  case,
suppose events occur at time ti,  t2,..., and
suppose that there are no tied events.  Let
where  nij(t^)  is  the number of  animals'in
group  (i,j)  exposed at time c^-0,  ru-+(tk) -
zijnij (ck)> and  the failure occurred in Kroup
(i,j).  The  statistic for testing  Finney's
hypothesis of  independent action is:

                  Z Zk
               (I
which has  approximately N(0,l) distribution.
The  terms  in the numerator,  {Zfc},  have
conditional expectation zero under the null
model, and In this, are unique up  to a multi-
plicative  factor.
   If there are ties in the data, Korn and Liu
suggest breaking them at random.   Note that
with the test discussed in this paper, it is
possible to use the general maximum likeli-
hood solution to the proportional  hazards
model, and thus ties do not pose a problem
(see Machado and Bailey).
   Korn and Liu's statistic was calculated for
the  data of Examples 1 and 2:

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Example 1.

Z • 2.45 (o • 0.014) compared with /X •
/6.21 • 2.49 (o - 0.013).  Mote that In
their paper, they obtained Z - 2.48, a minor
difference, but due to arbitrariness in dealing
with ties. For this example, the likelihood
ratio test and Z are very close.

Example 2.

  With the data ordered by treatment group:
Controls, DES, DMBA, DES plus DMBA, Z - -1.87
(a-  .06); with the data ordered in reverse,
i.e., DES plus DMBA, DMBA, DES, Controls,
Z • -1.645 (a- 0.10). There are a lot of  ties
in the data, which give rise to the difference
between  these  two values of Z.  These values  are
rather different from  /8.77 - 2.96 (a » 0.004)
from  the  time-dependent likelihood ratio test,  and
/6.61 •  2.57  (a - 0.01) of the time-independent
test.   It  is  difficult to see why  the Z values
should  indicate less evidence of  the  synergism
between  DES  and DMBA.
  Once  a consistent approach to dealing with
ties  is  found,  the Z statistic should prove
useful,  since it is reasonably simple  to
compute, especially  for situations  in which one
 fee.ls uneasy about assuming proportionality of
hazards.

9.  2*2*2 FACTORIAL EXPERIMENTS

   Let the three  treatments of  interest  be  A,
 B and C and  suppose  that  there are njifc animals
 in the group (i,J,k)  receiving a  total  treatment
 dose of (idA + jdB + kdc),  for l,J,k-0,l.  Let
 IllVt^)' Hjk(c)  ke defined in an analogous
 way to qij(t), ^ij(t)  of  Section  2.   Further,
 under the proportional hazards assumption, let
 Xljfc(t) " ^OOO^t^ijk. »here  fijfc is  Inde-
 pendent of t.  Let  q(t)  be the vector of the
 {qijkXt)} -with the  subscripts  in the order
 (000,100,010,110,001,101,011,111) and let  f  be
 the vector of  {fijk)  with  the  subscripts
 in the same order.   Let the  columns of the
 design matrix


            (i -11  ®  i:  11  ®  (i -i

 be labelled in order,  VQ, VA,  VB, vy^j, vc, VAC ,
 VBC> VABC'  Then the null hypothesis of inde-
 pendence of action of the three treatments has
 four parts, corresponding, respectively to the
 interactions between A and B, A and C, B and C,
 and A, B and C.:

'   HO:     VAB'log(q) ' 0 , (11) vAC'log(q) - 0,
       (iii)  vjc'logU) - 0,  (iv) vABC'log(q) - 0

   Under  the  proportional hazards assumption, as
  in  the  previous section, HQ becomes:
        (i)   VAu'f ' 0
      (iii)   vBC'f ' 0
(11)   VAc'f - 0
(lv)  v^c'f - 0
with the saturated model,  f  • WT7(B);  Che
resulting likelihood ratio statistic  has
approximately x*  distribution.   One  choice
for W is to take as its columns  VQ, VA,  vg,  VQ.
Sequential tests for single  Interactions are
made by modelling the specific constraints by
suitable choice of V.  for example, to test  for
the three-way Interaction, model f by W^CB),
where Wj Is 8 x 7, orthogonal to ?ABC» *n(* Bake
a one degree of freedom comparison with the
saturated model.  Further, to then test  for  the
AB interaction, model f by W2T5(6), where
V/2 is 8x6, orthogonal to v^g and Vp&c, and
compare with the model WjTgCS).   Leaving out
one constraint at a time  leads to a series of
one-degree of freedom comparisons in  the usual
way.

10. EXAMPLE 3

  The data for this 2x2x2 exatnole are  frow
a one-year experiment to  assess  possible inter-
action between DKBA, procarbazine (PCZ)  and
x-irradiation (X-ray) in  the induction of mam-
mary adenocarcinomas or fibroadenomas in feniale
Sprague-Dawley rats.  The experiment  was con-
ducted by Dr. C.J. Shellabarger  and  colleagues
at Brookhaven National Laboratory who kindly
made the data available.   The rats were treated
at about 3 months of age  and examined weekly
for the appearance of mammary tumors.  Summary
information on nuobers of rats with  one or  -nore
tumors is in Table 1.

Table  1.  Summary information on numbers of
          female rats treated with DMBA, PCZ
          and/or X-ray which developed mammary
          tumors.
  Treatment  group     Number
                       exposed

  Control               35
  DMBA                  37
  PCZ                   37
  DMBA & PCZ            37
  X-ray                 37
  DMBA & X-ray          37
  PCZ & X-ray           36
  DMBA, PCZ  & X-ray     37
                                                           Number with at
                                                           least one tumor

                                                                 2
                                                                14
                                                                 8
                                                                20
                                                                11
                                                                20
                                                                14
                                                                25
    A joint, test  of  the four parts of HQ is made
  by comparing the constrained model f • WT3< B),
  where W is an 8 x  4 matrix of constants with
  columns orthogonal to v^g,
   The maximized log-likelihoods were:
 LS - 510.981, LC • -511.422 resulting in
 test statistic for overall interaction, Xi  "
 0.822, which is far from significance.  There
 were no two-way or three-way interactions
 between DMBA, PCZ and X-ray since the single
 degree of freedom chi-square tests for Indivi-
 dual interactions are all bounded by such a
 small number.  For comparison, the likelihood
 ratio statistic from the time-independent test
 is 0.682, also very small.  The estimates of
 the multipliers {fijfe} from the saturated
 and constrained models are very similar:

 ?- (1,2.19,1.47,2.66,1.90,2.85,2.18,3.05)' and
 f - (1,2.42,1.53,2.70,1.96,2.85,2.37,3.04)' .

   "Log-mlnus-log" plots showed close to constant
 separation between all of the curves and thus
 Is there no reason to doubt the proportionality

-------
                         •  -•   ••       .    1-•>•••'.%,.   - r«-.,,»^v^*- ,kV.   ^-x^.,,
                     ^
of the hatard function*. Thus the three treat-
ments act Independently In the Induction of one
or acre mammary tumors in this species of rat.
  For this data, the time to the appearance
of second tumors was also recorded.  This is a
much leȤ understood measure of carcinogenesis,
and the analysis is summarized here only for
Illustration of the method.  Table 2 gives the
numbers of animals in each treatment group which
developed 2 or more mammary tumors.

Table 2.  Summary  information  on  numbers  of  female
          rats treated with DMBA, PCZ, and/or
          X-ray which developed 2 or more mammary
          tumors.
  Treatment  group       Number
                        exposed

     Control                35
     DMBA                  37
     PCZ                   37
     DMBA & PCZ            37
     X-ray                 37
     DMBA & X-ray          37
     PCZ and  X-ray          36
     DMBA, PCZ 4 X-ray      37
Number with 2
or more tumors

     0
     5
     1
    15
     1
    10
     3
    19
   The likelihood ratio  test  for, no two-way or
 three-way interactions  was   XA" " 1°-48 (<* '
 0.035), indicating the  presence of some inter-
 action.  The test for no DMBA  and X-ray, or PCZ
 and X-ray, or DMBA,  PCZ and  X-ray interactions
 was X3  • 2.77 (a >  0.1), which is far from
 significance.
   The xi  test for the  DMBA  and PCZ  interaction
 was 7.71 (a - 0.006), which  is highly signifi-
 cant.  Examination of the statistics showed this
 to be due to synergiso  betwen  DMBA and PCZ, in
 the induction of multiple mammary tumors.
 Although the biological implications of this are
 not clear, this example shows  that the test can
 identify which pair of  agents  contributed to the
 overall departure from  the null model.
11. REFERENCES

 Finney, D. J. (1971).  Problt  analysis,  3rd ed.
Cambridge:  Cambridge University Press.
 Hogan, M.D., Kupper, L.L., Most, B.M., Baseman,
J.K. (1978).  Alternatives to Rothman's approach
to assessing synerglsm (or antagonism) In cohort
studies.  American Journal of Epidemiology 108,
60-67.
 Kalbfleisch, J. 0. and Prentice, R.  L. (1980).
The statistical analysis of  failure time  data.
New TorVc:  Wiley.
 Korn, E. L. and Liu, P. (1983).  Interactive
effects of mixtures of stimuli  in life table
analysis.  Biometrika 70,  103-10.
 Machado, S.G. and Bailey, K.R. (1985).
Assessment of interaction between carcinogens
in long-term factorially designed animal
experiments.  Biometrics 41,  539-545.
 Peto, R., Pike, M.C., Day,  N.E., Gray, R.G.,
Lee, P.N., Parish, S., Peto,  J., Richards, S.,
Wanrendorf, J. (1980).  Guidelines  for siaole,
sensitive sisnificance tests  for carcinozenic
effects in long-tera animal  experiments.   Annex
to Long-term and Short-term  Screening Assays
for Carcinogens:  A Critical Appraisal.   Inter-
national  Agency for Research on Cancer.   Lyon.
 Prentice, R. L., Yoshimoto,  f. and Mason,
M. W.  (1983).  Relationship  of  cigarette
smoking and  radiation exposure  to  cancer
mortality in Hiroshima and Nagasaki.  Journal
of  the National Cancer Institute 70,  611-22.
 Shellabarger, C. J., McKnight, 3., Scone,
J.  P.   and Holt?mann, S.   (1980).   Interaction
of  Dimethylbenzanthracene  and Diethylstil-
bestrol on mammary adenocarcinoma  formation in
female ACI  rats.  Cancer Research 40, 1808-11.
 Thomas,  D.  C. (1981).  General relative-risk
models for  survival time and matched case-
control analysis.  Biometrics 37, 673-86.
 Wahrendorf, J., Zentgraf, R. and Brown,  C. C.
 (1981).   Optimal designs  for the analysis of
 interactive  effects of two carcinogens  or other
 toxicants.   Biometrics 37,  45-54.

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                               &&&£9^i*^i&&^&?M&^^
UW
-0.5 •
-1.0 *
-1.5
 -20-
 -2.5
 -3.0 •
 •3.5
              1t  Concroli
              2:  UTA
              3:  MKNG
              *:  NTA 4 HNNC
                                            FIGURE 1
                                  FLOT OF   loi
                                  KOBH-LIU DATA
                                                                 33    4
                                                                       4
                                                                3      4
                                                               3      4
                                                             3    4
                                                          3     422
                                                          3    4    2
                                                         43        2
                                                                        1 I
                                                         i      2
                                                   2    4
                                                      3
 -4.0
 •4.5 •
                                                   1
       5.5
 UOC I
  I .0 •
  0.5
  a.e
 -o.s
 •i.o
 •i.s
 -2.0
 -2.5
 -3.0
 -3.5
5.6    5.7     SB     5.9     6.0     6.1     6.2     63    64    65
                                IT

                               FIGURE 2
                        PLOT OF   lo»{-lo»(q(t))  v..  log(t)
                        DES, DKSA EXPEXIMtNT

   2:  DES                                             *        *
   3:  DHBA                                            4
   4:  DES t DKBA                                 *
                                        4     2
                                                            222

                                           2      2 2  22 22 2
                                                          33   333
                                                         3
                                                  3     3
                                             3   3
                                           3 3
                                   3    33
                                                                                       6 6
       4.0       4.2       4.4      4.6      4.S      S.O       5.2       5.4       5.6
                                              IT
                                                                          S.O

-------
                                              DISCUSSION
   Fro* Machado '• presentation we have seen that
quantities
               Chao W. Chen, U.S. Environmental Protection Agency

                                            is equivalent to the null hypothesis

                                                               HQ:  bu  • 0.
olay a key role In the hypothesis  testing  of  no
interaction when  time-to-event  data  are avail-
able!  Th« null hypothesis of no  interaction, for
the case  of a  2x2 design, is
 for every c.   For the case of a 2x2x2 design, the
 null hypothesis of independent action of the
 three treatments has four parts corresponGisg ,o
 the three two-way and one three-way interactions.
    Given the above background, my discussion
 will be on the following three issues:

     1   Application of the above null hypothesis
      '  to test the multiplicative effect of  syn-
        ergism under  the theory of the Armitage-
        Doll multistage model.  Under  the multi-
        stage model,  if each  of the  two  (or more)
        carcinogens  affects a single hut different
        stage of  the  carcinogenesis  process,  a
        synergistic  effect  (in a  multiplicative
        sense)  will  result.

     2.  The  implication  of the alternative null
  hypothesis  of  no Interaction,
      H02:  TlO»  TQI, and  T^ be random variables
                                             representing, respectively,  time to  cancer  death
                                             of  an  animal exposed to dose rate 0,  dA,  dg and
                                             (dA+d]j) under the condition  of no competing risk.
                                             Let MJ_J •  min(Tij ,  L) where  L is the time when
                                             the study  terminated.   It  follows that the expec-
                                             ted time to cancer death (latent period) is given
                                             by
                                                              E(Mi1) • /  qt1(t)dt
                                                                        0

                                              and the life-shortening due to the exposure is


                                                       RiJ	* E(Mo0-Mtj)
                                                              L
                                                              L     0

                                              Therefore, the null  hypothesis  of  no  interaction
                                              can be defined as

                                                             HQ: RH  - RIO + ROI

                                              which is  equivalent  to  the null hypothesis

                                                        L
                                                  HO:  / l  *  IQOM - qiO
-------
 '''( '**» -»•]'.,

 ^^^^^
vhcr*
    tit  ••••  tn  **• the time to Cancer death,
    to • 0, and
   q\j • Kaplan-Meier estimate of qj_ j .

Th« estimate  of  standard deviation Is given by
              u  - »„<«.,! -
The statistic

          Z - (eu
                         - e10 - e01)
               (311 * 900 + 310 * 301}

is  asymptotically distributed as a  standard nor-
mal under HQ.

3.   Implication  of the Hull Hypothesis used by
     Machado in a 2x2x2 Design

    The  null hypothesis used by Machado is

                   HQ: V'f - 0

                               ir are colunn vec-
where V  - V^, VAC, VBC, or
tors given below:

              VAB      VAC
                                                       Similarly,
                                                       and
                                                                                            - o
                                        Mow,  synergistic and antagonistic effects should
                                        be as follows:

                                          EABC • E(A,B,C) - E(A) - E(B) - E(C)

                                               " (PHI  * POOO> - (PIOO - POOO)  "
                                                 (POIO  ~ POOO) ~ (POOI - POOO)

                                               " Pill ~ PIOO " POIO ~ P001 *  2POOO

                                        This  shows  that the null hypothesis used  by Ma-
                                        chado is stronger than the test of no synergetic
                                        or antagonistic effects as defined by S^gc-   If
                                        there is no pairvise interaction [i.e.," E(AB)C»0
                                        • 0,  etc.],  the null hypothesis HQ of Machaco is
                                        equivalent  to the null hypothesis of  E^c " 0-
                                        However, in general, E^c - 0 does not  imply HQ
                                        to be true.
                                           Further  research is needed in testing  the null
                                        hypothesis  of no synergistic or antagonistic ef-
                                        fects on 2x2x2  design, without assuming that the
                                        pairvise interactions are zero.  It seeas  intul-
                                        ively true  that if all pairvise interactions are
                                        positive, there must be a synergistic effect.  On
                                        the other hand, if all the pairwise interactions
                                        are negative, there must be a synergistic  effect.
                                          'ABC
      o
      A
      B
     AB
      C
     AC
     BC
    ABC
               1
             -1
             -1
               1
               1
             -1
             -1
               1
                 1
                 1
                -1
                -1
                -1
                -1
                 1
                 1
Pijk>

  or
                        J. k - 0, 1

                         as used by Machado.
 Let
 denote
    The null hypothesis HQ corresponds to four
 parts :

 HABC " Pill ~ PllO - P011 " P101 * P100 +
        P010 * P001 * POOO
              ' PlOl ~ P011 * P001> ~
        (PI 10 ~ PIOO • POIO * POOO)
 where (EAB)C«I  and  (ExB^C-0 *re defined by the
 last equality of  the equation and represent,
 respectively, the effect due to treatment A and
 B when C is held  at level  1 or 0.

 HAB " (Pill * P011  ~ PlOl * P001> *
       
-------
                 &fe£^^
                           CONCENTRATION AND RESPONSE ADDITION OF MIXTURES OF TOXICANTS
                                 USING LETHALITY,  GROWTH AND ORGAN SYSTEM STUDIES

                  Lavern Weber,  Perry Anderson,  Carl  Muska, Elizabeth Yinger and Dennis Shelton
                                             Oregon  State University
   Dose response relationships are the most
single unifying concept to the nany branches of
pharmacology and toxicology.  Quantative
methodology to describe dose response
relationships began with the work of Trevan
(1927) and Gaddum (1933).  The theoretical basis
of joint toxicant action was first systematically
discussed by Bliss (1939).  Bliss recognized
three types of joint action:  (1)  Independent
joint action - the chemicals act independently
and have different modes of action; (2)  Similar
joint action - the chemicals produce similar but
i ndependent effects, one component can be
substituted at a constant proportion for the
other.  Susceptibility to the chemical components
are completely correlated; (3)  Synergistic (or
antagonistic) action - the effectiveness or tne
mixture cannot be assessed from the individual
chemicals.  Bliss' approach was modified by
Finney  (1942) to develop a logical relationship
between the mathematical expressions for the
different types of joint action.
   Plackett and Hewlett (1948, 1952) and Hewlett
and Plackett (1952, 1959) proposed a two-way
classification scheme of Bliss' model in an
effort  to provide a less restrictive form.  The
following diagram is their scheme with four
distinct'types of action.
 Non-
 Interactive
Similar
Simple similar
Dissimilar
Independent
 Interactive    Complex  similar     Dependent

    They defined toxicant mixtures as "similar" or
 "dissimilar"  according  to whether the toxicants
 acted upon the same  of  different biological
 systems and as "interactive"  or "non-interactive"
 according to'whether one toxicant influenced the
 "biological action"  of  the  other toxicants.
 "Simple similar" and "independent action" were
 regarded as special  cases  in  a continuum of
 biological possibilities and  the mathematical
 models proposed for  complex similar and dependent
 were generalizations of the models proposed for
 "simple similar" and "independent action"
 respectively.
    Their mathematical models, particularly for
 the quantal responses to mixtures of
 "Interactive" toxicants, are  very complex and
 require the knowledge of certain parameters which
 are normally unattainable when evaluating the
 effects of toxicant  mixtures  on whole organism
 performances.  However, Hewlett and Plackett's
 models for "joint action"  are useful for
 elucidating the limitations of and the
 assumptions required for the  special cases of
 "simple similar" and "independent joint action".
 The present discussion  only considers the special
 cases of "noninteractive"  toxicant mixtures.
    The difficulty of understanding complex
 mixtures and the Interactive  role that Individual
 toxicants play is not easily  elucidated.  In 1970
 my laboratory began  to  investigate the toxicity
 of mixtures of chemicals.   In the following •
decade we  utilized  the concepts of early
Investigators such  as Bliss (1939), Finney  (1942)
and Hewlett and Plackett  (1959) to study  the
validity of their nodels  for studying toxicant
Interactions.  The  results to be discussed
involved tests of the model using lethality
(Anderson  and Weber 1977), growth (Kolkemeister
and Weber  1979; Muska and Weber 1977, Weber and
Muska 1977) and on  an organ system (Shelton and
Weber 1981).
   The regulatory agents  at the beginning of  our
work essentially followed the National Technical
Advisory Committee's recommendation that  the  sum
of the ratios of tne measured concentration of
the permissible level of  each toxicant snould not
be greater than one.  This basically follows  the
concept of a "Toxic Unit".  The "Toxic Unit"
method arbitrarily  assigns a value of one to  that
concentration which induces particular response,
such as LCc0.  The  concentration of each  toxicant
in a mixture is then expressed as a fraction  of
its corresponding LCrQ value.  The fractions  are
added and  if the resulting quantity is equal  to
the toxic  unit (1)  than a 50% response is
predicted  for the mixture.  The basic assumption
of the "toxic unit" is that each toxicant
contributes to a common effect in proportion  to
its relative potency.  In Bliss' model this wouia
be "similar joint action" or in Hewlett and
Plackett's it would be "simple similar".
   A multitude of terms have been suggested to
describe the various types oi-combined toxicant
effects.   Ariens (1972) and Fedeli et al. (1972)
reviewed the various terminologies that have  been
used.  As  Sprague (1970)  and Warren (1971)  point
out, the nomenclature is  confusing particularly
since certain terms have  been defined in  more
than one way by different authors.  Furthermore,
terminology describing the mechanisms of  toxicant
action is  not appropriate for studies evaluating
the effects of toxicant mixtures on whole
organism performances without knowledge of  the
action of  the individual  toxicants.  To avoid
both ambiguities in terminology and assumptions
implying knowledge  of sites and mechanisms  of
toxicant action, Anderson and Weber (1977)
introduced the terms concentration and response
addition which are  mathematically analogous to
the "simple similar" and  "independent action"
defined by Plackett and Hewlett (1952).
   Concentration addition is mathematically
defined as the additive effect determined by  the
summation  of the concentrations of the individual
constituents 1n a mixture after adjusting for
differences in their respective potencies.  The
primary assumption  governing this type of
addition  is that the toxicants in a mixture act
upon similar biological systems and contribute to
a  common  response in proportion to their
respective potencies.  Bliss (1939) and others
have assumed that if two  toxicants act similarly
the variations in susceptibility of individual
organisms  to the toxicants are completely
correlated.  As a consequence, the dose response
curves for the components and the mixture are
parallel.   This has been  observed for some

-------
                                                              vf^-mtt-
                        *^  . ^:.-.-.-^M'^l^'^^^^iAJ^l\^iK^ ri
toxicant mixtures; however, Plackett and Hewlett
(1952) presented examples of chemically related
insecticides which gave nonparallel lines.  They
and other toxicologists (Ariens and Simonis,
1961) have stated, and we believe rightfully so,
that parallelism and hence complete correlation
of Individual susceptibilities 1s not a necessary
prerequisite for this type of addition.
   In cases where the dose response curves for
the  Individual toxicants  in a mixture are
parallel, a dose response curve for the mixture
can  be calculated based upon the assumption of
concentration addition.   With the regression
equations for the individual toxicants in the
form of y » a + b log x (where y is the "»
response to each toxicant and x is its concen-
tration), the regression  equation for a binary
mixture  can be represented by (Finney, 1971):
             b log
                                 b  log  Z
:D
 where,
     y  = * response to the mixture
     aT = y intercept of_the first toxicant
     b  « common slope
     IT. * proportion of the first toxicant in the
mixturi
     HP - proportion of the second toxicant 1n
the mixture
     p  * potency of the second toxicant relative
to the first
     Z  = concentration of the mixture

   This equation can readily be adapted to
represent mixtures containing more than two
toxicants.  It should be noted that equation (1)
for concentration addition is similar in
principle to the toxic unit method used by Lloyd
(1961) and Brown 91968).  Whereas the toxic unit
method measures the toxicity of mixtures only at
particular levels of response (LD.g, LC5Q, etc.),
equation  (1) incorporates the entire do5e
response  curve.
   Response addition is the additive effect
determined by the summation of the responses of
the organism to each toxicant in a mixture.  This
form of addition is based on the assumption that
the toxic constituents of a mixture act upon
different biological systems within the organism.
Each organism in a population 1s assumed to have
a  tolerance for each of the toxicants in a
mixture and will only show a response to a
toxicant  if the concentration exceeds Its
tolerance.  Consequently the responses to a
binary mixture are additive only if the
concentrations of both toxicants are above their
respective tolerance thresholds.  However, for
quanta1 responses the tolerances to the toxicants
in a mixture may vary from one Individual to
another  in a population; therefore, the response
of the test animals depends also upon the
correlation between the susceptibilities of the
 individual organisms to the discrete toxicants.
For  example, in order to predict the proportion
of organisms killed by a binary mixture, it 1s
necessary to know not only the proportion that
would be  killed by each toxicant alone but also
 to what degree  the susceptibility of organisms to
one  toxicant 1s correlated with their
 susceptibility  to the other toxicant.
    Plackett and Hewlett  (1948) recognized this
                                                       statistical concept  and  developed mathematical
                                                       models that accounted  for the  correlation of
                                                       Individual tolerances  ranging  from total  negative
                                                       to total positive  correlation.   If the
                                                       correlation is completely negative (r - -1)  so
                                                       that the organisms most  susceptible to one
                                                       toxicant (A) are least susceptible to the other
                                                       (B), then the proportion of  Individuals
                                                       responding to the mixture (P ) can be represented
                                                       by;
                                                          pm m PA + PR 1f (P, " Ph  *  1)              2a
                                                           m    A    B      a     b
                                                       where P. and Pg are  the  respective proportion of
                                                       organisms responding to  the  individual  toxicants
                                                       A and B.  With no correlation  (r  * 0)  in
                                                       susceptibility the relationship  is expressed by:
     p  = P
     Pm   KA
                                                                             - PA)
                                                     (2b)
   In the limiting case of complete and  positive
correlation  (r = 1), individuals very  susceptible
to toxicant  A in comparison with the peculation
will be correspondingly very susceptible  to
toxicant B.  In this situation the proportion  of
animals responding the the mixture is  ecual  to
the response to the most toxic constituent  in  the
mixture.  Mathematically this is represented by:
              PM ' PA  1f PA * PB

              PM-PB  ifPB*PA
                                                                                                     (2c)
                                                        For response addition, no significance can be
                                                        placed on the slope of the dose response curves
                                                        because the toxicants in a mixture are acting
                                                        primarily upon different biological systems with
                                                        varying degrees of susceptibility between
                                                        organisms.  Even if the regression equations for
                                                        the constituents in a mixture are parallel for
                                                        toxicants acting in this manner, the dose
                                                        response curve for the mixture will not be linear
                                                        (Finney, 1971).  This will be illustrated later
                                                        for two hypothetical toxicants whose dose
                                                        response curves are parallel.  Although the
                                                        mathematical equations (2a, b, c) representing
                                                        response addition are relatively simple, the
                                                        statistical consequences of this type of adoition
                                                        are more complicated than those of concentration
                                                        addition (Finney, 1971).
                                                           Terms such as supra- and Infra-addition are
                                                        used to describe toxicant Interactions which are
                                                        greater or less than those predicted on the basis
                                                        of either concentration or response addition.

                                                        LETHALITY STUDIES:  Anderson and Weber (1975)

                                                           Our first efforts were before we fully
                                                        recognized all the assumptions in the two models
                                                        we wished to use.  We felt we could simply use
                                                        fish of one species and we began our work.  Our
                                                        first lesson was that although we were
                                                        environmentally exposing the fish to toxicants,
                                                        the actual concentration to give a particular
                                                        response varied greatly with changes in size and
                                                        stock of fish (Anderson and Weber, 1975).  In the
                                                        initial dose response curve we corrected for size
                                                        by an exponent function of weight.  This approach
                                                        was developed by Bliss (1936).  Bliss used the
                                                        following formula:
                                                              Y « a + b log M/Wh
                                                     31

-------
   This expresses a linear function between
survival time and dose of different sizes of silk
worm larvae.  M represents weight, Y the
dependent variable (death of fish in our case)
and M the aean dally toxicant concentration.  The
Y Intercept "a* and the slope "b" of each dose
response was calculated.  An h factor of a best
fitting regression and highest correlation
 coefficient of the dose response curve was
 determined using a computer program.   The
 toxicants we used and their corresponding h
 factor 1s found in Table 1.
    Slight changes in the normal  distribution  of a
 species also was recognized as having  a
 significant effect on the slope  of any
 dose-response curve.   We therefore attempted  to
 control all these factors by using an  inbred
 species of guppy. We avoided sex difference  by
 using only males.

 Table 1. Toxicants and their corresponding  h
 factor.
           Toxicant
                   n Factor
      Copper chloride                '0.72
      Nickel chloride                 0.67
      Zinc chloride                   0.3
      Dieldrin                        0.81
      Potassium pentachlorophenate     0.72
 MIXTURES:  Anderson 4 Weber(  1977)

    Our first attempts were  with  five  mixtures:
 copper.-nickel; dieldrin-pentachlorophenate-
 copper-nickel; copper-zinc; pentachlorophenate-
 cyanide; and dieldrin-pentachlorophenate.
 Statistically there was an  apparent parallelism
 between the lethal response curves for copper and
              CONCENTRATION ADDITION
     . 6
    -M
    2
     o
     u
    0,
     i
     >. 5
     O
                     Copper
                            nickel.  We assumed that as  constituents of a
                            binary mixture, copper and nickel  would
                            contribute to the mixture's  toxldty In
                            proportion to their lethal potency.   We tested
                            organisms to a series of binary mixtures of
                            copper and nickel.  The linear function computed
                            for the observed results for the Mixture was
                            compared to the predicted linear regression by an
                            X' test for goodness of fit  (Figure  1).  The test
                            for goodness of fit between  the observed and
                            predicted was significant at P«0.05.  Our model
                            appeared to have predicted the strictly additive
                            action of Cu and N1.
                               The slopes of the response  curves for dieldrin
                            (HEOD) and potassium pentachlorophenate (KPCR)
                            were found not to be parallel.   Binary mixtures
                            of HEOO and KPCP were tested according to the
                            model of response addition.  There was a good fit
                            (P<.05) to this model.  (See Table 2)
                               An interesting temporal relationshio between
                            the lethal effects of pentacholorphenate ana
                            dieldrin supported (Bliss 1929}  the  "independent
                            joint action" or response addition hypothesis.
                            All mortality of fisn exposed  to
                            pentachlorophenate occurred  in  36  hours.   The
                            effects of dieldrin after another  10 hours.   The
                            predicted lethality of response  addition for each
                            toxicant very closely aligned  themselves  to  each
                            time period, that is, death  from
                            pentachlorophenate before 36 hours and death from
                            dieldrin after 45 hours with the total  equal  that
                            predicted for response addition.
                               Mixtures of copper and zinc  produced results
                            that did not fit either response or  concentration
                            addition.  The individual dose  response curves
                            for copper and zinc were not parallel.   We
                            initially tested the mixtures  on the assumption
                            that they would be response  additive.   The
                            responses were greater than  predicted.
                            Literature knowledge of the  actions  of mixtures
                            of  copper and zinc suggested that  they were
                            additive, a  test for concentration addition  was
                            made.  The numbers dead were again greater than
                                                                      Cu-Ni mixture
                                         Predicted
        -1.0
-0.9
-0.8
-0.7
0.9
                                                          1.0
1.1
1.2
1.3
1.4
                                                  log(M/Wn)
         Figure 1.  Lethal response curves for copper, nickel  and their mixtures.  The predicted
         regression line is based on the relative,observed proportion of Cu  (.006) and Mi (0.994)
         and a relative potency (p) of 6.58 x 10" .

-------
      Ttklt 2.  Toxfclty study of »upp1tt oxpoud t» Mixtures of ttf Md HEOD
Assayed
level
of HCOO
1.0
4.45
(.3
6.4
4.8
6.9
Assayed
level
Of KP
•I/I
0.2*
0.40
0.31
0.40
0.29
0.41
Independent
variable..
log M/r"
forHOO
-2.44
-2.46
-2.46
-2.40
-2.46
-2.37
S Mortality
predicted
or HEOO
17
IS
27
27
16
35
Independent
variable,,
log IVW7Z
for KP
•0.713
-0.639
-0.685
-O.S8
-0.668
-0.594
* Mortality
predicted
for KP
11
34
17
Cl
24
54
S Mortality
predicted
for aliture
2f
44
39
72
36
70
Observed
S
,) Mortality
10
40
SO
60
70
80
      Fron Anderson i Ueber, 1977

predicted.  A ratio of observed to predicted
values represented a relative measure of the
increased effect or what we considered to be
super addition.   The super addition was found to
be 2.5 times above that predicted on the
assumption of concentration addition.
   The real challenge was to use a mixture of
four chemicals,  two inorganic and two organic.
Nickel and copper were used as a pair that we had
shown to be concentration additive.  Oieldrin and
pentachlorophenate were response additive.
Combining the nickel and copper as a single
(concentration additive) component, we treated
the mixture as three response additive components
(Table 3).  Using the response additive approach
the predicted and observed results provided a
nice fit.  In the case of pentachlorophenate and
cyanide they were tested and found to be response
additive.
   We concluded that using the two forms of
addition, concentration and response, we were
able to describe four of the five combinations
adequately using lethality as an end point.  One
of these mixtures contained four, two inorganic
and two organic,  toxicants.   In  the  case  of the
one aberrant  binary mixture,  cooper  and zinc,  we
were able  to  clearly describe an  interaction
which is super-additv/e.

HYPOTHETICAL  QUANTAL DOSE  RESPONSE RELATIONSHIPS:
Muska and  Weber (1977)

   Completion of  these  quanta! studies brought us
to a better understanding  of  the  assumptions with
which we were working.  To illustrate graphically
the relationship  between concentration and
response addition,  lyoothetical  dose response
curves for two toxicants  (A and  B) are plotteo in
Figure 2 expressing percent response in probits
as a function of  the logarithm of total
concentration.   In  this example  the  dose  response
curves for the discrete toxicants are parallel
with A being  100  times  more toxic than B.   We
could have also chosen  non-parallel  curves;
however, for  these  cases equation (1) for
concentration is  not appropriate.  Hewlett  and
Plackett (1959) have developed a  more generalized
model (from which equation (1) can be deduced)
Table 3.  Determination, using mean daily assayed concentrations,  of the predicted mortality  of fish
exposed to mixtures of HEOD, KPCP, Cu and Ni.
Predicted Mortality Proportion
                                                       Observed
                                                       Mortality
                                                      Proportion
                Predicted
                 Numbers
                 Killed
Observed
Numbers
Killed
1 -
1 -
1 -
1 -
1 -
1 -
(1
(1
(1
(1
(1
(1
- .316){1 -
- .4)(l - .:
- .212)(1 -
- .268)(1 -
- .758)(1 -
-.655X1 -
.22X1 . - .057) =•
36)(1 - .136) - 0
.045)(1 - .023)
.184)(1 - 0.045)
.198)(1 - .084)
.277)(1 - 0.081)
0.50
.66
- 0.
= 0
* 0.
- 0.

27
.43
82
89
0
0
0
0
0
0
.30
.60
.60
.60
.80
.90
5
6.
2.
4.
8.
8.

6
7
3
2
9
3
6
6
6
8
9
                                                                            r *  5.57

                                                                           d.f. « 4.0

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                      d
                                                                                                       Si.
which does not depend on the assumption of
parallel dose response curves.  (See Figure 2)
   Dose response curves for mixtures of toxicant
A and B are obtained when the total concentration
1s varied and the ratio of the concentrations
for the Individual toxicants 1s kept constant.
Using the equations (1 and 2a, b, c) for
concentration (C.A.) and response addition (R.A),
dose response curves were calculated for
different mixtures containing fixed proportions
of toxicants A:B (1:10, 1:100, 1:1000).  In
Figure 2, the responses to the mixtures are shown
graphically 1n relation to the dose response
curves of toxicants A and B.
   Several observations can be made from the
relationships between the dose response curves in
Figure 2.  As should be expected, the relative
toxicity of the mixture depends on the ratio of
its constituents.  In Figure 2, a 1:10 mixture is
more  toxic than the other mixtures depicted
because  of the greater proportion of the more
toxic  component - toxicant A.  At certain ratios,
regardless of the correlation of susceptibility
 (rj,  the relative potencies of the mixtures
acting  in either a concentration or a response
additive manner are very similar.  This is
observed in Figure 2 for fixed proportions of
 1:10  and 1:1000.  Furthermore, for any one ratio
the  relative potency of the dose response curves
for  concentration and response addition (r » 1,
0,  -1)  depends on the level of response.
 Focusing on the dose response curves for mixtures
 in  the  ration of  1:100, it can be noted that at
 low  levels of response (i.e., at the probit of 2
which corresponds to approximately a 0%
 response), the mixtures acting in a concentration
 additive manner are considerably more toxic than
 those acting by response addition regardless of
               the degree of correlation (r).  This Is due to a
               fundamental difference 1n the two .types of
               addition.  At threshold or below threshold
               concentrations of toxicants A and B, a mixture
               acting 1n a concentration additive manner can
               elicit a measurable effect because both toxicants
               are acting upon similar biological systems.
               Therefore, their concentrations can sum to
               produce a concentration for the mixture which 1s
               above the threshold level.  However, the
               responses to toxicants acting upon different
               biological systems (response addition) are only
               additive If each toxicant in a binary mixture 1s
               present in concentrations above their respective
               threshold levels.  For similar reasons, as the
               concentrations for the toxicants in a 1:100
               mixture increase, the dose response curves for
               response addition (except in the special limiting
               case where r = 1) become progressively more toxic
               relative to the dose response curve for
               concentration addition.  It is even possible that
               a nign levels of res:cnse (in tnis example, for
               resconses greater tf.jn 84% prooit of 6.0)
               mixtures acting in a response adaitive manner
               with negative correlation of susceptibility (r =
               -1) can be more toxic than those acting on the
               basis of concentration addition.
                  These factors (the type of interaction, the
               ratio of the toxicants in a mixture, and the
               level of response) must also be considered along
               with the toxic properties of the individual
               toxicants in assessing the relative toxicity of a
               mixture.  The failure to recognize these factors
               can potentially lead to erroneous conclusions
               concerning the-nature of the interaction of
               multiple toxicants.
                  It is difficult to visualize the relationships
               between the dose response curves in Figure 2
                              TOXICANT A
                              (r - 5 + 4x>
                                                                                            TOXICANT 3
                                                                                            (Y - I r4X)
     -2.0
                 -1.5
                             -1.0
-0.5         0.0          0.5

 LOG TOTAL CONCENTRATION (X)
                                                                             1.0
                                                                                         1.5
  Figure 2.   Hypothetical dose  response curves  for toxicant A (1:0). toxicant B (0:1 I  and their mixture
              containing the fixed proportions (1:10, 1:100,  1:1000).  See text for  explanation.

-------
primarily due  to  the number of curves presented.
However, the relationships between the
hypothetical curve  1n Figure 2 can be readily
conceptualized with Isobole diagrams, a technique
Introduced by  Loewe (19Z8, 1953).  Isoboles are
lints of equivalent response.  They are
constructed by plotting on a two-dimensional
diagram the concentrations of a binary mixture of
toxicants that produce a  quantitatively defined
response, I.e. a  101, SOX or 90X lethal response.
It should be noted  that an Isobole diagram can be
constructed for any level or response and the
relationship between the  isoboles may vary
depending upon the  response level selected.
   The Isobole diagram for the SOX level of" ~
response of the hypothetical dose response curves
in Figure 2 is present in Figure 3.  The x and y
axes  in this diagram represent the concentrations
of toxicant B  and A respectively.  The radiating
dashed lines or mixing rays correspond to a
series of mixtures  (A:B)  of fixed proportions.
If the 50% response is produced by combinations
of two toxicants  represented by points inside the
square area, the  toxicants are additive.
Antagonistic  interactions are  represented by
combinations of concentrations falling outside
the  square.
   The  Isoboles for concentration and response
addition  are determined  from the'concentrations
of the  two  toxicants which correspond to the
points  of intersection between the 50% response
line (Figure  2)  and the  respective hypothetical
                                            dose response curves.   These concentrations are
                                            plotted In Figure 3 on the appropriate mixing
                                            ray.  The lines connecting these points define
                                            the course of the Isobole.  Concentration
                                            addition 1s represented by the diagonal Isobole.
                                            For quantal data, response addition Is defined by
                                            the curved Isoboles for complete negative (r *
                                            -1) and for no correlation (r • 0) In
                                            susceptibility.  The upper and right boundaries
                                            of the square correspond to the limiting case of
                                            response addition with complete positive
                                            correlation (r » 1).

                                            QUANTITATIVE (GRADED)  RESPONSE:

                                               A consideration of  the nature of the dose
                                            response curves for quantal and graded responses
                                            shows that the effects they express are quite
                                            different.  Quantal dose response curves exoress
                                            the incidence of an all-or-none effect (usually
                                            death) when varying concentrations are appliec to
                                            a group of organisms.   The curve is derives Dy
                                            observing the number of organisms which respond
                                            or fail to respond at  various concentrations.
                                            Consequently, the slopes of these curves
                                            primarily express the  individual variation of the
                                            population to a particular toxicant.  Graded dose
                                            response curves characterize the relationship
                                            between the concentration of a toxicant and the
                                            magnitude of the effect under consideration.  The
                                            dose response curve can be derived by measuring
                                            on a continuous scale  the average response of a
                                            group of organisms at  each concentration.
                1:0
                      IMO
                                             1:50
                                                                           I: 100
              .10
              .08
            t- .06
o
£

K
>-
              .04
              .02
                                      / AOOITION (r • I)	/
                                                                     *+
                                                                             1:200
                        2.0      4.0      6.0      8.0
                        CONCENTRATION OF TOXICANT 8
                                              10.0
                                                                              1:1000
                                                                             0:1
                  Figure 3.   Uobole diagram for quantal response data.  Isoboles (or
                             concentration and respoose addition were determined from
                             hypothetical.dose response curves in Figure 1.

-------
                          •„».
   As Clark (1937)  and  others have pointed out,
4t 1s possible to represent any graded  response
as a quanta1 response provided that  the response
of each Individual  organism can be measured.
However, this procedure if adopted Is at the
expense of some "loss of Information" (Gaddum,
1953). • Quantal response data reveal only the
number of organisms that respond or fall to
respond at some particular concentration.  On the
Other hand, graded response data not only tell  us
whether or not a group of organisms respond but
also how much they respond.
    The mathematical  equations (2a,b,c)  for the
 response  addition  are  not appropriate  for graded
 effects  for two  reasons.  First, there 1s a
 difference in the  way  the two types  of data are
 measured.   For quanta!  responses, the  proportion
 of organisms responding to any concentration is
 determined by the ratio of number of organisms
 showing the response to the  total number
 subjected to the concentration.   For graded
 responses, the mean response to  each dose  is
 measured but in general the  maximum possible
 effect is not known, no proportional response  can
 be calculated.  This is particularly true for
 growth experiments  where an organism's response
 can potentially range from growth enhancement to
 negative  growth depending on the concentration of
  a particular toxicant.  Secondly, the statistical
  concept of correlation  between  the  suscep-
  tibilities of the organisms to  the  discrete
  toxicants in a mixture is not appropriate for
  graded responses measured in the manner described
  earlier.  Graded response data  represent  the
                           1:0
average response of a group of organisms.
Therefore, the response of each Individual
organisms to the toxicants 1s not known.  To be
sure, the tolerances of the Individuals  In the
group will vary for the different toxicants In a
mixture; however, this factor will not alter the
relative toxlclty of the mixture because the
range of tolerances of the population Is
theoretically represented In the sample  of
organisms from this population.
   For graded response data, we have represented
the  combined response to a mixture of toxicants
acting in a response additive manner as  simply
the  sum of the  intensities of response which  each
component toxicant  produces when administered
alone.  A similar relationship was  defined  by
Loewe  (1953).   Concentration  addition  can  be
predicted for a  toxicant mixture  using  equation
 (1)  if  the  component  toxicants  exhibit  parallel
 dose response  curves.   Figure 4  represents  an
 isobole  diagram for a  graded  response.   The
 isoboles  for  concentration  and  response afioition
 were determined with  the appropriate mathematical
 equations discussed.
    The relatively simple types of isoboles
 represented in Figures 3 and 4 should only be
 expected for relatively simple in vitro systems
 or  in situations where there is a clear-cut
 relationship between dose and effect.   Given the
 complexity and interdependency of physiological
 systems, it is reasonable to suooose a  priori
 that the special types of additivity as
 represented by strict concentration and response
 addition will be approximated only  occasionally
                                      2.0      4.0     6,0     8.0      10.0

                                CONCENTRATION OF  TOXICANT B

                                Figure 4.  Uobole diagram {or graded rtiponie data.
                                                                                           0:1

-------
1n the responses of whole organisms to mixtures
of environmental toxicants.  Furthermore, as
mentioned earlier, the relative toxicity of a
mixture depends on several factors which Include
the level of  response (i.e., 10%, 50%, 90%
response),  the ratio of toxicants in a mixture
(1.t. 1:10, 1:100, 1:1000), and the nature of the
response Itself.  It should be noted that the
type of addition can only be described in
relation to the response under consideration.
With the same mixture of toxicants, different
types of toxicant interaction might be expected
for different responses (i.e. survival, growth,
reproduction).  However, these special types of
toxicant interaction do provide a frame of
reference  for evaluating the effects of toxicant
mixtures on whole organisms performances.
    Isobole  diagrams are useful for visualizing
the  relationship  between different types of
toxicant  interactions ana  for ael ineaf.ng the
various  factors whicn can  influence the  relative
toxicity  of multiple  tox'cants.   However, in
pracfce,  isoboles  a^e iif'icult  to cerive
 requiring  a series  of cose  response curves  for
 the  mixture at  different  ratios of the component
toxicants.   Furthermore, there are no statistical
criteria which might be used to distinguish
between  one form  of interaction and another
 (Plackett  and Hewlett, 1952).

GROWTH STUDIES:   Musxa and  Weber  (1977);
 Koike^eister and  Weber (1978)

    Growth  was selected as  the graded  response  for
 this  study because  it represents  a performance  of
 the  integrated  activities  of the  whole organism
 and  as such is  often  a sensitive  indicator  of  the
 suitability of  the  environment (Warren,  1971).
 Two of the ways  environmental toxicants  can
 affect the growth of  an organisms  are:   (1)
 alter its  ability to  assimilate and convert food
 material  into body  tissue,  and/or (2)  change  its
 rate of food consumption.   To determine  the
 manner in which toxicants  affect  the  growth of an
 organism,  both  processes  were  investigated
 separately.
    Juvenile  guppies were  fed daily a  restricted
 ration of  tubificid worms  to determine  the  effect
 of the toxicants on the  gross growth  efficiency
 and relative growth rate  (as defined  by  Warren,
 1971) of the fish.   The  effect of the individual
 toxicants  and their mixture on food consumption
 was investigated by feeding groups of fish  an
 unrestricted ration and  measuring the amount of
 worms consumed.
    Parallel  dose response (growth) curves  were
 found for  copper and nickel.  Concentration
 addition was predicted as in the lethality
 studies (Anderson and Weber,  1977).   On the basis
 of the mathematical model for  concentration
 addition,  the predicted  dose response curves were
 calculated and statistically compared to the
 regression equations experimentally  determined
 for the mixture.  The results  indicate that the
 effects of the copper and nickel  mixture on the
 gross  growth efficiency of the fish  subjected to
 both  the  restricted (Figure 5)  and unrestricted
  (Figure 6) feeding regimes.  However, the dose
 response  curves for the mixture  representing the
 effects of the toxicants on the  food consumption
 of  the fish  was supra-additive relative.to the
dose response curve predicted on the basis of
concentration, Figure  7.
   Dose response curves  for dieldrin and nickel
were accessed. Theslopes  of the dose response
curves for their individual  effects on growth
proved to statistically  parallel.  We judged that
these compounds might  Interact in a response
additive manner.  Based  on  existing knowledge we
assumed they should act  toxicologically by
different mechanisms of  action.   As we know the
parallelism of curves  1s  only a suggestion, not
an absolute criterion  for predicting either the
occurrence of concentration or for the negating
possible response addition.   Regardless of the
growth parameters we looked at the dieldrin and
nickel studies were inconclusive.  The reasons of
course could be many.  The  simple model we
proposed did not discriminate adequately to
classify the interaction  of these two chemicals.
   Mixtures of zinc and  nickel  were tested
(Koikemeiste1- and Weber,  1978).  Our assumption
basea upon available data anc parallel growtr
dose response curves was  that tney woula be
concentration adaitive.   Mixtures proved to be
inf ra-concenfation additive.
    In summary the  graded results indicate that
the  assumption of  concentration addition
adequately predicts  the  effects of a
copper-nickel mixture  on both the survival and
gross growth efficiency  of  guppies.  The dose
response curves  for  the  mixture representing the
effects of tne toxicants on the food consult-on
of  the  fisn was  supra-aocitive relative to tie
dose response curve  predicted on the basis of
concentration addition.   An explanation for the
differences  in these  two responses to the mixture
was  beyond the scope  of  the study.  However, it
is  reasonable to assume  that the effects of tne
toxicants on  the metabolic  processes involved in
the  conversion of  food material into body tissue
are  different from their effects on the
biological processes  regulating the consumption
of  food.
    In our studies  we  found  that the mathematical
model for'concentration  addition predicted the
responses of  guppies  to  both lethal and sub'ethal
concentrations of  a  copper  anc nickel mixture.
However,  it  should not be inferred from these
results that  the type  of joint toxicity observed
when organisms are subjected to high, rapidly
lethal  concentrations  of mixtures will
necessarily  occur  in  cases  where animals are
subjected to  low concentrations of the same
toxicants.   Furthermore, the nature of toxicant
interaction  can  only  be  meaningfully described in
relation  to  the  particular  response under
consideration.   For  example, we found that
mixtures  of  copper and nickel were concentration
additive  in  experiments  evaluating their effects
on  the  gross  growth  efficiency of the guppies;
however,  in  the  food  consumption studies, the
same mixture  at  similar  concentrations produced  a
more toxic response  than was predicted on the
assumption of concentration addition.
    Although  each of  our  mixtures were not
accurately predicted,  we must recognize that this
is  a simplistic  model.  The complexities of
physiological systems  from  pharmokinetics to
actual  receptor  interactions certainly makes the
real world much  more  complex.  The model does
allow a specific reference  point to evaluate and
                                                    37

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                                                                             NICKEL
               COPPER
                                                      Cu
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                                                                  OBSERVED
o:
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    -6.5   -6.0
                                 I
                                                 I
     -5.5    -5.0   -4.5     0.5     1.0      1.5

             LN  TOTAL CONCENTRATION  (mg/l
2.0
2.5
3.0
                Figure 5.  Doae respor.ae curves showing effects of copper, ruckei, ar.c their mixtures

                        (oDierved and predicted) on gross growth efficiency norma..zec to responses

                        of controls (restricted ration study).
 o
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                      Cu-Ni MIXTURE
                                                                               O  NICKEL
COPPER
                                                                1
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       -"6.0   -5.5    -5.0   -4.5     0.5      1.0      1.5      2.0

                             LN  TOTAL  CONCENTRATION  (mg/l)
                                                                                2.5
                                                                         3.0
                Figure 6. Dose response curves showing effects of copper,  nickel, and their mixtures

                         (observed and predicted) on gross growth efficiency normalized to responses

                         of controls (unrestricted ration study).
                                              38

-------
                            •».'
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a.   70

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         \
                                                                         NICKEL
                 COPPER
                                                          OBSERVED
                                             \
                                             PREDICTED
      -6.0   -5.5    -5.0    -4.5
         0.5
1.0
1.5
2.0
2.5
3.0
                            LN TOTAL  CONCENTRATION  (mg/l)
             Figure 7.  Dote response curve* showing effects of copper, nickel and their mixtures
                      (observed and predicted) on food conmrr.pt ion rate normalized to responses
                      of controls {unrestricted ration study).
 identify mixtures  which deviate from the model
 and the direction, infra or supra-additive, from
 either concentration or response addition.  To
 insure the success of a species in nature, it is
 necessary to evaluate the effects of potentially
 hazardous toxicant mixtures on the performances
 of whole organisms.  This approach provides a
 methodology for assessing the toxicity of
 mixtures of environmental toxicants at this level
 of biological  organization.

 ORGAN SYSTEMS:   Shelton and Weber (1981)

    Mortality (quantal response) and growth
 (quantitative)  respon-se were to this point used
 to evaluate the concentration and response
 additive models for mixtures.  A third test of
 the usefulness  of  this model was done using an
 organ system level  of toxicity.  The decision was
 made to test the model in a mammal (mice) rather
 than a teleost. Liver damage was the specific
 organ system response chosen.  Plasma alanine
 aminotransferase (ALT, formerly GPT) activity
 has been shown  to  be a sensitive indicator of
 liver damage in mice (Klaasan and Place, 1966)
 and plasma elevation of ALT correlate well with
 the severity of damage  (Balazs et al., 1961).
    The type of joint action observed for a binary
 mixture can be influenced by the degree of
 separation in the  duration of onset of toxic
 action for the respective toxicants in that
 mixture, Turner and Bliss (1953).  For that
 reason, the temporal effects of the selected
 hepatotoxicants used in the study on plasma ALT
                      were exanined.   All  experiments were perfomec
                      using male albino mice of the CF-1 strain reared
                      in our own breeding  colony and housed at five per
                      cage.   The animals  weighed 25-35 g and were
                      maintained on laboratory pellet diet and wate*- ad
                      libitum.  The animal room was maintained at a
                      12-hour light/dark cycle with an ambient
                      temperature of 70-72°F.
                         The toxicants were carbon tetrachloride
                      (CC1,), xcncchlorooenzene (MCE! and acetamnoznen
                      (ACEi).  The CC1, and MCB were dissolved in corn
                      oil and ACET was dissolved in 0.9% N.aCl at <10°C.
                      The toxicants were diluted to deliver the proper
                      dosage in a final volume of 0.01 ml per gram of
                      animal weight.   These compounds were administered
                      intraperitoneally between 11:00 a.m. and 1:00
                      p.m. each day.
                         Liver damage was  assessed by measuring plasma
                      ALT activity.  Relative plasma ALT elevations
                      were determined at 2, 4 8, 16, 24, 48 and 72
                      hours following the  administration of each
                      toxicant.  An optimum time interval was
                      determined and  used  in the toxicant mixture
                      study.  The plasma ALT determination of Reitman
                      and Frankel (1957) was used and the results are
                      reported as international  units (ID) per liter.
                         Single-component  dose response curves were
                      initially developed  for each hepatotoxicant.
                      Characteristics derived from these curves (i.e.,
                      slope, potency  ration, TD50) are shown in Table
                      4.  The TD5Q's  were  used to calculate the potency
                      ratios forjlhe  toxicants.   HCB and ACET were
                      found to be approximately equipotent in producing
                      liver damage whereas CC14 appeared about 35 times

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          Tabl« 4. Dost response characteristics of selected h*patotox1cants on pi ISM lUnlnc anlnotraniftrase1 activity In Hi
          lib'to "let.
m fc
""SO rV>te«£y,.
Toxicant
1.
2.
3.
Carton tttnchlorl*
(CC14)
Nonoehlorobcnzm* '
(«8)
tkttimlnophen
(ACH)
•9/kg
U.9
(14.2 . 19.9)
428
(395 • 466)
558
(485 - 643}
Mnl*/k9 ratio *
109.5 1.0
(92.5 . 129.6)
3807 34.8
(3505 - 4136)
3694 33.7
(3209 - 4252)
Slope
(±SO) t value4
6.57
(*!.»)
8.40 -0.45
(tl.24)
8.31 -0.77
Ul.15)
P value'
—
0.667
0.471
*A positive responie 1s defined as a plasma lUnlne anlnotransferase elevation *3 SO above the control meal (10 t 2 IU)

^Determined fro» the dose response regression eqoatlon.  Parentheses Indicate 95X CI.

CTDCC(HCS or ACrn/TDcjiCC
             CC
                                 unole/kg comoar'son
           T value determined when slope of dose response curve for KC3 or ACE* is corcsred to  that of carbon tetracnlorlde.

           'in eacn csse slopes were not significantly differs--, fror parallel at tt'e '  value (nd1catec.
           Fror Shelton and Weee' (1981)

more  toxic than either MCB or ACET."
Consequently,  we decided to test the joint
hepatotoxic effects of the mixtures CC1.  +  MCB
and CC1.  + ACET.  The large potency ratio between
the constituents in the tested mixtures allowed
greated resolution in differentiating the
possible  types of joint action resulting  from
them.  When the slopes of the MCB and ACET curves
were  each compared to that of CCL.  (t test),  no
significant deviation from parallelism was
apparent  (Table 4).  It was assumed from  these
findings  that  concentration addition would  be the
most  likely effect for the mixtures CC1.  +  MCB
and CC1.  + ACET predicted in each case.
   CC14 + MCS  Mixture.  A theoretical dose
response  curve for the binary mixture of  CC1, and
MCB at a  molar oose ratio of 1:38 was predicted
using Finney's (1971) equation for concentration
addition.  The development of this curve  involved
utilization of data from the single component
dose  response  regression equations as well  as a
common regression coefficient determined  by
analysis  of covariance.  This curve is shown
plotted in Figure 8 along with the curves for
CC1., MCB, and the observed curve for the 1:38
mixture.   The  results show no difference  between
the two curves at P > 0.975.
   CC1* + ACET Mixture.  The predictive equation
for the mixture of CC1, + ACET was developed  for
a  molar dose ratio of  i:36.6 (CC1,.:ACET).  This
curve is shown plotted  in Figure 9  along  with the
observe dose-response curve for the mixture as
well  as those for the singly applied CCl. and
ACET.  The test of comparison revealed that the
predicted and observed  curves for the CC1.  +  ACET
mixture differ  (P < 0.0005).  The observed  joint
effect for the mixture  can thus be  categorized as
 infra-additive on the basis of concentration
addition.
    To determine if response additivity might more
adequately describe the observed joint effect for
the CC1. + ACET mixture, the observe points were
statistically compared  to those predicted on  the
basis of response additivity.  The  findings show
that the observed and predicted curves again
differ (X     = 40.6; P <0.0005).
                                                  The results  of  the organ system  investigation
                                               suggest that the toxicity of the mixtures  can  be
                                               predicted and classified by examining  the
                                               single-constituent toxicities.  The  joint  effects
                                               observed for the CC14 + MCB mixture  we"6  clearly
                                               predicted by the equation for corcent'afon
                                               addition.   It is evident that the res:cnse :*  a
                                               given aose  of a CCld  + MCB mixture  is  not  merely
                                               the surr, of  the  toxic  effects of the  CC1. and MCB
                                               given singly.   Instead, the addition c* the
                                               effects follows a  log-linear relationship  wth
                                               respect to  the  total  concentration of  ooth CC1.
                                               and MCB in  the  mixture.
                                                  The interpretation of the joint effects of  the
                                               CC14 -i- ACET mixture is more difficult.  There  is
                                               an apparent antagonism exhibited with  a resultant
                                               infra-additivity.   Since present knowledge of  the
                                               toxic mechanisms for  both MCB and ACET does net
                                               present any striking  differences between the two,
                                               any observed differences in joint action,  when
                                               ccmoined witn CCl^, is largely unexplained.  It
                                               has been inferred  that acetaminopnen ~ay dar.age
                                               the hepatic endoplasmic reticulum (Thoreirsson et
                                               al., 1973).  If this  is the case, then this could
                                               affect the  activation of CC1., with  resulting
                                               infra-additivity..

                                               CONCLUSIONS

                                                  When we  began this work we hoped  to use
                                               pharmacological models already developed and
                                               apply them  to problems of environmental
                                               toxicology.  The desire was to have a model that
                                               had applicability  to  environmental  problems and
                                               was rich enough in its information to lead us
                                               into an understanding of the chemicals with which
                                               we had concern.  The  most careful  analysis of the
                                               mixtures would  involve a factorially designed
                                               experiment.  Using multiple toxicants and  among
                                               doses a factorial  design would become impossible.
                                               So the desire was  to  use existing  knowledge and
                                               utilize the model  to  expand our knowledge  about
                                               the toxicants.  It was also our wish to have the
                                               model serve truly  multiple mixtures  and not just
                                               binary mixtures."   We  did successfully use  it in a
                                               mixture of  four toxicants, two metals and  two

-------
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            £4
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                    • CARBON TETRACHLORIDE  , ,
                    O MONOCHLOROBENZENE
                    » CCI4:MCB (observed)
                   — CCI^MCB (prtdiclcd)
               10
                        10                     10"
                            DOSE (^mole/Kg)
                                                                              10
Figure 8. Dose response curves illustrating the effects of carbon tetrachloride (CC1.) , mono~
          chlorobenzene (MCB) and the 1:38 mixture (CC1.:MCB)  on the percent of animals
          (expressed as probit) responding with significant plasma alanine amrnotransferase
          elevations.  Both the predicted and the observed curves for the mixture are shown.
          Each point represents a treatment of a minimum of ten animals.
               7 -
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             0.
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             o
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         • CARBON TETRACHLORIDE
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         » CCI4: ACET (observed)
       	CC14:ACET (predicted)
                10
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                                        DOSE (>imole/Kq)
 Figure 9.
Dose response curves illustrating the effects of carbon tetrachloride (CC1.), acetaminoph
(ACET) and the Ii36,6 mixture (CC^iACET) on the percent of animals (expressed as
probit) responding with significant plasma alanine aminotransferase elevations,  Both the
predicted and the observed curves for the mixture are shown.  Each point represents a
treatment of a minimum of ten animals,
                                                                                               en
                                               41

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organic chemicals; on the other hand most of our
tests used only binary mixtures.  The basic model
has been theoretically expanded upon, Christensen
and Chen, 1985.  It has also been reduced to such
a simplistic form that it lacks any richness
other jthan Its description of direction from
concentration addition, Marking, 1985.
   Our own efforts using Plackett's and Hewett's
(1948) noninteractive scheme was not always on
the mark.  The concentration (simple similar) and
response  (independent) joint action has the
richness  to  describe  possibilities of
correlations between  susceptibility of animals,
 interaction  of infra  or  supra-addition compared
 to the concentration  and response  addition.   The
 formulation  of isobole diagrams plotting  both
 concentration and response  addition  defined
 mixtures which would  allow  the greatest
 statistical  opportunity  to  differentiate  between
 the two noninteractive  possibilities.
    Our approac^ aooears  to  offer a method for
 evaluating the effects  of coramec toxicants.  We
 were successful in describing  the types  of
 interaction for binary and of  a mixture  of four
 toxicants in the case of lethality.   The model
 also was successful  in describing interactive
 effects  of  binary mixtures on  growth and on an
organ, liver.  Although it didn't describe all
Interactions accurately, it did provide insight
into possible questions which if answered might
help solve the complexities of the interaction.
The limitations of the model should  not be
overlooked and one of the major limitations  is
Inherent to all statistical explanations.
Statistically 1t 1s possible to state whether the
observed responses to the mixture agree with
those predicted within the  limits of sampling
error.  The statistical analysis can only  provide
contradictory or permissive evidence, but  not
indicative evidence (Hewlett and Plackett,  1950).
For example, an implication of  the mathematical
model for concentration addition is  that  the
toxicants in a mixture  act  primarily upon  similar
biological  systems.   Statistical agreement of the
observed  dose  response  curves  to the curves
predicted on  the basis  of  concentration  jccition
does  not  necessarily  mean  that  tne  toxicants act
upon  similar  biolcc'cal  systems, but on"_,
they  appear  to oo  sc.
 Partial  support provided by NIH Grant ES-00210
                                                  REFERENCES
  Anderson, P. 0. and Weber,  L.  J.  (1975)   Toxic
     response as a quantitative  function of body
     weight.  Tox. Appl.  Pharm.  3J_:471-483.

  Anderson, P. D. and Weber,  L.  J.   (1977)  The
     toxicity to aquatic  populations of mixtures
     containing  certain heavy metals.  Proc. Int.
     Conf.  Heavy Metals.   2,  933-953.

  Ariens,  E.  J.  and Simonis,  A.  M.   (1961).
      "Analysis of the action  ofdrugs and drug
      combinations", Quantitative Methods in
      Pharmacology, H. de Jonge  uditor),
      North-Holland Publishing Company, Amsterdam,
      p. 286-311.

   Ariens, E.  M.   (1972)  "Adverse drug
      interactions—interactions of  drugs on the
      pharmacodynairric  level", Proceedings of the
      European Society for  the Study of  Drug
      Toxicity, 13:137-163.

   Balazs, T., Murray, T. K., McLaughlan,  J. M.,  and
      Grice, H. C.  (1961)  Hepatic  tests  in
      toxicity studies on rats.  Toxicol.  Appl.
      Pharmacol. 3:71-79.

   Bliss,  C.  I.  (1936)  The  size factor in the
      action  of  arsenic upon  silkworm larvae.   J.
      Exptl.  Biol.  13:95-110.

   Bliss,  C.  I.   (1939)  The  toxicity of poisons
      applied jointly.  Ann.  Appl.  Biol. 26:585-615.

   Brown,  U.  M.   (1968)  "The calculation of the
      acute toxicity  of mixtures of poisons to
       rainbow trout", Water Research, 2(10):723-733.
  Christensen,  E.  R.  and  Chen,  C-Y.   (198:}   A
     general  noninteractive  multiple  toxic'ty model
     including  probit,  logil -a-nd  weibull
     transformations.   Biometrics 41:711-725.

  Clark,  A.  J.   (1937)   "General  Pharmacology",
     Heffler's  Handbuch der  Experimentel'en
     Pharmakologie, i*.  Heuoner  and J. Scp.uller
     (Editors) /Verlag  von Julius Springer,  Berlin,
     Volume 4.

  Fedeli, L., Meneghim,  L., Sangiovanni,  V.,
     Scrollini, F. and  Gori , L.(1972)
     "Quantitative evaluation  of joint drug
     action", Proceedings of the European  Society
     for the Study of Drug Toxicity. 13:231-245.

  Finney, D. J. (1942)   The analysis of toxicity
     tests on mixtures  of poisons.  Ann.  ADD!.
     Biol. 29:82-94.

  Finney, D. J.   (1971)  Probit Analysis,  3rd Ed.,
     Cambridge Univ. Press, London and New York.

  Gaddum, J. H. (1933)  Reports on biological
     standards.   III.  Methods of biological  assay
     depending on  a quanta! response.  Spec.  Rep.
     Ser., Med. Res. Coun., Lond., no. 183.   His
     Majesty's Stationary Office.

  Gaddum, J. H.   (1953)  "Bioassays  and
     mathematics", Pharmacological Reviews.
     5(1):87-134.

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Hewlett, P. S. and Plackett, R. L.  (1950)
   Statistical aspects of independent joint
   actions of poisons, particularly insecticides.
   II.  Examination of data for agreement with
   the hypothesis.  Ann. Appl. Biol. 37:527-552.

Hewlett, P. S. and Plackett, R. L. (1952)
   Similar joint action of insecticides.  Nature.
   169:198-199.

Hewlett, P. S. and Plackett, R. L.  (1959)  A
   unified theory for quanta! responses to
   mixtures of drugs:  non-interactive action.
   Biometrics 15:591-610.

Klaassen, C. D. and Plaa, G. L.   (1966)  Relative
   effects of various chlorinated hydrocarbons on
   liver and kidney function in mice.  Toxicol.
   Appl. Pharmacol. 9:139-151.

Koikerne1!ster, E. A. and Weber, L. J.   (1978)  An
   effect of zinc anc mc
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           DISCUSSION OF THE CONTRIBUTION OF L.J. WEBER AND ASSOCIATES OF
            THE MARr HATFIELD SCIENCE CENTER AT OREGON STATE UNIVERSITY

                            Bertram D. Litt, EPA/OPP/HED
    The paper by Or. Neber and his
associates reviews their work of the
past 12 years.  They have made
significant contributions to the
literature on formal study of multiple
simutaneous exposures of fixed mixtures
of 2-4 compounds in fish and mice.

    They have shown that there are
limited and unpredictable circum-
stances in which the dose/response
relationship observed fit traditional
matnematical models.  More important,
tney have demonstrated that neither
structure—activity-relations hips (SAR)
nor parallelism of the allcmetric
responses of  incividual chemicals or
pairs  of chemicals provide sufficient
information to accurately predict the
activity patterns of simultaneous
exposure to three, four, or more chemi-
cals.  This is an important finding
because modern man lives in an environ-
ment in which he is everywhere exposed
to  sopnisticated combinations of chemi-
cal residues  in the air he breathes,
the fooc he eats, and frequently even
the water he  drinks.  Weoer's paper  has
shown  that combination of toxicants  can
result in either superadaitivity or
reduction in  toxicity below that pre-
dicted by simple adjiitivity in teleosts
and mammalian experimental models.
This complexity precluded the use of
simple strategies for dealing with
complex mixtures on a routine basis.
At  this particular  time EPA is issuing
guidance recommending the use of  the
additivity as the fallback position  for
estimating cancer risk of mixtures when
adequate data on the mixture  is not
available.
The work just summarized by Or. Weber
could be used as the first step for an
ordered strategy to evaluate both fixed
complex mixtures, such as pesticides
where the source mixture remains con-
stant, and varying mixtures, where
concentration and constituents of
pollutants vary with respect to time or
distance from the source of contami-
nation.  The approach to both problems
may be unified by first studying the
morbidity and mortality effects of the
chemical mixture at the source concen-
tration at time tQ.  A second study
would repeat the initial tQ effects as
part of a series of ocservations to
evaluate a; dirr.-nitior. of effects due
to temporal and/or spatial distance or
b) selective deactivation of the
chemical mixture.

Weber's work provides teleost and
rodent models which could be used to
perform rapid experiments suitable for
screening chemical mixtures for the
identification of components wnich
are reinforcing and tr.ose wnich show
antagonistic tcxicity enc-pcihts.
Secondly these studies should De usea
to identify LCso levels as the initial
step of evaluating the toxicity of the
mixture at the source concentration
and at lower concentrations of interest
in rodents.

Following the screening procedure,
the EPA Office of Pesticide Programs,
Section F Guidelines could be followed
for studying subchronic, chronic and/or
teratogenic effects using the source
concentration and at lower doses.  The
selection of lower doses snould be
keyed to levels indicated by pnysiolo-
gical and environmental factors rather
than the considerations listed in the
guidelines for technical grade
chemicals.

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             JOINT AQUATIC TOXIC1TY  OF  CHEMICAL  MIXTURES AND STRUCTURE-TOXI CITY RELATIONSHIPS

                                Steven J. Broderius, U.S. EPA, ERL-Duluth
                  INTRODUCTION

  Most  studies  evaluating  the toxicitv of
environmental pollutants to  various  aquatic
organises and systems  have involved  exposures
to  separate toxicants.  Relatively few
investigations  have  defined  the  adverse
effects of Bixtures  of two or more toxicants.
Effluents, leachates,  and  natural waters,
however,  frequently  contain  several  toxic  or
potentially toxic substances.  The zones of
'influence from  point source  pollution night
also overlap.   As waste treatment technology
is  advanced and implemented,  nonpoint
pollution from  sources such  as agriculture and
atmospheric deposition will  contribute to  a
»reater degree  to the  overall pollutant  load
received by aquatic  ecosystems   Therefore,  in
assessing the  effects  of  toxicants on aquatic
communities and to insure  their  success,
consideration  should be given to the
 likelihood that a wide variety of chemicals
might be present simultaneously  and  that joint
 toxicity  is quite likely  the reason  for
 adverse  impacts of pollutants on aquatic
 environments.
   Water  quality criteria  should  insure  that
 the discharge   levels of separate toxic
 cnemicals and mixtures are not  deleterious to
 either the distribution or abundance of
 important aquatic populations.   The  setting  of
 current  water quality  standards  has  been
 developed  from criteria based on "no-effect
 levels"  of single toxicants.  Such  a practice
 may be  inadequate to protect aquatic organisms
 exposed  to mixtures of chemical  pollutants
 (Spehar  and Fiandt,  1986).  This practice,
 however,  is becoming  firaly established out  of
 necessity for  lack  of  a better  approach.   A
 tentative proposed  approach to  incorporate the
 effects  of joint  toxicity has been  to assume
 strictly additive action  for even diverse
 toxicants  Standards  using this procedure,
 however, have  generally not been set because
 formulating regulations on  such a basis may be
 premature,  since  other forms of less toxic
  interaction are  not uncommon
    It  would be  desirable to  be able  to predict
 or even estimate  the  probable toxic components
 and response for  an effluent, leachate,  or a
 water  body solely from a  knowledge  of its
  important individual  toxic  and  relatively
 non-toxic chemical  constituents.  If such a
 predictive approach is valid, it would be
 possible to determine  the relative
 contribution of  each  toxicant to the overall
  toxicity.  One could  then take  the  appropriate
  action necessary to effectively reduce the
  toxicity of  the  waste  water.  Such  an approach
  would better enable regulatory  agencies to
  provide rationale for determining and
  predicting the effects of chemical
  combinations to  valued aquatic  organisms,  and
  defining high  hazard  situations where more
  than one toxic substance  is known to exist.
    Defining the toxicity of  mixtures is a  major
  problem at both the theoretical and practical
 level.   There  has  not  been sufficient research
 to  establish whether there is any widely
 applicable  rationale and workable approach to
 evaluate and possibly  predict the joint action
 of  toxicants in the aquatic environment.
 There  are a few publications (i.e.,  Sprigue,
 1970;  Anderson and Weber,  1975;  Marking. 1977;
 Muska  and Weber,  1977;  EIFAC,  1980;  Alabaster
 and Lloyd,  1980;  Calanari  and Alabaster, 1980:
 Konemann,  198lb;  Hermens et al.,  1985;.
 Broderius and  Kahl, 1985;  Spehar and Fiandt,
 1986)  that  summarise and' review much of the
 information on combined effects  of mixtures of
 toxicants on aquatic organisms  and approaches
 jsed to  evaluate  these  effects.   It  is
 apparent from  these articles that additional
 work must be conducted  to  characterize the
 joint  action of mu.t.ple toxicants,  especially
 at  sublethal levels    This paper summarizes ar.
 approach to explore basic  principles wruch
 govern the  toxicol05.ca1 issues  pertaining to
 the joint action of multiple toxicants

             TERMINOLOGY AND MODELS

   Various terms and schemes for classifying
 and naming  effects of  chemicals  to describe
 the response of test organisms  to two or tfore
 toxicants,  as  predicted from the separate
 toxicity of the individual substances, have
'been recommended   The  different forms of
 joint  action have been graphically illustratec
 and discussed  by Sprague (1970), Muska and
 Weber  (1977),  and Calaman and Alabaster
 (1980).
   The  development of  predictive methodology to
 describe the joint action of multiple
 toxicants has  been approached in two distinct
 ways (Marubini and Boranomi. 1970).   The first
 approach has been to  describe responses
 resulting from constituent  interaction and  to
 try to give them a mathematical  expression
 based on statistical  considerations   The
 second approach has  been to postulate a
 physical mechanism of   interaction at  receptor
 sites, to derive theoretical response curves
 on  the basis of assumed primary mechanisms.
 and to relate  experimental and theoretical
 results.   It  is the  general belief that the
 first approach is more  suitable for a broad
 and quantitative evaluation of the joint
 toxicity of chemical  mixtures to whole
 organisms.   Given the  complexity and
 interdependency of physiological systems,
 however, it is reasonable to suppose that a
 classification of the   interactions between
 environmental  toxicants into various  types  of
 responses  for whole organisms will not always
 be  possible.  The real  value of designating
 special  types  of toxicant  interaction  is that
 they  provide  a frame  of reference for the
 systematic  documentation  and empirical
 evaluation  of multiple  chemical effects.
    Central  to  an analysis  of joint action are
 the- concepts  of similarity  and  interaction.
 These  ideas were  first  proposed by  Bliss
 (1939)  for  two substances  and later developed
                                                     45

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by PUckett and Hewlett (1948, 1952,  1967) »nd
Hewlett «nd PUckett (1959).  Conridering
these general biological phenomena,  the
different types of combined effects  ctn be
identified from the relative toxicities of the
individual constituents.  The types  of joint
action are defined as similar or dissimilar
depending on whether the sites of  primary
action to the organisms are the sane or
different, and as interactive or
non-interactive depending on whether one
toxicant does or does not influence  the
biological action of another.
   Interactive joint toxicity  is not  directly
predictable  from the toxicity of the separate
components.  Models describing quantal
responses to mixtures of interactive toxicants
are  very complex and are not described by
simple  formulas  '.Hewlett and P'.ackett, 1359,
1964     Certain  parameters  required  for the
.Tiodels  are  also  normally unattainab.e when
evaluatin"  the  effect of a  number  of toncants
on wnoie  organism  responses   Therefore.
virtually  all  investigators  evaluating the
effects  of  toxicant mixtures  on  parameters
 such as  survival,  growth, and  reproduction  of
 aquatic  organisms,  only  consider  the special
 cases of non-interactive joint  action.
   These concepts  in conjunction  with
 concentration- response  curves  and  isobole
 diagrams of joint  action have  been used  in  an
 approach to study  the. lethal  and  suciethai
 toxicity of mixt'ures  to freshwater organisms.
 The resulting models  are named  concentration
 and response addition (Anderson and Weber,
 1975), which correspond to  the  previous
 terminology of simple similar and independent
 joint action (Bliss,  1939),  respectively
   With concentration addition the toxicants
 act  independently but produce similar effects
 so  that one component can  be expressed in
 terms of the other after adjusting  for
 differences in their respective potencies.
 Even  sub-threshold levels  for mixtures of many
 toxicants can combine to produce  a  measurable
 effect.  Since the toxicants act  upon the aame
 or  a  very  similar system of receptors within
 an  organism, the toxicants are completely
 correlated  so  that no coefficient of
 association need be determined.   Therefore,
  for homogeneous populations
  concentration-response curves for  individuals
 exposed to  separate toxic constituents and
  corresponding  mixtures of similar chemicals,
  or  ones which  act  similarly, are  expected to
  be  parallel  or  similar  in shape.  Parallelism
  of  concentration-response curves  and complete
  correlation of  individual susceptibilities,
  however,  are  not a requirement for  this type
  of  interaction.   In cases where the
  concentration-response  curve for  the
  individual  toxicants are parallel,  Finney
  (1971)  and Anderson and Weber  (1975)  have
  provided a procedure to predict a
  concentration-response  curve for  the mixture
  based upon the assumption  of concentration
  addition.   The toxic unit model  (Sprague,
  1970),  which measures  the  toxicity  of mixtures
  only at particular  levels  of  response, can  be
  considered a simplification  of the  con-
  centration addition model.  This  special case
of the general model assumes that a mixture
should be at a particular magnitude of  toxic
response when the sum of the concentrations of
all toxicants expressed as fractions  of  each
toxicant's effect concentration equals  unity.
  A second model of joint action, response
addition, is predicted when each toxic
component of a mixture primarily acts upon
different vital biological systems within an
organism or affects differently the same
systems.  Each toxicant neither enhances nor
interferes with one another and contributes to
a common response only if its concentration
reaches or exceeds a certain tolerance
threshold.  Therefore, multiple toxicity
effects cannot be expected when each  of  a
mixture's components  is below its respective
response threshold.  The tolerance of
individuals exposed t: a mixture of toxicants
acting  independently  -ay or may not be
correlated   Therefo-?  tr.e resior.se  c'jr'-e%
for each toxicar.t of  ± mxture -ay or ma;.  -;-
be parallel or  similar  .n sr.ape    if  the
response curves  for ccnpcjnds  in a mixture  are
dissimilar or  if  the  T.odes of toxic action  are
known  to be different  for toxicants which  have
similar  response  curves, then  it  is proposed
that  the degree  of  response to the mixture  can
be predicted  by  sumiring  in various ways each
response  produced by  the  separate  toxicants.
The proportion of ind.viduals of  a group that
are expected  to  rest:"d  or the  degree of
response  for  each individual  organism exposea
to specific  components  and combinations
exerting  response acc^tion depend  upon  the
 responses  to  the  inc.vidual  compounds and  the
correlation  between tne  susceptibilities  of
the individual  organisms to each  toxicant
For mixtures  of  two chemicals this tolerance
correlation  can vary  from completely  positive
to completely  negative   Three models have
been  proposed  (Hewlett  and Plackett,  1959;  for
 correlation  of individual tolerances  of -I.  0,
 and +1,   For  mixtures of  many  chemicals the
correlation  coefficient  ir)  is  expected to
 vary  from 0  to *1 i Xor.enann.  1981b.    Response
 addition is  less likely  to occur  than other
 types of action because  an organism  is  a
 coordinated  system (Plackett  and  Hewlett,
 1967).  Nevertheless,  response  addition is
 important theoretically  for  it  leads  to a
 limiting mathematical model.
   The application of  concentration and
 response additive models  to  mixture  toxicity
 data  has not been extensive  nor  have  the
 models proven to be useful  in all  cases.
 Also, when applying these  classifications  to
 mixtures of more than two  chemicals,  problems
 might arise because the  joint action  of the
 different groups can  fall  under  different
 models as additional  joint  actions  are
 possible between the  groups    Therefore,  a
 mathematical description of  the  joint toxicity
 of a  mixture of  greater than two  compounds is
 probably only possible for  special  situations
 where non-interactive joint  action seems to  be
 a prerequisite.
   The more-than-strictly additive
 (synergistic) and less-than-no  addition
 (antagonistic) joint  actions  are  characterized
 by a  toxicity that is either greater or less

-------
than predicted from studies on the  separate
toxicants.   With these  situations the
effectiveness of »  mixture cannot be assessed
from that of the individual toxicants.   The
response depends upon a knowledge of the
combined effect which is usually only
experimentally determined.
  The different forms of Joint action  can  be
graphically illustrated by an itobole  diagram
as presented in Figure 1.   Isoboles are  lines
of equal response and can be  determined  for
different mixtures of two toxicants where  the
concentration of one toxicant for a
quantitatively  defined response  (i.e., 96-h
LC50)  is plotted against the  corresponding
concentration of the other toxicant.   Mixtures
of  toxicants  A  and B in different ratios are
identified by lines (mixing rays) radiating
from the origin of the  isobole diagram.  The
relationship between the  isoboles may vary
depending upon the response level selected.
Combinations of the two toxicants represented
by points within the square area correspond to
responses that display  joint  addition.  An
enhanced effect to that which Is strictly
additive, as represented  by the diagonal
isobole, is more-than-strictly additive.  A
lessened effect to that predicted for
summation is less-than-strictly additive, or
shows no addition.  Response  addition for two
toxicants with parallel concentration-
response curves is defined by  the curved
isoboles for complete negative (r=-l) and for
no correlation (r=0) in susceptibility.
                                                                                           1:200
                                                                                         -I 4OO
                          2.0        4.0        6.0        8.0

                                 TOXICANT B (mM)
                10.0
              Figure 1.   tsobole diagram depicting various types of lethal responses for the
                         joint action of two toxicants displaying parallel concentration-
                         response curves.   (From  Muska and Weber, 1977).

-------
                                                  K&%£&£f;&3:^,::;: wvvms^i*^. i-.i;^-„• :^.
Combinations falling exactly on the  upper  and
right boundary of the square correspond  to
ureas of no addition or the limiting case  of
response addition with complete positive
correlation (r=l).  Areas outside of the
square represent antagonistic responses
(less-than-no-addition) where one toxicant
counteracts or opposes the action of another
beyond that expected for the individual
toxicants.  In a similar Banner to that
presented  above,  isobole surfaces can be
defined  for three toxicants.  This terminology
and  classification  scheme for the toncity of
two  chemicals can,  with certain modifications,
be extended to chemical mixtures containing
several  toxicants.

              EXPERIMENTAL APPROACH

   A  mechanistic  approach  incorporating
toxicant-receptor  theory  to  assess jo . nt
action  has not been pursued  during our
 research project  because  of  the  difficulty in
determining  and  lack of understanding of the
primary  mechanisms  by  which  toxicants eiert
their effects.   Instead,  the general approach
 has  been to  study the  relationships between
 toxicant concentrations and  whole organism
 responses which  can be  observed  and measured.
   It is  proposed that  general  elementary
 principles and  models  describing  responses
 resulting from  toxicants  having  simi.ar or
 different modes-of-action  can  guide the design
 of realistic  and practical  experiments  that
 will provide  insight into  joint  action  of
 multiple toxicants.  By  designating special
 types of toxicant interaction, a  frame  of-
 reference for the systematic documentation and
 quantitative  evaluation of  such  effects for
 chemical mixtures is provided.   It  should be
 noted that the  nature  of  each  type  of joint
 action can only  be  described in  relation  to
 the  particular  response  being  considered.  The
 special  case of  non-interactive  joint action
 has been  investigated  as  a  first  and
 predominant approach to evaluating  the  effects
 of toxicant mixtures.
   The specific  forms of multiple  toxicity that
 are of particular concern from an
 environmental point of view are  characterized
 by  those  with effects  either greater than or
 equal to  that which would be expected if  each
 toxicant  contributes to the overall  effect
 according to some function of  its respective
 potency.  Therefore, experiments  were designed
 to differentiate between  no addition,
 less-than, more-than,  and strictly  additive
 joint action.
   The actual approach to  studying the joint
 action of chemical  mixtures must be more
 quantitative than qualitative.   This may  pose
 complicated statistical questions related to
 experimental design and analysis.   To date
 this  issue has not been adequately  addressed.
 The  experimental design should involve  a  wide
 range of  concentrations (sub-threshold  to
 effect  levels for  lethal  and sublethal
 endpoints) of the  toxicants alone and  if
 possible  at  various proportions  of  chemicals
  in  mixtures.  To accommodate this need,  our
 primary  approach has been to conduct
experiments to define  the  joint acute toxicity
of binary mixtures as  determined from isobole
diagrams.  Additional  work is planned with
sublethal endpoints.
  Our initial experimentation has included the
testing of mixtures which  are expected to
produce a concentration- addition response.
This type of joint action  in which the
constituents act independently but similarity
is predicted when the  quantal or graded
concentration-response curves for the separate
component toxicants and all  mixtures are
parallel or similar in shape, or when the
primary mode of toxic  action for the test
chemicals is expected  to be  similar.  Response
addition, the joint action in which the
constituents act independently but diversely.
can be predicted if the quantal or graded
conce nt rat i on-re spc~.se curves for tr.e separate
component toxicants and all  mixtures are
non-parallel : r are c ; 3 s i m i 1 a r in snape    7w,. s
type of  joint act.or :s also predicted  ;f
triere  is a kr.swn difference  in toxic action
between  the constituents   initially,
experiments were conducted with mixtures of
only two toxicants with multi-chemical
mixtures tested wher, confidence in
interpreting the sispler systems was obtained
or when  a need for such information became
apparent
  To allow for a nore  comprehensive
interpretation anc extrapolation of  limited
test results, a multiple toxicant study should
rely on  fundamental relationships between
biological activity and selectivity, and the
chemical nature of toxicants   Such an
approach, based on quantitative
structure-activity relationships (QSAR's)
where  toxicity is predicted  from models
incorporating molecular descriptors derived
from structure, has been proposed by Konemann
(I981b)  and Hermens and Leeuwangh (1982).
With this approach it  can be initially
presumed that chemicals causing a specific
effect by a primary and common mode-of-action
ii.e , narcosis, respiratory uncoupling.
acetocholinesterase inhibition, etc.) can  be
modeled  by a single high quality
structure-toxicity relationship.   Each
different type of toxic action (selectivity)
should thus be characterized by a different
empirically derived QSAR,  and concentration
addition would be expected for toxicants
within each relationship.

             MATERIALS AND METHODS

Testing  Conditions, Apparatus,  and Procedure

  The  96-h acute toxicity  tests with 30-day
old  laboratory cultured juvenile fathead
minnows  (Pimepha1es prome1 as} were conducted
according to test conditions and with an
apparatus described by Brodenus and Kahl
(1985).  The testing procedure was according
to ASTM  (1980).  Tests were  initially
conducted with individual  toxicants, and
subsequently expanded  to test solutions
containing tip to 21 toxicants.   Seven ratios
of two test chemicals  were used to define  the
binary  isobolograms with four concentrations

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following »n 807. dilution factor  »t  each
•ixture ratio.
  The values for the n-oct»nol/w»ter partition
coefficients (log P) wert taken from Hansch
and Leo (1979). Veith et al.  (1979), and Veith
et al. (1985) or as calculated fro*  the ClogP
version 3.2 computer program developed by the
Pomona College Medicinal Chemistry project
(Leo and Welninger. 1984; see Leo, 1985).

Data Analysis

  Data were analyzed using several statistical
procedures.  Estimates of the concentration of
toxicant most  likely to cause 50% mortality
(LC5Q) and their 957. confidence limits were
determined from relationships fitted
mathematically by the trimmed Spearman-Karber
method ''Hamilton et al  , 197"'   Concentra-
tion-response  slopes were determined by a
least  squares  linear regression program
  The  manner  in which the comoined effects
of mixtures of two  or more toxicants are
calculated by  the quantitative tone unit,
additive  index, and mixture toxicity index
approach  have  been  outlined by Sprague (1970),
Marking  (1977), and Koneraann (I981b).
respectively.  The  procedures used to analyze
results  by concentration and/or response
addition  models are according to those
proposed  by Anderson and Weber (19751   A
statistical procedure to determine  if binary
test  data  are  better described by a  straight
(strictly  additive) or  curved  isobole has been
described  by  Broderius  and Kahl (1985)

             RESULTS AND DISCUSSION

Concentration-Response  Curves

  Acute  lethality tests  were conducted with
juvenile  fathead minnows in order to define
the  toiicity  of  individual chemicals alone and
 in  combination with certain other test
compounds.   A plot was  made of percentage
mortality in probit values as a function of
loj molar toxicant concentration (Log M) for
individual  treatment levels from experiments
conducted with several  chemicals and for each
of three suspected different modes of toxic
action.   An example of  one plot is presented
In Figure 2a.   The slopes of the concentration
response curves for each separate mode appear
to be reasonably parallel and therefore can be
characterized by a single slope.  Plots were
made of  these data for  the Narcosis I,
Narcosis II,  and uncoupler of oxidative
phosphorylation model  relationships, as
normalized  according to the potency (Log M
96-h LC50)  of that for  1-octanol,  phenol, and
2,4-dinitrophenol.  respectively.  An example
of one normalized  plot  is presented in Figure
2b   The slope for the  normalized response for
each different mode is  quite similar and
ranges from 12 8 to 15.1 for the Narcosis !I
and jncoupler chemicals  respectively liable
1)   Therefore, it is  apparent that the slope
of acute lethality concentration-response
curves cajinot necessarily be used to separate
chemicals by their mode of toxic action.

1 sobo1e  Pi agrams

  Acute  toiicity tests  are also conducted in
order to define isobole diagrams for binary
mixtures.  The test concentrations  of two
toxicants are combined  in various fixed ratios
to provide  seven 96-h LC50 values that define
an isobologram   Results from these tests.
representing three types of responses, are
presented in Figures 3-5.  Mixtures of
1-octanol and 2-octanone display a  strictly
additive type of joint action over  the entire
mixture  ratio range.  This  is apparent in
Figure 3 from a plot of the 96-h LC50 values
and 95" confidence limits for the binary
mixtures at 7 test ratios.  A statistical
analysis of these- test data that establishes a
           Table 1.  Percentage mortality in probit values (Y)  as a function of  log molar
                     toxicant concentrations (X) for 96-h acute tests with juvenile fathead
                     minnows
                                                        Normalized concentration-response
                                                        relationship (Y=a+bX)
Mode of
toxic action
Narcosis I
Narcosis 11
Uncoupler
oxidative
phosphorylation
Reference chemical
1-Octanol
Phenol
2,4-Dinitro-
phenol
Intercept
59.1
±3.18
50.8
+4.26
87.1
±6.28
Slope r*
13.5 0.724
±0.792
12.8 0.562
±1.20
15.1 0.730
±1.52
N
113
60
36

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                                                      _                      .
       I0r-
        Ol   I   1   I   I  I   I   I   I   I  I   I   I   l   I  I   I   I   1   I
       101-
        8-
    «••
    15   .
    o
                                                                B
        41—
     O

        2
        Y'59 I   +  13 5X
          t3.!8   ±0792
        R-0.724
        N-H3
                                i   i   I  i   i   i   i
-5
-4
-3
                                                  -2
                  Toxicant Concentration  (Log M)
Fi»ure 2.  (A)  Percentage mortality  in probit values as a function of  log
          toxicant  concentration for treatment  levels from Narcosis  I  test
          chemicals.   Up and down arrows represent  100 and 0 % mortality,
          respectively.  (B) Normalized plot of data in Part (A)  as  adjusted
          according to the potency  of 1-octanol.   (From Broderius and  Kahl,
          1985).

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strictly additive  joint toiicity was tlso
conducted (Broderius and Kahl,  1985).
  Because of the difference  in symptoms
associated with fish dying when exposed to
1-octanol and 2.4-pentanedione. a response
•dditi»e type of joint action would be
predicted for binary Mixture* of these
cheaicals.  Test results, however, were
definitely not strictly additive but did show
joint action that  was  less than strictly
additive but apparently greater than response
addition with r=-l (Figure 4).  Therefore,
results from this  binary raisturc acute test
did not fit either the concentration or
response additive  joint action models.  A more
hazardous joint action than  response addition
was observed.
  The tests with binary aixtures of  1-octanol
and 2-chloroethanol  provided  interesting  luc
explainable results.  The  96-h  LC50  for
1-octanol  was unchanjed up to the  LC50  level
for 2-chlorocthanol.  The  toiicity of
2-chloroethanol, however,  was Markedly  reduced
by the presence of octanol.   It is proposed
that the presence of  octanol  inhibits the
metabolise of 2-chloroethanol to a nor*  toxic
metabolite and thus  results  in  a complex
isobole diagram.  From  the approximately  75
isobole diagram relationships thai we have
jenerated, the majority display a  rcsvonse
exemplified by the  first  two  digrams (figure
3 and 4).   The complex  type  of  joint action,
as exemplified by 1-octanol  and
2-chloroethanol, was observed  in oiily a  few  of
           0.60r-
                         0.02     0.04     0.06     0.08     0. ft     0. 12     0~l 4
                                            -Octanol   (mM)
             Figure 3.   Isobole diagram depicting the  96-h  LC50 values and confidence  limits
                        for  juvenile fathead minnows  exposed  to different mixtures of
                        1-octanol and. 2-octanone.

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the  isobole  type tests.   These Utter  tests
frequently  included primary aromatic amines
(Aniline derivatives) us one of the test
chemicals.   In only one instance has a
Markedly "ore than strictly additive type
joint action been observed in binary Mixtures
of industrial organic*.

OSAR and Joint  Tnxicitr - Narcosis 1 Chfmu.-a 1 •<

  If the results from joint  tuxicity tests arc
 to make  an  important contribution  to aquatic
 toxicology, a certain  basic  understandinj as
 to  how  chemicals jointly  act must  be
provided.   Tests must  also be conducted in
such a manner that there  is a predictive
nature to our findings.   To address these
joals. our Mixture testing effort is related
to an acute toxicity data base that is being
systematically (enerated  for a projran to
evaluate aquatic toxicity of organic chemicals
from  a structure-activity approach.  This data
base  for juvenile fathead Minnows  is being
developed at the U.S.  Environmental Protection
Agency, Environmental  Research Laboratory-
Duluth.  Some of  this data  has been tabulated
by  the Center for Lake Superior  Environmental
Studies  (CLSES  1984,  1985).   A plot of  our
           °'8.0      0.02      0.04     0.06     0.08     0.
                                           1-Octanol   CtnM)
                           0.12      0.14
               Figure 4.  Isobole diagram depicting the  96-h  LC50 values and confidence  limits
                         for juvenile fathead minnows exposed  to different mixtures of
                         1-octanol and 2,4-pentanedione (-.-«- predicted relationship
                         for response addition with r=-l).

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acute toxicity data base for approximately 600
industrial orj»nic chemicals is presented in
Figure 6.  The solid  square data points define
»n approximate water  solubility line above
»hich there »re very  few observed dat»
point*.  This line, therefore, defines a xone
beyond which an acute response is not expected
in a four day test.   It is apparent that the
data do  not fall into siany obvious patterns
•hen the  acute response is plotted only with
lot P-   Virtually all of the test data fall
within a lo| P ran|e  of about  -1 to 8 and the
acute  toiicity is in  general directly related
to  log P.  Ve'ith et  al.  (1985)  observed that
almost 507. of the 20,000 discrete organic
                               industrial chemicals currently in production
                               have log P value* less than 2.0.   Therefore,
                               since our data base is representative of the
                               TSCA chemicals, the 96-h LC50 to juvenile
                               fathead minnows of nost industrial chemicals
                               is expected to be approximately 10  M or
                               greater.  There also appears to be a base line
                               toxicity  (Figure 8) below which a chemical can
                               not be  less toxic.  This is most apparent for
                               chemicals with a log P of less than about 4.0.
                                Because it  is difficult to make any specific
                               conclusions from such a plot the data were
                               divided  into  smaller units and plotted
                               according to  chemical class or subgroupings.
                               An example of  one  such unit was for the
           t  .4r-
            0.
0.02
                                  0.04     0.06    0.08
                                         1-Octanol   (mM)
             j
0.12    0.14
              Fijure 5.  Isobole diagram depicting  the 96-h LC50 values  and confidence  limits
                        for juvenile fathead minnows exposed to different mixtures  of
                        1-octanol and 2-chloroethanol (-«-.- predicted  relationship for
                        response addition with r=-l).  Vertical arrows  indicate  greater than
                        v;. I ues.

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                   ^
ketones (Figure 7).  From this plot it is
apparent that the majority of the tested
Icetones conform to a response model line that
Veith et al.  (1983) have characterized by a
•ode of toxic action called Narcosis  I.  This
procedure was repeated  for 22 other chemical
groupings and it  was observed that greater
than SOX of the  industrial organic chemicals
that we have tested  conform  to  this
non-reactive or  baseline  mode of  acute  toxic
action.  Therefore,  the majority  of organic
 industrial  chemicals apparently do not  have
 specific  structural  features which allow them
 to be  biologically active by specific
 mechanisms.  This nonspecific or general
 membrane perturbation mode of toxic  action
 called Narcosis  results from the reversible
 retardation of eytoplasmic activity  as a
 result of  the absorption of foreign molecules
 into biological  membranes   The  environmental
 concentration necessary  to  pro;Jce this
 response  is independent  of  Tc.ecular  structure
and is linearly related to log P.  This  is
only true, however, if no metabolic
alterations result in more toxic metabolites
and steady state equilibrium is attained.
  If test chemicals are conforming to  a  QSAR
that defines a suspected mode of toxic action,
then one Bight expect that chemicals defining
this Mode will be  strictly additive  in their
joint toxicity.  To test this premise, isobole
diagrams  were  generated for binary mixtures of
1-octanol  (e.g., Narcosis  I reference
chemical)  and  a second chemical  from each of
seven different chemical groupings that  in
general  conform to the Narcosis  1  model  line.
The  results of these  tests, as normalized to
the  potency of  1-octanol,  are presented  in
Figure  8 (Broderius  and Kahl.  1985).   It is
apparent that  the  iscsoles  are  in  genera!
characterized  Dy  a :.agonal  line  that
describes a  strictly  additive  type  of  joint
 action    This  su;g» = ts  tr.at  trie  fathead  ,TM nr.ow
 perceives these  che-  :a.s  as  having  the  same
            2r-
           -2
        o
        0
       O  -4
       to
       o
        -C
        s   -6
            -8
                               O      _J
                                                                                    o
                                                                                    o
                                                                                               -2
                                                                                               -4
                                             a
                                             o
                                        -6
                                             -Q
                                              L
                                              0
                                        -8
                                                                                             8.0
                                                                                                ~to
                                                    Log  P
                 Figure 6.  Acute toxicity to  the  fathead  minnow of  approximately 600 industrial
                           organic chemicals  is  related  to  the  octanol/water partition
                           coefficient (Log P).   Water  solubility of  allcyl  alcohols Indicated by
                           square data points.

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or a very similar »ode of  toxic  action.
  k iecond type of experiment  has  been
conducted to document the  joint  toxicity  of
Mixtures containing two or more  Narcosis  1
toxicants.  An attempt was »ade  to prepare
tett concentrations of these mixtures on  an
equal proportion basis, based  on LC50
concentrations of the individual chemicals.
Usinf the mixture toxicity index (MT1)  scale
(Konemann. 1981b), it was  observed that the
joint action for the tested mixtures
containing 2 to 21 chemicals is  in general
characterized by strict additivity (i.e.,  MT1
-  1).  Therefore, a concentration addition
type of joint action has not only been
demonstrated for chemicals fron the same  class
but also  for chemicals from seven different
:iasses and  in equitoxic nixfures containing
^r  to 2'.  chemicals (Brodenus  and Kahl, 1985)
   We have conducted acute  toxicity tests  with
severa: alky! alcohols, which produce  a
c'. ass.oa! narcosis type of tone act:on   The
                             acute  toxicity  of  these  alcohols has been
                             observed  to  increase  »ith  increasing log P and
                             decreasing water  solubility.  The  relationship
                             is apparently  linear  for the  homo logs  tested,
                             with the  acute  response  covarying  with water
                             solubility at  lof  P values  less than 4.0.
                               The  allcyl  alcohols  apparently define a QSAR
                             series when  log P  is  used as  the only
                             independent  variable.  Veith  et al. (1983)
                             have proposed a bilinear QSAR model for
                             physical  narcosis  that is based on a
                             relationship derived  from about 65 common
                             industrial chemicals  (e.g., alcohols,  alkyl
                             halides,  ethers, ketones, benzenes).   These
                             data indicate that chemicals  exerting  a common
                             narcosis  node  of action, characterized by
                             membrane  expansion, may  be  modeled jointly,
                             even though  ethers, ketones and benzenes are
                             in general siijntiy more toxic tnan alcohols
                             The joint action  of test chemicals associated
                             with the  Narcosis  1 SAR  were  expected  to be
                             characterized  bv  the  concent rat.on addition
         2r-
         0-
        -2 -
     O)
     O
    O  -A
    \r>
    o
    .c
     i
    
-------
model. Our results indicate thit this was true
for numerous binary and  equitoiic mixtures of
up to 21 chemicals.
  Konemann (I981a) conducted 7 or H day
equitoxic acute toxicity teits usini juppies
fPo>eiHa rettcnltti) and •ixturet containing-:1
up to 50 industrial chemicals.  Henaens et al.
(1984) conducted 48-h acute toxicity tests
with pfpbnia magna and Mixtures containing the
same 50 chemicals as tested with guppies.
When these data are plotted against the
Narcosis I bilinear SAR node!  line of Veith et
al. (1983) (Figure 9), a good  log P and
biological activity dependent  correlation is
noted among tJ1 three model lines.  This
suggests that the sensitivity  of different
fish  species and daphnids to non-specific
anaestnetic-iike chemicals is  similar since
the Narcosis I model relationships in Figure  9
are a.l quite similar   Schult: and Moulton
',1384   have recently reported  a similar
relationship with a different  activity  scale
between log ? and biological activity in
Tetrahvnena ovnformis for 43  aromatic
 industri al chemica1s
                                  The type of  joint  action that  Konemann
                                (1981b) an-* Hermens  et al. (1984)  observed  for
                                mixtures containing  numerous  lipophilic
                                organic compounds  can generally  be
                                characterized  by concentration addition.
                                Their MTI values were reported to  be  1.02 and
                                0.95, respectively.  This was even true for an
                                equitoxic mixture  containing  50  compounds at
                                0.02 of their  respective LCSO values.  This
                                apparent additivity  for  industrial  chemicals
                                characterized  by a narcosis type mode  of
                                action should  be of  particular  interest
                                because a proportionately  large  number of
                                chemicals from the TSCA  inventory  are  likely
                                to cause lethality through narcosis (Veith  et
                                al., 1983).
                                  Numerous authors (Ferguson,  1939;  Seeman.
                                19T2; Konemann.  1981a. and Hermens  and
                                Leeuwangn. 1982) have  suggested  that  phvs:
                                unspecific toxicitT  can  be minimally  expe:
                                of .nost nycrophob i c  organic cnemicals  at  •
                                concentrat,on.   This is  expected unless a
                                chemical is metaboUzed  or its  effect  is
                                masked by overwhelming irreversible and mo:
                                toilc effects  from specific structural
          0.12 —
         o.oo
            0.00
0.02
0.04
0.06
0.08
0.10
                                 l-Octanol(mM)
                                                                       TOXICANT H
                                                                     • l-Hexanol (fry)
                                                                     O l-Hexanol  (juveniles)
                                                                     A 2-Octanone
                                                                     a Dlisopropyl Ether
                                                                     * Tetrachloroethylene
                                                                     O 1.3-Dichlorobenzene
                                                                     4 n-Octyl Cyanide
                                                                     7 n,n-Dimethyl-P-Toluidine
0.12
            Figure  8.  A composite isobole  diagram  of 96-h LCSO values depicting  the  joint
                      toxic action for 1-octanol with seven other chemicals,  each  normalized
                      to the toxicity of  1-octanol.  (From Broderius and Kahl,  1985).

-------
characteristics.   In this case,  a specific
interaction with a receptor may be responsible
for the effect.   Therefore,  the joint toxicity
of mixtures of hydrophobic organic chemicals
with various actions is •inimilly based on
concentration addition of their Biniaal
untpeciflc toxicitj.  This contribution of a
compound In a nonionixable form can be
emulated fro. the Narcosis I  QSAR (Kone.ann.
I981b- Veith, 1983; and Hernens et al..
1984)   In mixtures with only a few compounds
with different specific and more toxic action
this unspecific toxicity mi|ht  not markedly
contribute to the observed response.  In a
mixture of numerous differently acting
compounds at equitoxic concentrations, the
specific tone effects mijht not be apparent
because the  concentration of the individual
members will be so  low.  The fractional
                                                  unspecific toxicity from hydrophobicity,
                                                  however,  will  persist  and this  tdditive effect
                                                  may markedly contribute  to the  observed
                                                  response.  Therefore,  organic chemicals in any
                                                  concentration are expected to contribute to
                                                  the toxicity of a mixture with  respect to the
                                                  non-specific common site of action.
                                                  OSAR and Joint Toiicitv - Narcosis 11  and
                                                  Uncoupler Chemicals
                                                      There is considerable evidence that
                                                  reversible narcosis might result from several
                                                  mechanisms.   Veith et al.,  1985 have suggested
                                                  that the comparatively non-specific narcosis
                                                  from membrane expansion might be separated by
                                                  a QSAR from narcosis by membrane depolar-
                                                  ization.  This latter more sensitive
                                                  mechanism, which is observed at chemical
          Or—
    -J  -2
     C
     O
 C
 0?
 O
 o
o
         -4
     o
     o
     'x
     |2
         -6
         -8
                                                                          B
            -2
                                                                                             8
                                                   Log   P
              Figure  9.   Acute  toxicity  to the guppy (0,7- or 14-day LC50)  and Daphnia ma?na
                         (t,  48-h  LC50)  of 50 and 19 industrial  chemicals,  respectively, as
                         related to  the  octanol/water partition coefficient (Log  P).  QSAR
                         model  lines  for physical narcosis were  determined  by  (A)  Konemann,
                         1981b;  (B)  Veith et al., 1963;  and (C)  Hermens et  al..  1984.  (From
                         Broderius and Kahl,  1985).
                                                   57

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           • '•"•• •'•-.->•""  ' .  v**.-y-.«•;• v-<• •.'•• ' "",'• "  •-  ' ..   .'   "•••-jV'x?-' "-"••
activities statistically lower than the
baseline narcosis  (Narcosis I),  is identified
by Veith et al.,  (1985) as Narcosis II.  One
major class of  chemicals thought to produce
narcosis by depolarizing membranes at chemical
actiritits lower  than baseline narcosis  is the
esters (Veith et  al.. 1985).  This group
includes the benzoates. adipates. phthalates,
simple salicylates,  and alkyl acid esters.  We
cannot, however,  confirm that these esters are
acting by this  second mode of toxic action.
In fact, we have  observed that nany of the
nonoesters are  approximately  strictly addi'tive
with  1-octanol  in their acute joint toxicity
and therefore  presumably act  by  a similar
Narcosis  I mode of toxic action.  Several
diesters  were  observed to be  less than
strictly  additive with  1-octanol  or phenol and
thus  assumed  to have a different mode  of
action  than  either reference  chemical.
   Additional  groups  of c-emica's that  we  have
tested   include the  sub;:. fjted  ana
halogenated  phenols    Tv.ese compounds  can
generally  be  thought of as  not  chemically  or
biologically  reactive.  However,  depending
upon  the  substituents  present  on the molecule
the  hydroxyl  derivative night  ionize to
various  degrees at different  test pH values.
The  hydroxy  substituent can also conjugate
with  electron-withdrawing groups by  resonance
through the  aromatic ring of  the molecule
(Hansch and  Leo  1973)  Therefore,  it  "as
anticipated  that  non-log ?  related  effects
might be  important in  determining their  toxic
response  and  thus not  mode led-by the Narcosis
!  QSAP.
   The results  of  our studies  have suggested
that  the  toxicity of phenolic  compounds  can  be
modeled by  three  QSAR's.  We  have observed
that  those  non-acidic  substituted and
halogenated  phenols  with a  log  P of  about  3  or
greater are  strictly additive  with  1-octanol
or phenol.   Those phenolic  compounds with high
log P values  are  highly hal ogenated' and/or
alkyl  substituted and  act chemically more  like
hydrocarbons  or halogenated hydrocarbons  than
phenols.   Those phenols with  a  log P of  -3 or
 less,  however,  are only strictly additive  with
phenol  and  not with  1-octanol.   Since  phenol
 is not  strictly additive with  1-octanol  we
feel  that we  have defined another mode of
tone action characterized  by  Veith et al .
 (1985)  as Narcosis I!.  These  polar chemicals
are  slightly more active than the baseline
toxicity of  non-ionic  narcotic  chemicals.
   Multiple  chemical  mixtures  consisting  of 11
phenolic compounds characterized by a  Narcosis
 II mode of  action have  been observed to  be
 strictly additive in their  joint acute
 toxicity to the guppy (Konemann  and Musch,
 1981).   Their test chemicals  consisted of
 phenolic compounds with  log P values of  both
 greater and less than 3.
   A third SAR grouping has  been identified and
 is characterized by acidic  phenols.  Chemicals
 in this group have activities lower  than that
 of Narcosis 1 and II SARs  and are structurally
 characterized as having strong electron
 withdrawing substituents  adjacent to  a
 hydrogen bonding  group.   Their mode of toxic
 action is thought to be that  of uncoupling of
oxidative phosphorylation.  In our experiments
we have designated 2,4-dinitrophenol  (2,4-DNP)
as the reference uncoupling agent for  this
mode of toxic action.  Acute toxicity  tests
have been conducted with 2,4-DNP and  chemicals
such as HCN or rotenone which are known to
inhibit electron transport in the Mitochondria
of cells.  These latter two chemicals  have
activities lower than those of the oxidative
phosphorylation nncouplers (Figure 10)  and  are
therefore thought to have a different  mode  of
toxic action.  When  rotenone was tested in
combination with 2,4-DNP, a less than additive
but more than response additive type  of joint
action was observed.  When HCN and rotenone
were tested in combination, however,  a  nearly
strictly additive joint acute action  was
observed   These results  are consistent with
the proposal that chemicals characterized  by
different QSARs do  indeed have different
primary modes of acute toxic action  ar.c s-.c'iij
not interact ;n a concentration aci . t .-•.-•
manner.  Those within a mode, however,  snouid
be  strictly additive in their joint  action
   It has been proposed that the QSARs  for
Narcosis II and uncoupling of oxidative
phosphory1 ation might be  improved through,  in
addition to log P   the use of molecular
decriptors such as.  electronic and steric
factors which reflect the polarity of  the
chemicals   The use  of pKa as an electronic
descriptor has seen  ased  extensively
  Hermens and  Leeuwangh 11962;  proposed  that
for mixtures  with  a  relatively  large  number of
chemicals with  diverse  modes  of  action  a
similar joint  toxicity  for the  different
mixtures will  result.  Thus,  mixtures
containing an equal  number of  chemicals  will
have MTI values  which are approximately  the
same.   This hypothesis  was tested by Hermens
and Leeuwangh (1982) «ith five  mixtures  of
eight chemicals  each, one mixture of 24
chemicals,  and  was demonstrated to be
approximately correct.   The joint response of
the mixtures  varied  from partially additive to
concentration additive.   It is  not likely that
this unexpected high joint response resuitea
from simple similar action,  because in  some
mixtures it is  most  probable that the
chemicals actually had different modes  of
action.  Hermens and Leeuwangh (1982) proposed
that the most plausible explanation for  their
experimental  results for lethal  tests  is that
dependent action is  the most likely type of
joint action to occur when dealing with
mixtures of numerous chemicals  with diverse
modes of action.   The fact that these mixtures
result  in a nearly constant MTI  value  is most
interesting but yet  unexplained.   It  is
important to determine how the  size of  a
mixture group would affect these results
Hermens and Leeuwangh (1982) and others  have
adequately demonstrated that organic chemicals
with diverse modes of action and at
concentrations about 0.1 of the LC50 values
and lower do contribute to the joint toxicity
of  mixtures.   Therefore, no effect levels of
separate chemicals may have little meaning for
mixtures and probably should be established
for groups of chemicals.

-------
                                             it.j.^.i»_A
Future Research

  The direction of future research in
evtluttin{ the environmental  hazards posed by
Multiple toxicants should include not  only the
•cute response bat also important chronic
endpoints such as growth and reproduction.
The effects of an accumulated total body
burden of toxic chemicals on reproductive
success and embryo-larval fish survival and
growth should be investigated.  In addition  to
these traditional endpoints,  future  research
might include the effects of multiple
chemicals on cytotoxic responses such  as
teratogenic and carcinogenic effects.
  Most aquatic multiple toxicant tests have
been  conducted mth daphnids, various
freshwater  fishes, and a few other organisms
The  incorporation of  new test organisms and
endpoints such as the African Clawed  Frog
i X e n o p u s  I a e v i s 1 to study teratoger.ic  effects
(Schultz  and  Duraont,  1984,   and the  rainbow
trout embryo  for carcinogenic effects  (BlacK
                         et al.. 1985) Bight be desirable.  Tests using
                         endpoints other than those obtained from whole
                         organism responses may also be instructive.
                         These  later tests may be of particular value
                         when it is suspected that mixtures are
                         displaying an interactive Joint action with
                         the metabolism of parent compounds playing a
                         major  role in defining observed responses.
                           The  type of tests that are needed in
                         multiple toxicant work include those that are
                         systematically conducted with individual
                         chemicals and various mixtures.  One cannot
                         over-emphasize the importance of a good data
                         base on diverse chemicals.  A specific test
                         that has proven most valuable is the binary
                         mixture test as conducted at several mixture
                         ratios.  Such data allows one to define
                         isobole diagrams of joint action   This
                         procedure has prove:, useful as a dis-
                         criminating  too! ir, identifying pairs of
                         chemicals that have a suspectea similar or
                         different mode of ::iic action   As testing
                         nas expanded  into -..tipie che-,;ca
           Or-
          -2
          -A
       O
       o
      O  -6
      10
      o
      JC
      s  -8
         -10
              NARCOSIS  i:
                MODEL
       -NARCOSIS  I  MODEL
               2.4-01nftroph«no
                     HCN
                                          UNCOUPLER  MODEL
ELECTRON  TRANSPORT
    MODEL
                                                   -^

                                                    °   Antlwycln A
                                             _L
                            0.0
               2.0             4.0
                      Log P
                                                                              6.0
8.0
              Figure 10.  Acute  toxicity to the fathead minnow  as  related to the octanol/water
                         partition coefficient (Log P) for  chemicals thought to be uncouplers
                         of  oxidative phosphorylation (•) or that  inhibit electron transport
                         and thus the metabolism of oxygen  (0).

-------

it has been the tradition*! approach to teat
equitoxic Mixtures.  In future testing it
might be desirable to plan experiments
accordinf to a multifactorial design.  With
this approach all combinations of sereral
diffcrtnt ieti of no-effect and effect level
treatments or measurements of all possible
joint interactions can be tested without
examining all possible combinations.  The size
of such  studies can  thus be  reduced  by
assuming that certain  interactions between  the
concentrations  and the responses are
nejli gible.
   Our selection of  test  chemicals  has  been
{Uided  by  principles established using  a  QSAR
approach.   This is  done  to  optimize  our
evaluation of  how chemicals  jointly  act  and to
broaden the application  of  test  results    We
 nave attempted to test chemicals  within and
 between different QSAR's,  assuming that we  are
 estaoiishing how chen. ca 1 s jointly act w,th
 si-ilar and different modes of action
 Reference chemicals have been usec  to
 represent various modes of toxic action
 Future  experiments  will  include those
 chemicals that have a "more specific" mode  of
 tone  action and which might display different
  levels  of  electrophi1ic reactivity.  We have
  also  separated our  testing  of organic
  chemicals  from that of metals.  It  would be
  desirable  to  combine  organic  and  inorganic
  chemicals  into  mixt-jres when  an understanding
  is  obtained of  how  each group acts  separately.
    The  statistical  analysis  of our  test  data
  has been  minimal,   We have  used  standard
  statistical techniques  as  previously  described
  by Sprague (1970),  Marking  (1977),  and
  Konemann (I981b).   More  sophisticated
  techniques as reported  by Durkin  (1981)  or
  Chnstensen and Chen (1985) might be more
   instructive in defining the degree of joint
   action and similarity among chemicals in
   mixtures.
     Various  relationships have been  derived
   between  toxicity and the octano1/water
   part;t;on  coefficient as the aotiinant
   parameter.  This has proven adequate to
   describe  the  relationships for non-specific
   organic  toxicants  but might  be inadequate  for
   chemicals with  more  specific primary
   aodes-of-act ion.   An untested but  potentially
   powerful  approach  to predicting  joint  toxicity
   of mixtures  deals  with N-space analysis where
   the "likeness"  of  tested  and untested
   chemicals, and certain benchmark chemicals,
   can be quantitatively  described.   With this
   approach  it would be assumed that  if  the
   structural properties  of a chemical  can be
   described with N  factors and plotted in an
   N-dimensional structure  space,  the chemical
   and  biological properties of a chemical should
   be similar to  its "nearest neighbors" for
    which data are available.  This approach might
    allow one to cluster compounds that show a
    similar  mode of toxic action and  thus display
    a concentration-addition  type of  joint toxic
    action.   The  type of joint action displayed by
    chemicals in different clusters light be
    characterized  by  a  form of  response addition.
    It is also  quite  probable that  the type  of
    joint action between chemicals  in different
clusters is too complicated to be presented by
simple models and will r.sed to be empirically
def ined.
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  Christensen,   E.R.  and C-Y Chen. 1985. A
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                           r VT*"' A* IrfVir'n -Vi—' '
                            * •
                                              .,.i.^^.iJ
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 Fertuson. J. 1939.  The  use  of  chemical
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 Finney.  D.J.  19V1. Probit Analysis. 3rd
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  Hansch   C.  and A   Leo   1979  Substituent
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             6 209-21'
  Hermens, J.,  H.  Canton. P. Janssen, and R. De
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   Hermens, J. and P. Leeuwangh.  1982.  Joint
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    Hewlett, P.S. and R.L. Plackett.  1959. A
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    Hewlett,  P.S.  and R.L. Plackett.  1964. A
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      Biometrics 20:566-575.

    Konemann,  H. 198la. Quantitative
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     Konemann, H. 1981b. Fish toxicity tests with
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     Konemann, H. and A.  Musch.  1981. Quantitative
       structure-activity  relationships in fish
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       pH on the QSAR of chlorophenols. Toxicol.
  19:223-228.

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  Environ. Health Perspect.  61:275-285.

Leo,  A.  and D. Weininjer.  1984.  ClofP Version
  3.2: Estimation of the  n-octanol/water
  partition coefficient  for  organics in the
  TSCA industrial inventory. Pomona Medicinal
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  Claremont,  CA  (mimeo).

Marking,  L.L.  1977.  Method for Assessing
  Addi tive Toxicitv  o£ Chemical  Mixtures,
  Aauat i c Toxi co1o?v.  and Hazard Evaluation.
  ASTM  STP 634,  F.L.  Mayer and J.L. Hamelink.
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 Marubini,  E.  and '_   Boranomi  1970
  Quantitative evaluation of toxicity tests
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 Wuska.  C.F.  and L.J. Weber   1977.  An  approach
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   Statistical  aspects of the independent joint
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    Roy. Statist. Soc., Series B. 14:141-154.

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i^v'^^

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                                               DISCUSSION
                      Q.  P.  Pmtil and C.  Taillie,  The Pennsylvania State University
        1.  DTTBODDCTION AND BACKGROUND
       author deserves Commendation for t>\m
      prepared for presentation at the ASA/SPA
           on Currant AM element of Combined
Toxicant Effects to •joint audience of
participants from various related disciplines.
As he puts it, "defining the toxicity of
Mixtures is a aajor problem at both the
theoretical and practical level ...  There has
not been sufficient research to establish
whether there is any widely applicable rationale
and approach for evaluating and possibly
predicting the joint action of toxicants in the
aquatic environment ...  The types of tests that
are needed in Multiple toxicant work include
those that are systematically conducted with
individual chemicals and various Mixtures.  One
cannot over—emphasize the importance of a good
data base on diverse chemicals.  A specific test
that has proven roost valuable is the binary
Mixture test as conducted at several Mixture
ratios.  Such data allows one to define isobole
diagrams of joint action.  This procedure has
proven most useful as a discriminating tool ID
identifying pairs of chemicals that have a
suspected similar or different mode of toxic
action.  As testing has expanded into multiple
chemical mixtures, it baa been the traditional
approach to test equitoxic Mixtures.  In future
testing, it May be desirable to plan experiments
according to a multifactorial design —  Our
selection of test chemicals has been guided by
principles established using a QSAR approach.
This  is done to optimize our evaluation of how
cheaicals jointly act and to broaden the
application of teat results.  We have attempted
to teat chemicals within and between different
QSAR'a assuming that we are establishing how
cheaicals jointly act with similar and different
aodes of action.  Reference cheaicals have been
used to represent various modes of toxic action
has been minimal.  We have utilized standard
statistical techniques ...  More sophisticated
techniques may be more instructive in defining
the degree of joint action and similarity among
cheaicals in mixtures  ...  An untested but
potentially powerful approach to predicting
joint  toxicity of mixtures deals with N-space
analysis where the 'likeness* of tested and
untested chemicals, and certain benchmark
cheaicals, can be quantitatively described.
With this approach, it would be assumed that if
the  structural properties of a chemical can be
described with N factors and plotted in an
N-dimensional structure space, the chemical and
biological properties of a chemical should be
similar to its 'nearest neighbors' for which
data are available.  This approach may allow one
to cluster compounds that show a similar mode of
 toxic action  and thus display a
 concentration-addition  type  of joint toxic
action.  The type  of joint action displayed by
 chemicals  in different clusters May be
 characterized by a form  of response addition.
 It is also quite probable  that the  type of joint
 action between cheaicals  in  different clusters
 is too complicated to be presented by siMple
 Models and will  need to  be empirically
 defined..."
    The author should be  complimented for  his
 effort in developing  these  complex problem  areas
 and in communicating  them  to the  substantive
 scientists,  statistical  sethodologists, and
 managers.   The paper  covers  a broad spectrum  of
 issues and approaches  pertaining to aquatic
 ecotoxicology.  risk assessment, monitoring  and
 •angement with particular emphasis on  matters
 relating to the perceptive isobole diagrams and
 the widely recognized QSAR techniques.
         2.   STATISTICAL CONSIDERATIONS

   We initially propose to briefly discuss and
 formulate MOM of the basic statistical aspects
 of the approach leading to the isobole diagrams,
 and  aubsequently offar a few remarks pertaining
 to their role and use for field situations.
 Let   X
    lat the toxicants be denoted by A, B, C,
denote the tolerances of
 an individual to the toxicants A, B, C, ...
 respectively.  Let BA, Eg, Eg, ...  denote the

 exposure/concentration levels of A, B, C, ... .

 2.1  Tolerance Distribution: Assume that a

 tolerance level can be associated with each
 individual organism.  Thus the organism shows a
 response if the exposure level exceeds its
 tolerance.   The distribution of tolerance levels
 across the population of individual organisms is
 said to be the tolerance distribution.

 2.2  Response Function:   This is the expected

 proportion of organisms that show a response at
 a given exposure level.   Note that in the case
 of one toxicant, the response function is the
 save as the cumulative distribution function of
 the tolerance distribution.  As we will shortly
 see,  this is not true for multiple toxicants.

    While the response function is directly
 observable, tolerances and tolerance
 distributions are concepts that may be useful in

 guiding one's thought processes.  However,
 situations arise where the tolerance concept may
 be faulty.   Whether a given organism exhibits a
 response depends upon numerous environmental
 factors.   To the extent that these factors and
 their interactions are not known or are not
 predictable, the organism's zero-one tolerance
 level needs to be replaced with a "fuzzy"
 tolerance,  i.e. there is a probability that the
 organism responds at a specified exposure level.
 The response function is then the average, taken
 over all exposed organisms, of these
 probabilities.  The effect is to increase the
 variance or, equivalently, to decrease the slope
 of the probit diagram.  The smaller slope is a
 major point of differentiation between field and
 laboratory investigations.

 2.3  Joint Tolerance Distribution:  For

 simplicity, we consider only pairs of chemicals
 and bivariate distributions.  To each individual
 is associated a pair  (X^,Xg)  of tolerances.
 Notice that each component tolerance,  XA  or

 Xg, determines whether the individual responds
 to the chemical,  A  or  B, when exposed to the
 chemical separately.  There are no combined
 effects involved at this point.  The
 distribution of the pairs  (X^.Xg) across all

 organisms in the population is the bivariate
 tolerance distribution.   If  A  and  B act upon
 similar receptor sites,  then the tolerances
 (X&,XB)  are expected to be positively

 correlated.  A correlation of zero is expected
 if the sites are dissimilar.  Negative
 correlation, while possible, appears to be
 unlikely.

 2.4  Mode of Action:  Unlike the univariate

.case, the bivariate tolerance distribution does
 not determine the response function.   To pass
 from tolerance to response, an additional
 concept is required, one that describes the
 effect of the chemicals when they act in
                                                   63

-------
                                                                                           .ffe
                                                                                                           •u «i.  Jc".
    imtioa with one toother.  Ut  (I^.lg) b«
the Joint concentration  (exposure lavel) of the
tno chemicals.  The Bode of action of  A  and  B
should determine, in terms of the organisms'
tolerances (XA,XB), which organisms will show a
         to (BA,BB).  Formally then, the •£& fif
      metioB can be defined aa a rule that

        to aach Joint anpoaura level (B^iBg)  •
ration in the two-dimensional plane  (XA,Xj)  of
possible tolerance value*,  a jiven organism
shows a response to  (BA,BB) if  and only if ita
tolerance pair falls within this region, which
we call the response region.
   Once the mode of action is specified, it is
easily seen that the bivariate response

function, evaluated at  (BA,BB),  is the integral
of the bivariate tolerance distribution over the
region associated with  (E^,Eg).  Both the joint

tolerance distribution  and the node of joint
action are needed to determine the joint
response function.  A central  issue is whether
and to what extent it is possible to infer
properties of the tolerance distribution
and/or the node of action from observations
made upon the response  function.
        Joint
      Tolerance
     Distribution
    The mode of joint action needs to
 satisfy at least the following requirements
 (where R is the response region associated with
 (BA,BB)):
           The point  (E^.Eg)  lies on the

           boundary of  R.   This requirement
           appears to rule out physical
           interactions between the
           chemicals.
(i)
     (ii)
     If
     and
                    is in
                     then
                               R and if
S X.
                                        is also
      A-V
      .B < XB.  then  (XA,XB)

in  R.  In words,  if an organism
shows a response then so will all  less
tolerant individuals.
(iii)   If  B^ < BA
                  and
                                < BD  then the
           response region associated with
           (BA,Bg) is a subset of the response
           region associated with (BA,Bg).
    Figure 1 shows some hypothetical response
 regions that meet these requirements.


 2.5  Examples of Modes of Action: The mode of
 action  is called concentration addition when a
 law of  simple linear substitution applies.  In
 other words, it is possible to reduce the
 concentration of  B  and produce identical
 results by making a corresponding increase in
 the concentration of  A.  The response region
 has for ita boundary a straight line with
 negative slope; the magnitude of the slope is
 the relative potency of the two toxicants.
 Notice  that ail points along this straight line
                                                                                      \\ V
                                                                *A
                                                                                           (at)
                                                    Figure 2.  Response regions (shaded) for three
                                                    different Bodes of action:  (ii) concentration
                                                    addition, (iii) response addition.   The mode of
                                                    action (i) has no standardized name but
                                                    represents a situation in which A and B act
                                                    upon different sites and these sites form a
                                                    parallel-system in the sense of reliability
                                                    theory.
                                                   determine the same response region and,
                                                   therefore, the sane value of the response
                                                   function.  It follows  that,  in  the case of
                                                   concentration addition,  the iaobolea (contours
                                                   of the response function)  are exactly the
                                                   boundaries of the response regions.
                                                   Concentration addition is often motivated by
                                                   supposing that the toxicants act upon the same
                                                   receptor sites, thereby implying a perfect
                                                   correlation in the tolerance distribution.  We
                                                   see that a perfect correlation  is not a
                                                   logically necessary condition for concentration
                                                   addition.  In fact,  there are infinitely many
                                                   tolerance distributions that assign the same
                                                   probabilities to the triangular response regions
                                                   and thereby determine  the same  response
                                                   function.
                                                      From a regulatory standpoint,  it is the law
                                                   of substitution that is important and it is the
                                                   linearity of that law  that makes for a simple
                                                   regulatory strategy.   One can easily envision
                                                   situations involving nonlinear  laws of
                                                   Substitution (Figure 2).   Let us define a mode
                                                   of action to be self-similar if every point on

                                                   the boundary of a response region  R  has  R  as
                                                   its resoonse region.   Concentration addition is
                                                   self-similar, as is any mode of action whose
                                                   response regions have  boundaries defined by
                                                   single equations such  as XA • XB = constant  or

                                                   X? -f X5 = constant.
                                                             :          of nonlinear law of
                                                     substitution..  Response regions have circular
                                                     area for their boundaries.  All chemical

                                                                         ne °f
                                                    64

-------
to iraanin with concentration addition,  all
eelf-eiailar modes of action have  the two
properties that (1) isoboles are the  boundaries
of the response regions and (ii)'there are
infinitely many different Joint tolerance
distributions that yield the same  response
function (in fact one can always find such a
joint tolerance distribution that  is  perfectly
correlated in the sense of concentrating its
probability MM on a ooe-diaensiooal subset of
the (IA.X.) plane).

   A aaconrt mode of joint action la known aa
         addition.  This occurs when  an organism

      .a response to  (BA,BB)  if and  only if it

would respond to  BA  acting alone or to  Bg

meting alone.  Response addition calls for the
simple regulatory strategy of setting separate
standards for each of the two toxicants.   The
response region for response addition is shown
in the third diagram of Figure 1.  The picture
reveals the aptness of the term "response
addition" since the total number of responses is
the sun of the responses to  A  and the
responses to  B  (after adjusting  for double
counting).
   Response addition is not a self- similar
mode of action; for example, the points  (B'^.Bg)

and  (BA>Bg)  in Figure 1 determine different

response regions.  The shapes of the  isoboles
depend upon the joint tolerance distribution.
By contrast, for a self-similar mode  of action,
we need the tolerance distribution to determine
the levels (LC50, LC80, etc.) but  not the shapes
of the isoboles.  Also, in the case of response
addition, the joint tolerance distribution is
uniquely determined by the response function.
Indeed, from Figure 1, the response function


evaluated at (E.,BB)  is  1 - F(BA,E,,)   where F
               AD              AD
is the survivor function of the tolerance
distribution.

3.  STATISTICAL ISSUES IN THE APPROACH OF
    BRODBRIUS

   This section hopes to identify  a few
statistical issues that seen to be implicit in
the approach that Broderius has presented.  This
is not an exhaustive  list, but only indicative
and preliminary.

3.1   Isoboles and the Nature of the Joint
Action:   Isoboles are the appropriately chosen
contours of the response function. They depend
upon  both  the mode of joint action and the joint

 tolerance  distribution.  Thus, it  is  impossible
to infer the nature of the joint action from the
examination of  the isoboles alone. It is
necessary  to know or  to assume a model for the
 joint tolerance distribution.  Broderius appears
 to assume  a joint probit model.  But  different
models could yield different conclusions
 regarding  the nature  of the joint  action.

 3.2  fcoboles and Levels of Isoboles: Broderius
 restricts  attention to LC50 isoboles.  Would the
conclusions be  qualitatively the same or
 different  if other levels were employed?  It
 should be helpful to  investigate these problems
 both in theory  and practice.
3.3  Biological Homogeneity in Broderius

Approach and Field Heterogeneity.   The

laboratory work described by Broderius
maintains a high degree of biological
                                                       homogeneity.  This results in'the steep slops*
                                                       in his probit diagrams and nearly degenerate
                                                       tolerance distributions.  Bven within  the
                                                       framework of probit model, the isoboles
                                                       corresponding to response addition are heavily
                                                       dependent upon the slopes.  It is not  apparent
                                                       that conclusions about modes of joint  actions
                                                       that are derived from laboratory studies under
                                                       regimes of strict biological control could be
                                                       extrapolated to field conditions, where
                                                       biological aa well aa environmental
                                                       heterogeneity prevails.

                                                       3.4  the Issue of Synchronous and

                                                       Asynchronous Exposures:  Fish are mobile,
                                                       sometimes highly so, and are exposed to a
                                                       variety of toxicants during their lifetimes.
                                                       Would the results from Brooerius study, which
                                                       assume synchronous exposure,  carry over to the
                                                       asynchronous exposure that is common under field
                                                       conditions?
             4.   CONCLUDING REMARKS

    Steve Broderius has  presented a very
 interesting and illuminating paper on a problem
 of current  practical  concern in aquatic
 ecotoxicology.   It reminds us of three workshops.
 on aquatic  toxicology and risk assessment held
 in the recent  past.
    The Northeast Fisheries Center of the
 NOAA/NMFS organized a workshop in 1983.  Issues
 involved definition of  water management zones,
 grouping of chemicals and endpoints with a view
 to be able  to  consider  representative chemicals
 and representative endpoints, and formulation of
 indicators  and field  based statistical indices
 leading to  a crystal  cube for coastal and
 estuarine degradation.
    The BPRI workshop  had emphasis on
 multivariate bioassay,  ecological risk
 assessment,  and relevant experimental designs.
    The NOAA Chesapeake  Bay Stock Assessment
 Committee has  had its thrust on partitioning
 fish mortality due to pollution  (multiple
 chemicals included),  environment, habitat,  and
 fishing that has involved multivariete multiple
 time aeries and categorical  regression  related
 tools.
   Broderius' paper develops  a promising
 approach to the contemporary issue of aultiple
 toxicants and raises several  challenging and
 fascinating technical problems such as:
 statistical graphics of combined effects,
 multivariate tolerance distributions, binary
mixtures and multivariate  results, synergism
 concepts for the  'whole' being  'more  than  the
 'sum', QSAR related chemical  species grouping
methods reminding one of ecological 'guilds' and
 functional groups, and so on.
   The multiple toxicants  'ball' is not just
 in a statistical court.   It  is in every other
 relevant court at the same time.  It will take a
timely interdisciplinary effort involving
simultaneous (and not sequential) collaboration
of various substantive players.  We wish to
congratulate Steve Broderius  for this
 interaction at this ASA/EPA Conference.

Acknowledgements:  The authors would

like to acknowledge the partial support for
this work received through a  NOAA research
grant to the Center for Statistical Ecology and
Environmental Statistics, Department of
Statistics,  The Pennsylvania  State University.
under the auspices of the Northeast Fisheries
Center of the National Marine Fisheries
Service, Woods Hole,  Massachusetts.
                                                  65

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                           DEVELOPMENT OF MODELS  FOR COMBINED TOXICANT EFFECTS

                         Erik  R.  Chrlstensen, University of Wisconsin—Milwaukee
                    ABSTRACT

   Adequate  unlvarlate  dose-response functions
•re necessary 1n order to develop a satisfactory
•ultlpl* toxldty model. We Investigate here the
use  of  unlvarlate Welbull and  problt distribu-
tions with literature data for  the  quanta! re-
sponse of  fathead minnows (Pimephales promelas)
to  27  different  organic  chemicals.  We  also
examine  fits of the  Welbull, proMt.  and loglt
models  to literature data  for  the  growth  rate
and  yield  of the  diatom  Navicula  Incerta Inhi-
bited  by  Cd.  Cu,  Pb. or In. The  Weibull model
appears  to provide a superior fit for both fish
and   algae,   thus   supporting    a   previously
developed   mechanistic-probabilistic   basis  in
terms  of  chemical  -eactions between  toxicant
molecules  and  receptors of the organisms.
   The   application   of   a  general   multiple
toxldty  model  is  demonstrated  using published
experimental  results  regarding  the  action  of
binary  combinations  of N1,  Cu,   potassium penta-
chlorophenate,  dieldrin,  and potassium  cyanide
on  male guppies  (Poecilia  reticulata).  We also
analyze  results of  our  own  experiments  regard-
ing   the   combined   effects   of  Ni2+  and  Zn2+
on  the growth  rate  based on  cell  volume of the
green  alga  Selenastrum  capricornutum.   Host  of
the  multiple  toxicity  data  are  fitted  well  by
the  model.
                   INTRODUCTION

    Aquatic Ecotoxicology  is  becoming a topic of
 major  concern  (1,2).   It deals  with  the  re-
 sponse of  aquatic organisms to  toxicants such
 as heavy  metals  and organics, both in natural
 waters  and  water   and   wastewater   treatment
 plants.  One important goal  is to protect aqua-
 tic  organisms  against  adverse  effects  from
 pollutants.
    Several factors complicate the evaluation -of
 the  toxic  response  of   aquatic  organisms  to
 specified  concentrations  of  pollutants.   For
 example,  the  chemical  form of heavy metals 1s
 Important.  It 1s well  known  that  the ionic form
 of metals such as Cd, Pb,  Ni, or Cu  is  generally
 more  toxic  than  the complexed  forms (3).  For
 organics,   e.g.,    polychlorinated    biphenyls
 (PCB's)   or  polycycllc   aromatic   hydrocarbons
 (PAH's), the octanol-water partition coefficient
 1s of  Interest.   This  1s  because  there  1s often
 a  correlation  between  this coefficient,  the
 UpophiHty. i.e.,  the solubility  in  fat,  and
 the   toxicity  (4).   Other factors include vola-
 tilization  to  the  atmosphere  and  partitioning
 to  particulate matter.   Considerations  related
 to the organisms  are exposure time,  biomagnifl-
 cation, age, and species  composition.
    The response obtained within a given time of
 exposure,  e.g.,  96h,  has  been   studied  for
•any  different  compounds   and  a  variety  of
organism  such as fish  and algae  (2.5).   How-
ever, In  most cases,  only one toxicant  has  been
considered  In any  given  experiment.   This  1s
obviously a  simplification  since actual  aquatic
systems  usually  have more  than  one  dominant
toxic  compound.   The  objective of  the  present
work   1s   to  Introduce  a  multiple   toxicity
dose-response  model  and  apply  it  to fish  and
algae.    Unlvarlate   dose-response   models   for
these organisms will also be examined.
           CLASSIFICATION OF 8IOASSAYS

   The  response  of  aquatic organisms  to  toxi-
cants  can  be evaluated from bioassays  conducted
in  the laboratory or  in the  field,  or  in  sorre
cases,  from the  observation  of  actual  ecosys-
tems.   Possible  forms  for  laboratory  bioassays
are  shown  in Table 1 (6).  For most  macroorgan-
1sms,   or  mixed  cultures  of  microorganisms
(Groups  I  and  II), there is a tolerance  distri-
bution  for  individual  organisms.    This  means
that  some  organisms with  high  tolerance  will
survive  at high  concentrations or long exposure
times  while  others with  low tolerance will  not.
In  contrast, organisms  from a  pure  culture  of
microorganisms  (Groups  III  and  IV)  originate
from  a  single  clone  and,   therefore,  have  the
same   genetic  material.    Thus,  there   is   no
tolerance  distribution   for   Individuals  which
will  respond 1n the  same way to  the toxicant.
   The  response  can  be quanta!  or  continuous.
An  example of a  quanta! response  is  death  for
Group  I organisms.  A  continuous response  can,
for  example, be  growth  rate based  on  biomass
(Groups  II.  IV).   For  Group I  organisms,  the
response   is  the  fraction  of  all   individuals
that  are  affected,  e.g.,  by death.  Similarly,
for  Group  III  organisms,  we may  consider  the
response  to be  the  fraction of   subsequent  cell
divisions  that  are  blocked.  This  is  the  same
as  the reduction  In  relative  growth rate  based
on  cell  number.  This  Interpretation  1s extended
to  apply  also to Group  IV organisms.
   DOSE-RESPONSE MODELS FOR ONE TOXIC  SUBSTANCE

    Dose-response  models  for  a  single  toxicant,
 assuming a  fixed  time of exposure,  e.g.,  96h.
 are  shown   1n  Table 2.  Of  these,  the  problt
 model (7)  Is  perhaps  the most  well  known.   It
 1s based on a normal  distribution  of the  re-
 sponse  as  a  function  of  log(z)  were  z  is  a
 toxicant concentration.   Other   useful  linear
 expressions are  the  logit  transformation  (8),
 and the  Welbull transformation (9).
    The  probit, loglt,  and Weibull  models  must
 be  considered  mainly  empirical   although  some

-------
                       TABLE 1.  Populations of Organisms  Considered In Bloassays
         6enet1c
     Characterization
                                                TYPE OF RESPONSE
                            Quanta! Response
                                               Continuous Response
 Tolerance Distribution
 for Individual Organisms
 All Organisms from a Single
 Clone
 (No tolerance distribution
 for individual organisms)
                      Broup I

                      Nacroorganlsms
                      Response: death of an organism
                      Classic problt analysis
                      Binomial  statistics

                      Group III

                      Pure  culture of microorganisms
                      Special case:  Synchronous
                      growth
                      Response: growth  rate based on
                      cell  number
                                         Srouo II

                                         Nacroorganlsms
                                         Nixed cultures of microorganisms
                                         Response:  growth rate, C-14
                                         uptake, respiration

                                         Group IV

                                         Pure culture of microorganisms
                                         General case
                                         Response: growth rate, C-14
                                         uptake, respiration
TABLE 2.  Comparison of the Weibull  Transforma-
          tion with the Probit  and  Loglt  Trans-
          formations
 Type
Transformation*
  Probability of
   Response or
Relative Inhibition
Weibull   u * In k + n In z    P»l- exp(-  e  )


Probit    Y « a + B log z      P « kl+erf(^))


Logit     I = 6 + $ In z       P » 1/(1  -t- e'1)


     *z is a toxicant concentration
      k, ti,.«, B, 6, $ are constants; A « In  k
theoretical basis has  been  claimed.   The  problt
model  1s based  on  the  often  found  log-normal
distribution  In  biological  systems.  The  legit
model  is  valid  for  certain  types of autocataly-
sls  and  enzyme kinetics  (Group III and  IV  or-
ganisms)  (10,  11).   The  parameter  *  1s  the
number of toxicant  molecules per  receptor.   It
appears that  the Weibull  model  may have a simi-
lar  Interpretation  so  that  n  would  be  the
number of toxicant molecules  reacting  per  re-
ceptor  molecule  (12,  13).   In  addition,  the
Weibull  model  1s  related  to  the  multistage
model  in  carcinogenesis  and  1s  Identical  to  the
single-hit model for n - 1 (14).

Applications
   F1sh.  To  Illustrate  differences  between  the
problt  and Weibull  models,  we shall  consider
the  experimental  results of Broderlus and Kahl
(15)   on   the  mortality  of   fathead  minnows
(Plmephales promelas)  in the  presence  of each
of  27 different  organic chemicals.   A plot  of
the  results obtained by  these  authors  is  shown
1n  F1g.  1, where  the toxldties have  been  nor-
malized   (M   96h   LCSO)  to   the   potency   of
1-octanol.   The normalized experimental  results
and  the  problt line  (o   = 59.1, B  -  13.5)  are
as  reported by Broderius  and Kahl.   In  addition,
we  have   included  a  Weibull  curve   (A  =  53.16,
n «  5.81)  fitted to the  experimental  points.
                                                             10r
                                                   8 -
                            £
                            CO
                            o
                            cc
                                              ^   6 -
                                              <
                                              cc
                                              o
                                                                             PROBIT
                                                     	WEIBULL
                                                   4 -
                                                                       -3
                                                               -2
                                              -1
                                                         TOXICANT CONCENTRATION (Log M)
                                             F1g.  1.   Mortality  vs.  toxicant   concentration
                                                      for  27  different  organic   chemicals.
                                                      The  mortalities  are  normalized  (96h
                                                      LC50)  to  that  of  1-octanol.   Experi-
                                                      mental  points  and  the  problt line  are
                                                      from  Broderlus   and   Kahl   (15).    In
                                                      addition,  we have  Included  a  Weibull
                                                      function  with  parameters  A =  53.16  and
                                                      n -  5.81  that  have  been  adjusted  to
                                                      fit the experimental data.

                                                Because  the  normalization   was  made  with
                                             respect  to  the LCSO-values only,  and not  the
                                             slopes,  1t 1s not entirely appropriate to  use a
                                             statistical   criterion  such  as  ch1-square   to
                                             compare  the goodness of fit of  the two  models.
                                                   67

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Howtvtr.  since  the  slopes  of  the  27 'dost-
rtsponse  curves were  fairly  similar,  and  the
Mortalities   were   fairly   evenly  distributed
between  0 and  100%.  a comparison may still be
wild.   From Fig.  1  It  1s  seen  that the  test
«ata tMd to follow the curved Helbull function
ratlttr  than the straight problt line.  Similar
observations  on other  bloassay data  were  made
previously  (9).
   The  problt  and  Welbull  models give compar-
able response rates for problt values between 4
and  6,  but  highly  diverging values  at the ex-
tremes.   The mortalities  from the Welbull func-
tion  Is  the  highest  In  both  ends.   This 1s
Important 1n the case  where response  functions
obtained  by  fitting  to Intermediate  test  mor-
talities  (e.g., between  10  and  90S)  are   used
for extrapolation to high or low concentrations.
   As may be  seen from Table 3,  the difference
between  the mortalities from  the  two models 1s
rather  trivial  for  values of log H between  -4.1
and  -3.9.  However, at log  H = -3.8,  the probit
model  predicts  that  255  out   of  105  organisms
will survive, while the corresponding  number for
the  Weibull function is  only  5.  Similarly, at
log  M  * -4.3, the  probit model Implies that al-
most no organisms are affected  (only 4), whereas
a  total  of  1230   are  killed  according  to the
Weibull  model.

TABLE  3.  Number of  Fish Killed  (fathead minnows,
          Pimeohales promelas)  out  of  an Initial
          Population of10s  as Predicted   from
          the Probit  and  Weibull  Models.   The
          parameters of  the  problt  model  (a  =
          59.1;  8 «  13.5) are from Broderius and
          Kahl   (15)  *  and those of  the Welbull
          model   (A  -   53.16;   n   -   5.81)   have
          been adjusted to  fit the experimental
          data of these  authors.
Toxicant
Concentration
(log M)
-3.6
-3.7
-3.8*
-3.9
-4.0*
-4.1
-4.2
-4.3*
-4.4
-4.5
-4.6
-4.7
-4.8
-4.9
-5.0
Model
Probit
100.000
99,998
99,745
92,645
53,983
10,565
466
4
0
0
0 -
0
0
0
0

Weibull
100.000
100,000
99,995
92,681
49,534
16,463
4.608
1.230
324
85
22
6
2
0
0
 Corresponding to broken vertical  lines  1n  F1g.  1
    Algae.   The  models for  one toxic  substance
 listed  In  Table  2  have  been applied  to  the
growth  of  the diatom Mavlcula  Inqerta exposed
to Cd,  Cu   Pb, and  Zn.   The raw  data are from
RachUn, Jensen, and Warkentine (16).
   The  results for  Mavlcula Incerta  are given
1n Tables 4 and F1g. 2.  From Table 4  It 1s seen
that tlit  Welbull  model  provides the better fit
compared to the problt and loglt models when the
number  of  degrees of  freedom art  two or more.
The  slope  *  appears  to assume  the  value  0.5
for Cu,  Pb. and  Zn when growth  rate  1s used as
a  parameter.   The  Interpretation  of  n  may be
the  number  of .toxicant  molecules  per receptor
of  the organisms,  and  the  Implication  1n  the
present  case   1s,  therefore, that each  of  the
metals  Cu,  Pb, and  Zn combines  with  two recep-
tors.
    A NONINTERACTIVE MULTIPLE TOXICITY MODEL

   We have  expanded  Hewlett and Plackett's P7)
bivariate normal model to include any mono-
tone   tolerance  distribution   for  individual
toxicants,  such  as  a logit or Weibull distribu-
tion, and  n toxicants (12).   Let  us consider a
general blvariate model.  Besides the parameters
characterizing   the   Individual   dose-response
curves (Table 2), there are two additional para-
meters:  a   similarity  parameter x  and  a corre-
lation  p   of   mortality  tolerances  (Group  I
organisms,  e.g.,  fish) or  cell  division toler-
ances (Group III or  IV organisms, e.g., algae).
   The   similarity    parameter   x   indicates
whether  the toxicants act  on similar  (x « 1),
different  (X =  0),  or  partially  similar  bio-
logical  systems  (0  <  X  <  1).   The other para-
meter  p is  a  measure of  the degree  of corre-
lation  of   the  susceptibility of  the organisms
(Group  I)  to the two  toxicants.   For full  cor-
relation  (p  »  1),  organisms  that  are  very
susceptible  to  one  toxicant  are  also very sus-
ceptible  to the other.   In the  case  of  full
negative,  correlation  (p  -  -1),   there is  an
Inverse  relationship  between  the  susceptibi-
lities, e.g., organisms  that are very suscepti-
ble  to  one  toxicant  are least  affected  by the
other.   Zero correlation   (p =  0)  means  that
there  is  no relationship  between  the suscepti-
bilities of the organisms  to the two toxicants,
and  all  other  values (-1  <  p <  1)  represent
intermediate cases.   For microorganisms,  it  is
hypothesized that  p  should be one  because  all
organisms are from the same clone and are in the
same  (Group III) or nearly the  same (Group IV)
physiological state.
   The  case of (x -  1;  p  - 1)   1s  charac-
terized  by the  term  concentration  addition
(C.A.),  and the case  of  (X -  0; p  «= 0)  by
the  term response  multiplication  (R.M.).   Com-
puter  programs   to   estimate  the  parameters  of
the  unlvariate  distributions  in   Table  2  are
available   (18).   Also,  the  general  noninter-
active  multiple  toxicity model  has  been formu-
lated  into  a computer program  MULTOX which may
be obtained from the same source (19).
Applications
   F1sh.   We  shall   here  analyze  the  results
obtained  by  Anderson  and  Weber  (20).   They
                                                    68

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TABLE 4.  Fit of  the Helbull, Problt.  and loo.1t Distribution  to Growth  tat* for  the  Diatom  Navlcula
          Incerta.  The  Raw Data are from Rathlln.  Jensen,  and Harkentlne (16).  Concentrations  are 1n
ttolbull
Para-
meter
Growth
Rate



Yield



Toxicant df*
Cd 1
Cu 2
Pb 3
Zn 6
Cd 1
Cu 2
Pb 2
Zn 6
A «
-2.13 0.895
±0.051
-2.59 0.561
±0.079
-2.26 0.561
±0.092
-1.75 0.431
±0.104
-1.'31 0.797
±0.088
-1.72 0.554
±0.084
-2.07 0.650
±0.098
-2.59 0.958
±0.109
X2
0.00031
0.0019
0.0071
0.0345
0.0021
0.0062
0.0031
0.0197
Problt
• B
3.71 1.53
±0.03
3.49 0.831
±0.127
3.74 0.807
±0.154
4.01 0.665
±0.166
4.21 1.66
±0.003
3.94 1.03
±0.16
3.83 1.01
±0.15
3.32 1.69
±0.21
X2
0.000044
0.0026
0.0104
0.0366
8.2xlO-7
0.0084
0.0040
0.027
Loatt
• *
-2.18 1.12
±0.004
-2.61 0.641
±0.093
-2.19 0.625
±0.116
-1.66 0.495
±0.123
-1.31 1.18
±0.01
-1.77 0.745
±0.114
-1.99 0.755
±0.120
-2.91 1.27
±0.15
X2
3.1X10-8
0.0021
0.0089
0.0356
0.000020
0.0075
0.0038
0.0206
 •Degrees of  Freedom
                  -3-2-10123456

                       LOG CONCENTRATION (ZINC, mg/l)
     -3-2-10    1    234    56
          LOG CONCENTRATION (ZINC, mg/l)
 F1g. 2.   Fit of the Welbull,  probH and  loglt models to growth parameters for Navlcula Incerta:
           (a) relative growth  rate,  and  (b) relative yield,  both based on  cell  number.   The raw  data
           are from RachHn,  Jensen,  and Warkentlne (16).
 considered  only  R.M.  and  C.A.  with  parallel
 dose-response curves.  I.e.,  Identical  B-values
 (Table 2), while we  shall allow any  correlation
 p   between   -1   and  +1,   partially   similar
 systems,  and   C.A.   with  non-parallel  dose-
 response  curves.   Also,  1n contrast  to their
 approach, we  Include not only  problt but also
 loglt and Welbull  transformations.
   Basic  probit  lines  for  the  action  of  the
Individual  toxicants  nickel  (N1),  copper (Cu).
potassium   pentachlorophenate   (POP),  dleldrln
(HEOD), and  potassium cyanide  (CN)  on male gup-
pies (Poec111a retlculata) are given 1n Table 5.
In this and  the following tables, the weight of
each lot of  fish  Is the  total  weight of the ten
fish  1n a  batch.  The  weight  W  modifies  the
                                                    69

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                                                                                                   fie.
concentration  H  of  a  toxicant  such  that  the
•affective  concentration*  M/Vf1  (h  -  0.67  -
0.81) remains  the same  either for a high actual
concentration  ftnd. high  average weight  of  Hsh
or a  low actual  concentration and a low average
weight  of fish.   In ether words, the  Important
quantity  Is concentration per  weight raised to
the  power  h  and  not just  concentration.   Or:
larger  fish can  tolerate  higher concentrations
for the same mortality rate.

TABLE 5.  Problt  of Mortality  to  Male Supples
          (Poecilia   reticulatal   for  Several
          Toxicants.  M(rag/l) 1s the Concentra-
          tion  of the Toxicant  and M(g) 1s the
          Weight   of  Each   Lot   of  Hsh.   The
          relationships  are  from  Anderson  and
          Weber  (20)
                                     6.5 - -  3.21 *  6.32 log
                               or
                                              34.36
                                   In   the  regression  of  the  linear  Melbull
                               transformation u  •  In  k  + *ln  z,  z  -  z-\  Is
                               then   34.36  and   «  •  u-i  Is  given  by  ui  *
                               ln(-ln(l-P)) - ln(-ln(0.067)) -  0.994.   Similar
                               points are  obtained  for  N.E.D.  *  1.0,  0.5,  0,
                               -0.5,  -1.0,  -1.5.   The  Intercept  A and  slope
                               n  are  then given  by:
  Toxicant
Probit of Mortality
N1
Cu
PCP

HEOO
CN
Y
Y
Y

Y
Y
- -3
- n
- n

= 20
- 14
.21
.4
.77

.33
.71
+ 6
+ 7
+ n

+ 6
+ 11
.32
.46
.23

.84
.65
log(M/W°-67)
log(M/W°'72)
log(H/W°'72)
n en
log(M/W°-B')
1og(H/W°-?2)
   n - (l/DHdw^dw^X^ - (I


where   0 » (Iw^Xlw.X2) - (Zw^

        1-1	N;  N - 7

       Xj - In Z1


For N we obtain A = -9.4 and n •
                                                                                          2.99  (Table 6).
                                                        Table  6.   Logit  ana  Weibull  Parameters  Corre-
                                                                   sponding  to  the  Probit  Relationships
                                                                   of  Table  5
    Logit  and  Weibull  parameters  corresponding
 to the  problt parameters  in  Table  5  should  pre-
 ferably be derived from the original  test data.
 However,  since  they  were  not  available,   we
 determined approximate  parameters  by a  fitting
 process, using the weighting:

                 °1         2
        w  = "       (1n Q}     Weibull
                                 logit
 where
 Qi  » the  survival fractions corresponding to
      M.E.O. values of -1.5, -1, -0.5, 1, 1.5

 YI  - probit of QI

 n<  - number of test organisms in trial 1 (10)
      1  -  1. 2	7

     The  logit and  Weibull  parameters  (Table 6)
 were  then calculated  using regressions based on
 the linear transformations  in  Table  2  and the
 weighting indicated  above.  As  an  example, the
 value   of M/W°-67 (Table  5)  is calculated  in
 the following manner, considering  Mi  at N.E.O.
 - 1.5:
                                Toxicant
                 Logit
                Parameters
 Weibull
Parameters
N1
Cu
PCP
HEOD
CN
-13.9
10.9
11.5
26.9
16.5
4.66
5.50
8.28
5.04
8.59
-9.4
6.5
6.9
16.8
10.1
2.99
3.53
5.31
3.24
5.51
                                   The  bivariate  fitting  was  carried  out  as
                                indicated  previously  (12),  except  that  we here
                                use  minimum ch1-square as  the  criterion  rather
                                than  maximum  likelihood.   However,  because  of
                                the  indirect  determination  of  the  logit  and
                                Weibull  parameters,  it  was  estimated  that  a
                                larger  stepsize.  I.e., 0.1,  was  sufficient for
                                both X and  p  in  search of the  global  minimum
                                for  x* which  is  calculated  according  to the
                                formula:
                                                   (Orq.)
                                where qj • experimental survival fractions,
                                           e.g.,  70X  in the first  case  and 20X
                                           in the second (Table 7).
                                      Q^ « calculated survival fractions.
                                                     70

-------
      iu • number of ttst organism  1n  trial
           1 (10).
       N - number of trials (6).

   Ut systematically  calculate x2  for  several
combinations -of x and ».  The pair producing  the
global minimum of if Is retained.
   TIM  results  for  the binary  mixtures  (N1,
Cu).  (rtP,  HEOO), and  (KP.  CM)  are  listed  In
Tables  (7-9)  and  summarized  In Table 10.   There
1s  little difference  between the  fits of  the
problt  and  loglt models,  both In terms of  the
optimum values of x and p and the  resulting  x2-
However,  the Welbull model shows some distinc-
tive differences.  It produces the best fits  for
the  (HI.  Cu)  and (PCP,  HEOO)  pairs.   For  the
(PCP,  CN)  pair the  problt or loglt models pro-
duce  minimum  x2  but  this would appear  to  be
less  Important  because  none  of  the   fits  are
particularly  good  in that case (P < 0.01)    The
X   values   are  the  same  and   the   p  values
nearly  so   for  a   given   binary  mixture  and
                                  different  models (Table  10).  The  reason  that
                                  the similarity  parameter  X  and  the  correla-
                                  tion  p  between  the  two  tolerances  are  rela-
                                  tively  Insensitive  to the  form of  the  mathema-
                                  tical  model  here  1s  that  there are only  ten
                                  fish  1n each experimental  batch  of the example
                                  (20).   Thus, the models  are essentially fitted
                                  to response  probabilities  between  10 and  90X.
                                  •nd 1n this range there Is not a great deal  of
                                  difference  between  the  fits  of  the  problt.
                                  loglt,  and  Uelbull models.  However, as Illus-
                                  trated  by Christensen and  Chen (12).  the situa-
                                  tion  1s  different when  high  or  low  response
                                  probabilities are Included.   In  that case,  not
                                  only  will   the  estimates  of  x  and  p  depend
                                  upon  the choke of model,  but the  problt model
                                  may not  fit at all.  The  advantage  of  using
                                  non-normal   blvariate  tolerance  models  will,
                                  therefore,  be particularly evident  when extreme
                                  response  probabilities are  encountered  as  for
                                  example  in models for  carcinogenesis.
 TABLE  7.   Evaluation of the Joint Action of Ni and Cu on Male Guppies Based on the Parameters
           of  Tables 5, 6 and the Computer Program HULTOX
Weight of
Each Lot
of Fish
(9)
1.53
1.07
1.27
1.30
1.23
' 1.23
Concentrations
(mq/1)
N1
12.23
15.56
14.17
10.77
15.15
H.79
Cu
0.049
0.082
0.084
0.063
0.071
0.058
Calculated Percent Mortality
for M1n. Chi-Sauare
Problt
(X-l; p-0.5)
16
83
66
28
64
53
Logit
(X-l; p-0.5)
16
83
67
28
65
53
Weibull
(X-l; p-0)
19
68
68
29
66
53
Observed
Percent
Mortality
(20)
30
— 80
70
30
80
80
 TABLE 8.  Evaluation of  the Joint Action of PCP and HEOO on Hale Guppies Based on the Parameters
           of Tables 5,6  and the Computer Program HULTOX
Weight of
Each Lot
of F1sh
(9)
Concentrations
(mq/1)
PCP HEOO
Calculated Percent Mortality
for M1n. Ch1-Sauare

Problt Loglt Welbull
(X-0.1; p~ 0.1) (X-0.1; p-0) (X-0.1; p— 0.2)
Observed
Percent
Mortality
(20)
      1.51

      2.15

      1.76

      1.79

      1.51

      1.94
0.26

0.40

0.31

0.40

0.29

0.41
0.005

0.00645

0.0063

0.0063

0.0048

0.0069
30

49

45

75

39

72
28

47

42

74

37

70
34

49

46

73

42

69
10

40

50

60

70

80
                                                    71

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TABU *.  Evaluation of  the  Joint Action of  PCP  and CN  on Male Supples Based on  the
          Parameters of  Tables 5.6  and the  Commiter  Program NULTOX.   The Height  of
          Each Lot of F1sh Has Been Set to l.SOg
   Concentrations
      PCP
0.233
0.231
0.257
0.257
0.246
0.177
0.135
0.146
0.175
0.169
0.153
0.139

Problt
(X-0.2; p— 0
8.5
13
54
48
22
4.6
Calculated Percent Mortality
for Mln. Chl-Souare

Loglt Weibull
1.8) (X-0.2; P— 0.8) (X-0.1; p— 0.8)
9.8
14
53
47
23
5.6
17
22
53
49
30
11
Observed
Percent
Mortality
(20)
10
10
40
100
10
10
 TABLE  10.
Chi-Square for  Binary  Mixtures of  Toxicants  Considered  1n  Tables  7-9  (four  degrees  of
freedom).
Toxicants
N1-CU
PCP-HEOD
PCP-CN
Model
Problt
5.79
(X-1,
7.83
(x-o.
13.2
(X=0.
P-0.5)
i; P— o.i)
2; p--0.8)
Loglt
5.72
(X-1;
8.12
(X-O.
13.2
(X-O.
P-0.5)
1; P-O
2; p-0.7)

Welbull
5.38
(X-1; p-0)
7.72
(X-0.1; p— 0
14.2
(X-0.2: p— 0



.2)
•8)
     The  estimation of  parameters  when  three  or
  more  toxicants are considered, using  the above
  method  with   x2  as   criterion,  is  very  cum-
  bersome  and  we  have  not attempted  it.   Other
  means  of  estimating  parameters  are   currently
  being  explored.
      Isobolograms  for  the  three  binary mixtures
  of  Table  10, based  on  the Weibull  model,  are
  shown  in  Figure  3.   The curves  are  drawn  for
  three  values of the  non-response probability Q:
  0.1.  0.5, and 0.9.   The  symbols  M  and W of the
  modified'concentration are  defined in Table 5,
  and h (0.67-00.81)  is the exponent  of  the weight
   of  each   lot  of  fish.    CT   and   C2 are  the
   values of  H/W**h  for each  toxicant   that will
   give the desired response when acting separate-
   ly    It  is  clear  that  the  isoboles for Cu and
   Ml (Figure 3a) are close to defining a straight-
   line  relationship  characteristic  of  C.A.   This
   might  be  expected  since X -  1;  and  although
   p  «  0   the variation  of  the  response for  p
   between  0 and 1  is  modest  (12).   The Isoboles
   for  HEOD and PCP  (Figure 3b) are  typical  when
   „  >  i  for  R.M.  which 1s   Indicated  by  the
   values  of   X  and  p  that   are  both  close  to
   zero    Except  for  an   Interchange of Indices,
   these curves are  1n fact similar  to the  curve
                                            labelled  1  1n Figure 3a  of  ref.  (12), which  is
                                            strictly  valid for R.M.
                                               Algae.    The   use   of  the  above   multiple
                                            toxicity  model  for  algal growth  rate based  on
                                            cell  number was  considered   previously (21,  22).
                                            The growth  rate of the  green alga  Selenastrum
                                            caprlcornuture and  the  blue-green  Synechococcus
                                            leopoliensis  was modeled as a  function of ionic
                                            concentrations of  Ni and  Zn.
                                               We consider  here  the growth  rate  of  Selen-
                                            astrum based  on  cell  volume.   The  experiment
                                            was  designed  such that  for  each point,  equi-
                                            toxlc concentrations  of  NI2* and  Zn2* would be
                                            combined.    The  culturing   methods  were   as
                                            described  by  the  U.S. Environmental  Protection
                                            Agency (23), and  Ionic  concentrations were cal-
                                            culated  by the  equilibrium  speciation  program
                                            MINEQL (24).  The  results  of  such an experiment
                                            are  shown  In Fig. 4.   Just as  for  growth rate
                                            based on cell number  (21),  the joint action ap-
                                            pears  to  be  close to  C.A.  (p  =1;  X  = 0.9).
                                            However,  here  the  best model  is  logit rather
                                            than Weibull.

                                                               CONCLUSIONS

                                                 The following conclusions may be  drawn from
                                            the  present  study:
                                                       72

-------
                                                                             (c)
                       MOEX  0
                       0
                           U

                           M
 C«l
 c,

O.MZ

0.144

•.004
                                      M
                                      c,
                                    ji.0
                                     11.1
                                             MOCX  a
                                                      HfOO
                                                       c,
                      •OCX  0
01
c,
rv
c,
0  0.1   0.0073   0.320
(D  O.S   1.00504  OJSS

(})  M
                                                                     0
                   e^C,
                    1.0
                                        MODIFIED CONCENTRATION  M(mg/l)/W(g)«*n/C,
Fig. 3.  Isobolograms  for the  effect of  (a)  (N1.  Cu).  (b)  (PCP,  HEOO),  and  (c)  (PCP,  CN)  on male
   •     gupples based on the  Weibull model  with optimum values of x and  p (Table 10).
                                   Wood 'op

                                     S,'2 .=26300 09506 m 2

                                     In'' '=2533* •• i576inZ
                                  1         0
                               in (Total ion Cancvnoon) f* *' & Zfl *
   2T3
          23
                 186
                                059 034
                               1 (Tom Concenmtfi) NI 4 in
 Fig.  4.
         Combined  effect   of  N12+  and  Zn2+
         on  the  growth   rate  based  on  cell
         volume  (biomass)  of  the  green  alga
         Selenastrum  capricornutum.   The  test
         results are best  fitted by a bivariate
         loglt model with X » 0.9 and p » 1.

(1)  The Weibull model  should  be given serious
     consideration  as  a  replacement  for   the
     probit  model   as  a  general  dose-response
     function for the quantal  response of macro-
     organisms  with  a  tolerance   distribution
     (Group  I  organisms).   The main  reason  is
     that  the  Weibull model  appears  to  give  a
     better  fit to experimental  data, and that
     1t,  therefore,  1s  more  likely to provide
     valid mortality estimates  by extrapolation,
     particularly  to  low  concentrations.    The
     better  fit of  this model  supports  a pre-
     viously suggested mechanistic-probabilistic
     basis  in  terms  of  chemical  reactions  be-
     tween toxicant molecules  and a  key  receptor
     of the organism.
(2)  Literature data  for the  growth  rate  and
     yield  of  the  diatom Navicula  incerta  Inhi-
     bited  by Cd,  Cu, Pb,  or Zn were fitted to
     the  univariate Weibull,  probit, and  loglt
               models.     The    Weibull   model   provides
               generally  the  best fit,  thus  supporting a
               basis  which  was  previously developed  for
               microorganisms  (Group III and  IV organisms)
               when  the growth  rate based  on cell  number
               was modeled as a  function of  toxicant con-
               centration.
          (3)  A  general  noninteractive  multiple toxicity
               model  was  applied  to  literature  data  for
               the toxlcity of  binary  mixtures of N1,  Cu,
               PCP,  HEOO,  and  CN to male guppies  (Poecilia
               reticulata).    We  confirm that  the  action
             ' of  (N1, Cu)  and (PCP,  HEOD)  Indeed  may  be
               approximately  characterized   by   C.A.   and
               R.M.,  respectively.   The estimates   of  the
               similarity  parameter  x  and  the   correla-
               tion  coefficient p  are  relatively  insen-
               sitive  to the  choice of model here because
               the  response   probabilities  mainly  are  in
               the  range  between 10 and 90X, and in this
               range there  is not much  difference between
               the   fits   provided  by  the three models.
               Nevertheless,  in  both of the  above  cases,
               the Weibull model gives minimum  chi-square.
          (4)  The   combined  effects of  N12+ and Zn2* on
               the  growth rate  based on cell volume (bio-
               mass)  of the  green alga  Selenastrum  capri-
               cornutum  were  approximately  according  to
               C.A..  with  X  -  0.9  and  p  -  1.   While
               previous  bioassays,  In  which  the  growth
                rate  was based on cell number, demonstrated
               that  the Weibull  model  was preferable,  the
               present  results,  based  on   cell volume.
                indicate  that  the  loglt  model   is best
                suited to describe the combined  response.

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                                                     73

-------
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     Biometrics ]£: 591-610.

(18) Christensen,  E.R.,  Chen,  C.-Y.,  and Fox,
     D.F.  (1986).  Computer programs  for fit-
     ting dose-response  curves  1n aquatic toxi-
     city   testing,   AEEP's  software  library,
     Dept.  of  Civil  Engineering,  Michigan Tech-
     nological University. Houghton, Michigan.

(19) Chen.  C.-Y.  and Christensen,  E.R.  (1986).
     MULTQX  a  computer  program for the calcula-
     tion of the  response  of  organisms  to mul-
     tiple   toxicants  and   multiple  limiting
     nutrients,  AEEP's  software  library,  Oept.
     of  Civil  Engineering,  Michigan Technologi-
     cal University,  Houghton, Michigan.

(20) Anderson.  P.O.  and  Weber.  L.J.  (1975).
     The  toxicity  to  aquatic  populations  of
     mixtures  containing  certain  heavy metals.
     In:  Proc. Int.  Conf.  Heavy Metals  in  the
     Environment,  pp.   933-953,   Institute  of
     Environmental    Studies,    University   of
     Toronto,  Toronto, Canada.

(21) Christensen,   E.R.,   Chen.   C.-Y.,    and
     Kannall,  J.  (1984).   The response of aqua-
     tic  organisms  to  mixtures  of  toxicants.
     lAWPRC's     12th    Biennial    Conference,
     Amsterdam.  Holland.  Water  Sci.  Tech.   1_7.
     1445-1446.

(22) Christensen.  E.R., Chen.  C.-Y.,end Fisher,
     N.S.  (1985).  Algal growth  under multiple
     toxicant  limiting,  conditions.  In:  Proc.
     Int. Conf.  Heavy Metals 1n the Environment
     held  1n  Athens, Greece,  Vol.  2,  pp. 327-
     329.   CEP  Consultants,  Edinburgh,   United
     Kingdom.

(23) U.S.    Environmental    Protection   Agency
     (1978).    The   Selenastrum   capricornutum
     Printz  Algal  Assay  Bottle Test. Report  No.
     EPA-600/9-78-018, Corvallis. Oregon.

(24) Westall.  J.C..   Zachary.  J.L..  and   Morel,
     F.H.M.  (1976).   MINEQL a  computer program
     for  the  calculation   of  chemical  equili-
     brium   composition   of   aqueous   systems.
     Technical  Note  No.  18,  Oept.  of  Civil
     Engineering,  Massachusetts   Institute  of
     Technology,  Cambridge, Massachusetts.
Acknowledgments.   This  work  was   sponsored  by
the  U.S.  National Science  Foundation  Grant No.
CEE8103650.   The  assistance  of  C.-Y.  Chen  in
carrying  out   the  numerical   calculations  is
gratefully acknowledged.
                                                   1A

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                                            DISCUSSION
                    Richard C. Hertzberg, U.S.  Environmental  Protection Agency
  Chrlstensen presents examples of the
application of a noninteraction multiple
toxicant Mdel to several data sets. Including
Mortality of fathead minnows and population
growth rate and yield of diatoms.  He
concludes that the Mixture Helbull Is a
preferred model for similar acting chemical
pairs and Infers the type of noninteraction
between the mixture components by the values
of the model's parameter estimates.  There are
three aspects of this work that should receive
critical attention:  the usefulness of the
mixture models as descriptors"of mixture
tox1c1ty, the biological Interpretation of the
He1 bull and Us parameters, and the future of
modeling binary mixtures.
  The feeling one gets 1n reading the paper Is
that  the results of the mixture models are
tantalizing yet Incomplete.  To his credit,
ChrUtensen's work does Include many desirable
characteristics:  multiple dose levels,
different types of toxicants, two very
different species, and well-defined biological
end  points.   But several Hems are missing:
the  models are not presented, the dose
adjustment model (divide dose by a power of
body weight)  has no statistics on Us
parameters that might suggest the validity of
such an adjustment, and the descriptions of
the  model fits do not Include significance
levels  or even graphs.  The latter 1s
Important since the Information that Is
provided  (ch1-square values)  shows only a
marginally better fH for the Welbull. which
1s  an Inadequate criterion for model
preference.
  Of more concern, perhaps, 1s the motivation
for  the models.  ChrUtensen  states that the
models  are to be considered empirical, yet he
then Infers,  biological meaning to  the value of
the Welbull  parameters.  The  biological
properties should have been established first
 (e.g.,  Cu and N1 are  tox1colog1cally similar)
and then  shown  to be consistent with the
model's  results  (e.g., lambda=l).  Two
 similarly acting toxicants are often
 characterized as being dilutions  or
 concentrations of one another so  that, once
 adjusted  for  potency  differences,  the  two
 chemicals  should have the same dose-response
 curves.   Because of  this, 1s  seems  that two
 similar chemicals  (lambda*!)  should also have
 the same  tolerance distributions  (rho-1).  The
 Inclusion  of this constraint, and  verification
 by actual  data, would Improve the  support  for
 ChMstensen's approach.  Without  such  support,
 Inferences  about toxic similarity  from
 parameter  values are  not believable.
   The use  of mortality as the toxldty
 Indicator  raises several  Issues.   First,
 mortality  1s usually  Interpreted  as a
 non-specific toxic end point, and  thus  1t
 provides  little  Information  on  toxic
 mechanism.   Consequently, the Inference about
 toxic similarity  Is  confusing.   The usual
 definition of toxic  similarity  (EPA, 1986)  1s
that the same  tissues  and  organs are affected,
and that the same  type of  damage or lesion
results.  In contrast, mortality usually
results from failure of several organs and  the
txact caust of death Is rartly Identified.
Second. Mortality  Is useful primarily for
assessment of  ecosystems.  Presence/absence
and population size of Indicator organisms
have been used successfully for years to
evaluate water quality of  lakes and streams.
But mortality  1s not particularly helpful for
human risk assessment.  Particularly for
systemic toxicants (chemicals with a toxic
threshold), the preferred  data would Include
doses showing  several  degrees of sublethal
effects along  with doses showing no effects.
   The problems with developing a general
mixture assessment methodology are only
touched on 1n  Chrlstensen's discussion.  These
Include having more  than two components 1n  the
mixture, multiple  end  points and varying
degrees of severity  for each end point.  The
extension of binary models, as has been done
for multistage cancer  models (Thorslund and
Charnley, 1986), 1s one approach for
evaluating several components, particularly
for single end points.  But the general
n-chem1cal model can become Intractable, as
Chrlstensen mentions,  even for one end point.
The extension  to multiple  end points by
traditional methods  seems  out of the question.
   One useful  approach we  are Investigating 1s
to combine expert  Judgment w1tti generalized
linear models.  We have adapted the work of
McCullagh (McCullagh and Nelder, 1983) to give
a multl-chemical model which uses Judgments of
the overall severity of the toxic reponse 1n
Ueu of response rates or  numerical Intensity
measures of specific effects.  In this way,
data describing several end points, even
purely qualitative descriptions, can be
modeled to give estimates  of an "acceptable"
dose or of a dose  corresponding to a low
risk.  Consider the  following data for
dieldrln-lnduced nephritis (FUzhugh et al.,
1964):
                     LESIONS
  DOSE   —		
  (ppm)   None      Slight  Moderate  Severe
 0,0
 0.5
   2
   10
   SO
 100
 150
  The multiple  response  curves  plotted against
dose (given  1n   Fig.  1)  are  not easily
Interpreted  In  terms  of  overall risk.  The
cumulative response  (F1g.  2)  separates the
severity groups  and  allows an estimation of
the probability  of  seeing  a  given  severity or
less for any given  dose.   The statistical
approach we  are  developing 1s similar.  The
                                                   75

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steps are as follows:

   1. The main covarlables (dose, duration.
      species, route) are represented by
      categories  (Intervals for the continuous
      variables).

   2. Tbt response  Is coded 1n term of a
      lexicologist's Judgment of overall
      severity to the animal.  This code will
      be from a  predefined set of categories.

   3. Apply HcCullagh's approach for ordered
      categorical data:

      a. Identify a link  function to transform
          the original response variable  Into
          one that 1s  linear  In the
          covarlables.  We  are  Investigating
          the log cumulative  odds:
          where:
               k-1
                  Pij
          For a single covarlate,  say  dose.
          then J Indexes severity,  1  Indexes
          dose, and q^j 1s the log  odds  of
          the severity being 1n category J  or
          less, given a dose In category 1.
          Here p^ 1s the fraction  of
          responses of severity k  at  dose 1.

       b. Regress q on the covaMates:
          q  * Ax t b

       c. Calculate the risk of response from
          the link function.  For  a dose d,  and
          severity s:

          rds= Pr [response at level  s or
                       less, given  dose d]

               =  exp(Ad+b)/(l * exp(Ad+b))

     The  primary advantage of this method 1s
  that the data constraints are minimal;
  virtually  any  type of  toxlclty data can be
•odeled to give doses that art 'acceptable* or
that correspond to low risk.  In addition.this
approach yields maximum likelihood  estimates.
The disadvantages are that Uttle  Indication
1s given of the mechanisms of toxldty.  and
that the dose-response relation Is  limited by
the precision of the dose and response
categories.  What remains to b* checked  Is the
numerical performance of this method,  and the
ease of determining a suitable link function.
Note that this approach also works  with
complex (say. for n > 20 chemicals) mixtures.
If the mixture Is relatively stable over time,
then 1t can be treated as a slnglechemlcal
entity and the severity judgment reflects the
Impact on the test animal of all effects from
all components.
   In summary, ChMstensen's work  appears most
applicable to ecosystem assessment  of  simple
mixtures.  The use of Welbull parameters to
Indicate  the nature of the  Interactions  Is
Intriguing, and should be pursued.  Including
validation by chemical pairs with  know-.
mechanisms of  toxic Interaction.   Further, one
must agree with his caution against the
habitual  preference for  the problt  model. For
human risk assessment, however,  1t  seems that
other approaches  such as  I  have  outlined will
be required, particularly those  which  place
fewer demands  on  the quality and  quantity of
the  data.

REFERENCES

FHzhugh, .0.,  A.  Nelson  and M.  Qualfe  (1964).
Chronic  oral  toxldty of  aldrln  and dleldrln
 1n  rats  and  dogs.   Fd. Cosmet.  Toxlcol.
2:551-562.

McCullagh, P.  and J. Nelder  (1983).
Generalized  linear  models.  Chapman and  Hall,
New  York. 261  pp.

Thorslund, T.  and G. Charnley  (1986).   Use of
 the  multistage model  to  predict  the
 carcinogenic  response associated with
 time-dependent exposure  to  multiple agents.
 These  Proceedings.

 U.S. EPA (1986).   Guidelines  for the health
 risk assessment of chemical mixtures.   Federal
 Register (In press).
                                                    76

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RESPONSE
                                DIELDRIN KIDNEY TOX1CITY
                          so
     100

DOSE (mg/kg)
                 FIGURE 1.  EXWPLE OF DIFFICULT INTERPRETATION OF STANDARD OVERLAID
                 DOSE-RESPONSE PLOTS OF MILTIPLE EFFECTS.  SOURCE: FlTZHJGH ET AL., 1S&4.
                                DIELDRIN  KIDNEY TOXICITY
                                                                                             200
CUMULATIVE RESPONSE
 100'
                          50
     too

DOSE (mg/kg)
150
                  FIGURE 2.  EXAMPLE OF PROVED INTERPRETATION DUE TO SEPARATION OF CURVES BY
                  USING OnJLATIVE RESPONSE. SOURCE: FlTZHUGH ET AL., 1964.
                                                 77
200

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                            MA
                                ««. !» «SESS«o at JoiHT «no« or
1. PnaODOCTION
   ladtvldaala are expo««u w  .—„	
toxic  chemicals  In  the environment.    The
aaaeaament  of  health  risks  fro*  the  exposure
becomea Increasingly important.  The construction,
of  mathematical models  for  predicting  joint
toxicity  by using only the Information about the
toxicity  of  individuals  is  difficult.    Dose-
addition  and  response addition frequently have
been mentioned for evaluating  the  joint effects
 of two toxicants, (Shelton and Weber, 1981; Reif,
 1984).   Two  chemicals  are  said    to  be  dose-
 additive  or  are  said  to have  "simple similar
 joint action" (Flaaey, 1971)  if  one  chemical acts
 exactly  as if it were  a dilution  of the  other.
 Response-addition  or   effect-addition  has been
 used  in  different contexts in the  literature;  the
 most  co=3on definition  for  response-addition  is
  that  cocbined effect  of  the mixture  is  equal  to
  the  sum  of  each  effect alone,  (Reif,  1984).
  Synergism  and  antagonism represent  a  deviation
  from  additivity  under the  null  model   of
  dose-additivity  or  response-additivity.
     Hamilton and  Hoel  (1980) distinguished  between
  two  purposes  for studying the  joint  actions of
  chemicals,  "those  studies  conducted  to  provide
  risk  estimates  from  the  joint  exposure  and those
  studies conducted to elucidate the mechanisms of
   Joint  toxicity."   In  this  paper,  we  propose a
  mathematical  model  for  presenting and  analyzing
   the  data  from mixture studies.  The dose-response
   function  is modeled  as a  function  of both  the
   proportions  of  chemicals in the mixture  and the
   total  concentration  of  the  chemicals.     A
   response-additivity  is  introduced  for assessing
   the Joint action of  chemicals.

   2. RESPONSE-ADDITIVE MODEL
      Let x.  be  the  proportion  of  chemical  C

    (1-1,2)  in  a mixture  with  total  concentration T.
    Then  t.»Tx.  represents  the  concentration of

    chemical C. in the  mixture.   Suppose  that F (t.)
    represents the  (dose)  response  function of
    chemical C. at a dose level t..   It  Is assumed
    that  the  response   of the  mixture,  R(t.,t-)   •>

    R(xltx2,T), can be  expressed  as
                       F2(T)
                                             (1)
The terms x^CT) and x^T) «ay  represent  the
-expected' responses produced by  administration

of  the  single  chemicals,  and  EU^.T)  then

represents  the   "excess"  of  the   response  over
   r  (i)
  1
                  Produced ^  the mixture-
       equation  (1),  the data  were
           total  concentrations  of  the mixture
                                                  Figure 1.   This model  was  first  Introduced  by
                                                  Scheffe (1958) for  atudying mixture experiments
                                                  with only one concentration.
                                                     The joint action of the  two chemicals  la  aald
                                                  to be 'response-additive' If EC^.x^.T) • 0,  for

                                                  all x^ Xj, and T with ^^"l-  That  1«, If the

                                                  joint action of two chemicals can be predicted by
                                                  response-additivity,  then  the  response  of  the
                                                  mixture  at the  (total)  concentration  T  can  be
                                                  represented  by  the weighted  average  of  the
                                                  responses  produced by the individual chemicals at
                                                  the  concentration  T  with  the  weights  for
                                                  individual   responses   being  equal   to   the
                                                  proportions  of the  cnemicals in the mixture.  For
                                                  a  fixed  concentration  T,  the  response-additive
                                                  model can be  expressed  as  a  linear function of
                                                   the proportion of a cnemical in the mixture.  An
                                                   example of a plot  of  response-additivity  is  shown
                                                   in  Figure 2.    For  a  fixed  concentration  the
                                                   response can be  represented by  a  straight line.
                                                      The response-additivity defined in  this  paper
                                                   is conceptually parallel  to  the  dose-additivlty.
                                                   The  joint action  of a  mixture  is  said  to  be
                                                    "dose-additive" or simple  similar  (Finney,  1971)
                                                    If
                                                       R(Xl,x2,T) - F^T+mx^) - F,(t^
                                                                                                 (2)
                                                    where m  represents  the relative potency  of the
                                                    second chemical  to  the first.   (A more general
                                                    form of dose-additivity allows m to  be  a function
                                                    of  T.)    A common  method to  present  the  dose-
                                                    additivity is to use  the  isobolographic analysis
                                                    which  shows  the various  combinations  of  dose
                                                    levels  of the  two  chemicals  which produce  the
                                                    same  level  of   response.    The isobologram  for
                                                    dose-additivity  can be represented by a   set  of
                                                    straight  lines.   An  example of a plot   of  the
                                                    isoboles  for dose-additivity is shown  in  Figure
                                                    3.   The  isobole  of  a  given response is a straight
                                                    line.
                                                        Without loss  of  generality,  assume  that F2(T)>

                                                    ?.(T),  i.e. m>l.   If  the  joint action of  C. and

                                                     C.  is dose-additive then
                                                        F2(T)
                                                                                                   (3)
                                                          That   is,   the  response   predicted  by  dose-
                                                          additivlty  is bounded by the  two  responses
                                                          produced  by  the  single  chemicals  of  the  same
                                                          total  concentration.  If the response function F.
                                                          is convex in  the (dose) interval (T,mT), then the
                                                          response  predicted  by  dose-additivity  is  less
                                                          than  that  predicted by  response-additivity.   On
                                                          the  other  hand, if  the response function F.  is

                                                          concave in  the interval   (T,mT),  then response
                                                          predicted  by dose-additivity is greater than that
                                                          predicted  by  response-additivity,  Figure  4.
                                                          Therefore,  a definition  for  "additive"  joint
                                                           action of  two chemicals can be
                                                   78

-------
   F2(T) > Kx^Xj.T) > F^T)                 (4)

for any *i»*2' •"* * ****** *i * *2 *  *"  *wo non~
additive  actions,  synergism  and antagonism, can
be  defined  by   using   equation   (4).    The
•ynmrslmtic (antagonistic) action occurs  If the
response  of  the mixture la  greater (leaa) than
the additive response, that la,

                           tj.T)  < Ft(T)]     (5)

This, definition agrees with that of Vendetti and
Goldin (196A) for studying the combination of two
drugs.

3. ASSESSMENT OF IKTERACTIVE  ACTIOH
   Suppose that the purpose of the experiment is
to  understand  the underlying  Joint  toxicity
(interaction) of  chemical  combinations.   Termi-
nologies  used for describing  the  joint  actions of
mixtures  are  interaction,  Independence,  syner-
gism, antagonism, and additivity. Unfortunately,
these  terms  mean  different  things  to different
authors (Kodell and Pounds, 1985).  Equations (4)
and   (5)   define  three  possible   models  for
characterizing the joint  action  of two  chemicals.
The  assumption  for the  response-additive model
defined   in  this   paper  is  that the sites  of
primary action of the two chemicals 'are the same;
this  type  of  action  is  called similar  joint
action  according  to  the  classification  of
Plackett  and Hewlett (1967).  The joint action of
two chemicals is simple  similar  or noninteractive
if  the presence of one  chemical does influence
the   action  of  the  other.    Dose-additivity
commonly  has  been  used  for  assessing  the
interactive  effects   between  two  drugs  in
pharmacology.   In this   section,  we  apply, the
response-additive  model  to  assess dose-additive
Joint action.
   Suppose  chemical  C_   is m times  more potent

than  C. at  dose T. To assess dose-additive, the
dose  measurement for chemical C.  is scaled as T'

» mT  so that both chemicals are  equipotent, i.e.,

F.(T')  •  F-(T').   At the  "concentration  T1" In
  i         z
the mixture,  the  response predicted  by response-
additivity, Equation (1)  is
R(x1,x2,T')
                                 ') - F^mT)
for any  "proportions" x^  and  x_.   The  response
predicted by dose-additivity, Equation (2) is
            M  - F1(x1T'-hnx2T') - F^mT)
That is, at the concentration T'  the response  of
the  mixture predicted  by dose-additivity and
response-additivity is constant regardless of the
proportions of  individual  chemicals  in the
mixture.
   A procedure  for testing dose-additivity can he
constructed.   Suppose  that doses mT of C.  and
                                                      dosaga T  of Cj  produce tha  same  level, p,  of
                                                      response,  e.g.,  SOZ  affect.    (This  can  ba

                                                      obtained by plotting the dose-response curves  of
                                                      each chemical.)  Tha concentrations mT of C.  and
                                                      T of C. will ba uaad aa tha standard preparations
                                                      for  constituting  various  mixtures   of  the
                                                      experiment.  Each  mixture will  contain  x.mT  of
                                                      compound C^ and XjT of chemical C2, where x.,x, >
                                                      0, and JL+X.-I.   Let  n.  denote  the number  of
                                                      subjects  in the  experiment  and  r.  denote the
                                                      observed   number   of  effects   in  the  J-th
                                                      preparation (mixture).   The hypothesis of  dose-
                                                      additivity  can  be  test by  using  the  chi-square
                                                      test for homogeneity
                                                              g
                                                         X "  I
                                                             3-1
                                                                     ? U-P
                                                      where g  is the number  of  preparatinos.    If  the
                                                      two  chemicals  are dose  additive, then  X has  a
                                                      chi-square  distribution  with  g  degrees   of
                                                      freedom.

                                                      4.  RESPONSE SURFACE  ANALYSIS
                                                          Suppose that the purpose of  an experiment  is
                                                      to study the relation between  the different dose
                                                      combinations   with   the   responses.      The
                                                      experimenter may  be  interested in  finding  a
                                                      suitable approximating  function  for the  purpose
                                                      of predicting  future responses over  a range  of
                                                      dosage,  or determining  what dose  combinations  (if
                                                      any)  can yield an optimum as far  as  the  response
                                                      concerned.   The  common approach  of  this  problem
                                                      is by a statistical  curve fitting  technique or  a
                                                      so-called response surface method.
                                                         Assume  that  the  observed  values y from  the
                                                      mixture  contain  variations   e,  the  mixture
                                                      responses can be written as
                                                       y - R(XI,XZ,T) + e
                                                                                                    (6)
The variations e are assumed to be  independently
and  normally  distributed  with  zero mean and
common  variance.    The  functions   F.(T)  and
E(x1,x2,T),  in general,  can be  represented by

polynomial forms;  that  is,  equation  (1)  can be
expressed as

   1(VVI> - x,
                                                                       b!2
                                                                                                Vt Jc,
                                                                                             l!2
                                                                                                   (7)
                                                   For practical purposes,  lower-degree  polynomials
                                                   are normally fitted.  For  example,  a  quadratic
                                                   response model,  a second degree  polynomial
                                                   function for  x  and  T,   for  the mixture  can be
                                                   expressed as
                                                  79

-------
where B^T), B20
provided  that  B^T)  + BU(T)  > B2(T)  and 32(T) +
B, (T)  >  BT(T)  for all T;  similarly,  the minimum
respo-se  occurs  in Che experimental dose range  if
B  ,(T)  <  0 provided  that B^T) +  B12(T) <  B2(T)
and  S2(T) + B12(T) < B^T).
   Equation (7), alternately,  can  be expressed  as
                        b2  "2  +
                                   !2
                                           T2
                                              (9)
 When the  experimental  dose levels are  coded  to
 have   zero  mean  (e.g.,   -1,0,1  for   three
 concentrations),    the    coefficients    have
 Interpretations. (Piepel and Cornell,  1985), e.g.,
 the  intercept  term  (b^ x^  +  b2  X2  +  b^2 ^i^
 represent  linear  and  nonlinear  effects  of  the
 proportions  in  the  mixture  at  the  average
 concentration of the experiment .

     If  b!2° "  b!2l "  b!22  "  °'  then  the J°lnt
 action of  the two  chemicals is response-additive;
 the response  is  linear with the  proportion of a
 given  chemical at each concentration (Figure 2) .
 Three  special situations are of Interest:
 1)  If  b^  • b21  -  bx2 - b22 - 0, then R(x1,x2;T)

     - (b1°x1 + b2°x2); the lines in Figure  2 are

     coincident, the chemical concentration  has no

     effect  on the response.
2) If b.   -
                and
   (b1°x1 + b2°x2) +
                            DZ  , then

                            +  b^T2;  the  lines  in

     Figure 2 are parallel, the response increases

     by  a   constant  amount   as  concentration

     increases .

  3) If b^ - b2° - b^ - b2l  - 0, then R(x1,x2;T)
                                                                                        th*  llne*  to
                                                        Figure 2 are not parallel, the response
                                                        increases    proportionally     vith     the
                                                        concentration.

                                                        Kquation  (9)  assumes that  the  experimental
                                                     variations are normally Independently distributed
                                                     with zero mean and common variance. However, many
                                                     data collected from the bioassay  experiments do
                                                     not follow  the model assumptions.   For example,
                                                     Snee and Irr (1981) found  that  mutagenesis data
                                                     collected from a mammalian  cell  assay system did
                                                     not  satisfy  the  assumptions  of  normality  and
                                                     constant  variance.   Various  transformations can
                                                     be  used to  achieve  the model assumptions.   For
                                                     analyzing  dose-response relationships  of  3uta-
                                                     genesis  data, Snee and Irr  (1981) suggested using
                                                     the Box-Cox  (196-*) power transformation model
                                                          y    -  [R(x, ,xO] A  + e  for    \  * 0;

                                                      log  y    •  log  [R(x1,x2)] + e  for  X  - 0
                                             (10)
where  X  is  the power  transformation  parameter to
be estimated  from the data.   An application of
the model is given In the next section.

5. EXAMPLE
   An  experiment was  conducted  to  study  the
effects  of  mixtures  of l-nitrobenzo(a)pyrene
(1-NBP)  and  3-KBP on  mutation  induction  in  the
Salmonella  reversion  assay.   Both chemicals  are
suspected environmental  contaminants and  are
potent  direct-acting  mutagens  in  Salmonella
without  exogenous activation (Pitts  et al., 1984;
Chou  et  al.,  1984).  Assays  were performed with
Salmonella  typhimurium tester strain  TA98  in the
absence  of  exogenous metabolic activation using
the methods  described in Mar on and  Ames  (1983).
1-NBP and  3-NBP  were  synthesized,. free  from
contaminating  isomers, by  the methods  of  Chou et
al.,  1984).    Mixtures of  the  two chemicals were
prepared using seven different proportions of the
two mutagens at  the fixed total concentrations of
0.1,  0.2,  and 0.4 ug of mutagen per  plate.  The
mixture  proportions  and  the  experimental results
are shown in Table 1.

TABLE  1. The number of mutants per plate produced
         by  mixtures  of  1-NBP and 3-NBP
1-NBP :3-NI
Ratio
1:0
4:1
2:1
1:1
1:2
1:4
0:1


150
219
204
206
213
255
194
Revertants per Plate
0.1
,171
,165
,197
,202
,237
,284
,176

,151
,196
,208
,196
,205
,275
,210

212
258
462
480
379
527
286
0
,213
,333
,393
,495
,418
,503
,264
.2
,183
,349
,418
,475
,389
,489
,289
0.4
216
339
604
660
612
471
315
,198
,328
,520
,621
,737
,660
,333
,237
,305
,490
,572
,491
,605
,305
                                                    80

-------
   The Box-Cox power  transformation was  used  to
ensure  that  tht assumptions  of  normality  and
homogeneous varlanct  of  experimental  error were
satisfied.   Using Che method  given by  Irr  and
Snee (1982) to calculate the power.parameter   X;
the estimated value of X was approximately 0.20.
Thus, the transformation y°*2 was used to fit the
dose-response functions for subsequent analyses.
The   fitted   equation   with  the  estimated
coefficient  standard  errors for  the data from
Table 1 is

   y - (183 1-NBP + 283 3-NBP + 760  1-NBP*3-NBP)
         (12.0)   .        (16.7)       (73.1)

    + (147 1-NBP + 435 3-NBP +  4190  1-NBP*3-NBP)T
         (99.0)          (139.4)       (610.9)

                                            (ID

where  T  is coded  as  -.0133,  -0.033,  and  0.166.
Note  that  the  coefficients  for   the  quadratic
function of T are not significant.   Equation (11)
shows mutagenic  responses  on 1-NBP and 3-NBP, and
the  responses  produced by each chemical are not
equal.   The  effect of  the total concentration of
the   mixture  is  linear  with  the  response.
Increasing  the total concentration affects both
the  linear terms, b.l -  147 and b,  •  435, and
                    ^  1           ^
 the  nonlinear term,  b
                      12
                           4190,  in  the mixture

components.   Moreover,  it can  be shown  that  a
aynergistic  Joint  action   between   the   two
chemicals  in  the  experimental dose  range, total
concentration from  0.1  and 0.4  ug/ml,  and  the
mixture with proportions of 1-NBP to 3-NBP about
0.43 to 0.57 at  total concentration. 0.4 ug/ml can
produce the strongest mutagenic effect.

REFERENCES
Ames,  B.,  MaCann,  J.,  and YamasakJ.,  E.  (1975).
   Methods for detecting carcinogens and mutagens
   with  the Salomella/maomalian  microsome muta-
   genicity test,  Mutation Research, 31, 347-364.
Box, G.E.P. and Cox, D.R.  (1964).  An analysis of
   transformations (with discussion),  Journal of
   Royal Statistical Society B, 26, 211-252.
Chou,  M.W.,  Heflich,  R.B.,  Casciano,  D.A.,
   Miller, D.W., Freedom,  J.P.,  Evans, F.E., and
   Fu,   P.P.  (1984).     Synthesis,   spectral
   analysis,  and  mutagenicity  of 1-,   3-,  and
   6-Nitrobenzo(a)pyrene,   Journal  of  Medicinal
   Chemistry, 27,  2256-2261.
Finney,  D.J.  (1971).    Probit   Analysis, Third
   Edition,  Chapter 11.   Cambridge  University
   Press,  Cambridge.
Hamilton, M.A. and Hoel,  D.C.  (1980).  .Quantita-
    tive  methods  for describing  interactive
    effects in toxicology.   Technical Report No.
    1-6-80,  Montana  State  University,  Bozeman,
    Montana.
Irr, J.D. and Snee, R.D. (1982).   A statistical
    method  for  the analysis  of mouse lymphoma
    L5178Y call Tt  locua  forward mutation assay:
    comparison  of  results  among three  labora-
    tories, Mutation Research, 97, 371-392.
Kodell,  R.L.   and   Pounds,   J.G.    (1985).
    Characterization of joint  action  of  two
    chemicals  in  an  in  vitro test  system,
    American Statistical  Association, Proceedings
    of Biopharmaceutical  Section, 48-53.
Maron,  D.M.  and Ames,   B.N.  (1983).    Revised
    methods for the Salmonella mutagenicity test,
    Mutation Research, 113,  173-215.
Plackett,  R.L.  and  Hewlett,   P.S.   (1967).    A
    comparison   of  two  approaches    to   the
    construction  of models  for quantal responses
    to mixtures of drugs, Biometrics, 23, 27-44.
Piepel,  G.F.  and Cornell,  J.A.  (1985).   Models
    for  mixture  experiments  when   the  response
    depends  on the total amount, Technometrics,
    27, 219-227.
Pitts,  J.N. Jr.,  Zielinska,  B.,  and  Harger,  W.P.
    (1984).   Isometric mononitrobenzo(a)pyrenes:
    synthesis,  Identification  and  mutagenic
    activities, Mutation  Research, 140, 81-85.
Reif, A.E.  (1984).   Synergism  in Carcinogenesis,
    Journal  of  National   Cancer  Institute,  73,
    25-39.
Scheffe,  H..  (1958).   Experiments with mixture,
    Journal of  the  Royal  Statistical Society, B,
    20, 344-360.
Shelton,  D.W. and Weber, L.J.  (1981).  Quantifi-
    cation  of the  joint  effects of  mixtures of
    hepatotoxic agents: evaluation of a theoreti-
    cal  model in mice,  Environmental  Research,
    26, 33-41.
Snee,  R.D.  and  Irr,  J.D. (1981).   Design  of  a
    statistical  method   for   the  analysis  of
    mutagenesis at the hypoxanthlne-guanine phos-
    phoribosyl  transferase locus  of cultured
    Chinese  hamster  ovary   cells,  Mutation
    Research, 85, 77-98.
Vendetti,  J.M.  and  Goldin,  A.  (1964).    Drug
    synergism in antinioplastic chemotherapy,
    Advance Chemotherapy, 1, 397-498.
                                                   81

-------
   FIGURE  1.  Mixture Design
  FIGURE 2.  Response-Addltivlty
         Dose of
                                                    Proportion in Mixture
   FIGURE 3.  Dose-Addltlvlty
FIGURE  4.  Response from Dose-Additivity
o

"o

w
o
Q
                                              M


                                              O
                                              o.
                                              (O
                                              0)
                                              DC
      F,(T)
 F, (MT)

= F2(T)
                                                T         Dose of C,           MT

                                                1:0    Proportion of C-, to C2       0:1

-------
                                               DISCUSSION
                      Elizabeth H. Margosches, U.  S.  Environmental Protection Agency
   X «• pleased to have Che opportunity  to  comment
on this paper.  When Dr. Chen first  sent M the
paper, several directions  for rgamuts came to
sdod.  At this conference  we  have already heard
•any speakers refer to  the properties of response
additivity.  Nevertheless > there are some special
points here.
   Chen  e_t  al. define a response additive model
for which  the  outcome is the  same as that  under
the dose additive assumption, and then proposes to
test  for dose  additivity.   This special case, how-
ever,  where E(xi,x2,T)  • 0,  is, as Dr. Kodell
pointed out yesterday, the exponential  case, which
has  many well defined properties.  One  of  the
problems here is that the method works  from a
 count of effects.  Among  the  pluses:  a  test sta-
 tistic is proposed; data  are  used on a  noncancer
 er.cpoint, mutagenicity.
    Chen st_ al. quote Hamilton and Hoel  (1980)
 regarding two purposes for studying  the joint  ac-
 tions of chemicals,  as shown in Table 1.   Studies
 may be  conducted to  provide  risk estimates from
 joint exposure and they may  be conducted to elu-
 cidate  mechanisms of Joint toxicity.  The  emphasis
 in Chen e_t  al. appears to be on the latter and, as
 we've heard from several  speakers,  this is an
  imortant  facet  of research.  As an EPA statis-
  tician, however, I must  admit my concern  is more
 with the  former, although our interest is  in both
  foci.
     Can we expect one study to assist us in both
  endeavors?  Probably not.  Can we find one method
  of  modeling to help in both?
     What are the modeling questions  asked  in these
  two perspectives?  In the first, we  assume the
  components are  unknown.  We  then try to predict
  the curve at some other  dose than  that studied.
  In the second,  we can assume the components  are •
  known.  Then we try to decide  if,  at some dose,
  there  is joint  action (or compounded effect).
  Any model  that  is chosen for use can only reflect
  the  extent of joint activity built into it.   Sim-
  ilarly,  the  shapes  at low doses, the thresholds,
  etc.,  depend on the underlying postulates, not
  necessarily  the true  state  of nature.
      Thus,  the two perspectives must have different
   analyses.   Providing risk  estimates from  joint
   exposure calls  for  procedures that are robust
   against  misspecification in the range of  interest.
   Elucidating mechanisms calls for tests of full
   versus reduced models like  those of which Dr.
   Machado spoke  earlier.  Chen et al. have provided
   conditions for maximum response and minimum re-
   sponse in the  experimental  range.  What  about in
   the low dose range where I  have to work  so often?
   Can the cancer model of which Dr.  Thorslund spoke
   earlier help with  transformed cell assay data?
      But it seems one  of  the  greatest limits we
   have  placed on ourselves so far is that  of  deal-
   ing with  substances  in  pairs.  As  Dr. Litt  de-
   scribed  yesterday, the  Agency must deal  on  a  daily
   basis with  toxicants combined in  both unidentified
   and unquantified mixtures, e.g.,  pesticides,  waste
   dumps.   We need methodology to take us beyond
   pairs.
       What are our barriers  to extension?   I won't
 pretend to have  identified all of  these, and I
 offer just a  few thoughts on ways  statisticians
 have already  extended themselves in other set-
• tings.  Three that cove  to mind nave entered into
 several papers at this symposium.   (l)Looking at
 all the cross-products:  this becomes quite cum-
 bersome with  more than two compounds in anything
 beyond near linear responses.  Let's consider
 adopting  a matrix notation,  so useful in the
 analogous leap in regression.  Or  consider, as
 Dr. Patil suggested  yesterday,  the multivariate
 distributions that may be  at work  to produce the
 phenomena we  see as  marginal distributions.
  (2)Looking at pairvise  isoboles:   again, we're
 bound by the  paper  plane.   *r.at  aoout colors,
  faces,  perspective,  etc.  It's almost ten  years
  since Gnanadesikan published his book on ways  to
  look  at multivariate events.  Let's  consiaer
  other grapnic devices,  enlist the computer.
  (3)Looking at complex biological systems:   while
  the organisms whose risk concerns us will  almost
  always be complex, whether as a human or  as an
  ecosystem, perhaps we can find other indicators
  of the likely response.  More work  needs  to be
  put into examining and developing  short term
  assay surrogates for prediction.
     In summary,  in this  paper, a narrowly  defined
  response  addition
     R(x1,x2,T) - XjFjCT) + x2F2(T)  + E(x1,x2,T),

  where E(x^,x2,T) »  0 for any x^,  X2, T  such that

  Xj + X2  » 1, namely, the special  case  of linear

  responses at fixed  concentrations,, permits (l)the
  construction of a test  statistic  and (2)the use
  of short term data.  Furthermore, it calls atten-
  tion to  the  literature  that uses  both composition
  and  concentration  to examine the  behavior of
  mixtures.

                      References

  Gnanadesikan,  R. 1977.  Methods  for  Statistical
  Data Analysis  of Multivariate Observations. John
  Wiley, New York.

  Hamilton, M. A., and Hoel, D. G.   1980. Quantita-
   tive methods for describing interactive effects
   in toxicology.  Technical Report No. 1-6-80,
  Montana State  University, Bozeman.

                      Table 1

     FuzpoM    provide risk e*tl*»te*   elucidate  Mchanlm*
               rro» Joint exposure     of Joint toxicity

     Situation   assume components ate   aiauae components are
                  unknown;              known;
               predict curve  at doae   decide If  joint action
               not studied           at dose studied

     Method*    procedure* robuit       tests of full vs
     need       *taln*t aUapeclflca-   reduced «odel«~
               tlon
                                                       83

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                     STATISTICAL DIRECTIONS TO ASSESS  EFFECTS OF COMBINED TOXICANTS

                           Ronald  E.  Wyzga,  Electric Power Research Institute
                     ABSTRACT

   tte papers presented  at  the  ASA/EPA Conference
on Current Assessment  of Combined Toxicant
Effects  are  discussed.  The papers Illustrate the
existence of screening methodologies to Indicate
when  "Interaction" between  toxicants la likely.
This  can help assess mixtures toxlclty for
mixtures of  a small number  of toxicants at dose
levels  In  the experimental  range, but additional
methods need be developed when extrapolation from
one  dose level to another Is required or when
more complex mixtures are assessed.  The
conference  provided some limited  guidance on the
use of  models  for  such  cases,  but greater
 statistical efforts are  needed.

                      KEYWORDS

 Joint  Action Models
 Complex Mixtures
 Combined Toxicant Effects
 Interaction

                   1.   INTRODUCTION

     This paper attempts  to  summarize the jse of
  statistics  to address the  toxicity of mixtures
  and to suggest alternative statistical approaches
  that might  be taken  to achieve further progress
  in addressing the  Issue.  Emphasis is given to
  the papers presented at this  conference.
     The toxicity  of mixtures Is clearly an
  Important  subject.   If  there  were  only  100
  potentially toxic  agents, the possibility of
  unusual or unexpected  combined  effects  is hardly
  trivial.   Taking combinations of two  agents  at  a
  time,  the  matrix of  combinations yield  4950
  cells.  If the  probability  of one  agent
   influencing the toxicity  of another were even as
   low as 0.01,  there would  still  be 49 combinations
  where the  toxicity of  the combined toxicants
   would be different from the sum of the toxicity
   of the Individual toxicants in assessing combined
   toxicant effects.  The real world of thousands of
   agents and mixtures, far more complex than
   binary, obviously has  considerable potential for
   a  large number  of "interactive" effects.
      One of  the problems in assessing combined
   toxicant  effects  is that there  is a whole range
   of Issues to be  resolved.  Mixtures can be
   defined at different  levels  of  complexity.   Much
   of the  research to  date and  of  the  research
   reported  here  has been performed  with  binary
   mixtures. This is  probably  due to  two  reasons.
   First of  all,  as Kodell  (1986)  pointed  out  in his
    introduction,  the earliest work was  performed
   with drugs and pesticides,  the objective  being to
    examine the  effectiveness  of one  of these
    substances  in the presence of  another.   Hence
    only simple combinations were  studied.   (The
    simultaneous presence of environmental and other
    agents was  ignored or assumed  to be unimportant.)
       Secondly, binary substances are a conceptual
    aid.  The best approach for understanding a
    mixture's toxicity profile  is  to consider simple
mixtures first.   This can provide Insights on how
to analyM mot*  complex mixtures, which realis-
tically reflect  exposure.  Environments are
complex; pur* mixtures do not exist.  We do not
Inhale, Ingest or absorb pure substances or even
a handful  of substances, but mixtures of numerous
substances.   If, perchance, exposure were to be
pure,  the  purity would cease once the substances
entered  the bloodstream.  One potential approach
to  assess  mixture toxicity  is to divide the
mixture  into its components and  to  study  these
singly and In combination to arrive somehow at  an
estimate  of  the  mixture's  toxicity.  Often,
however,  the mixture  is  ill-defined;  Its  compo-
nents cannot be  defined.   In  such  circumstances,
 one can only work with  the  total mixture  and/or
 its fractions.   Binary  experiments  are  still
 possible,  but assu-ne  a  different role  here  as  the
 experimental agents  Tiay be  mixtures themselves.
    Time complicates  the definition of  a mixture.
 Mixtures  and exposures  thereto can vary con-
 siderably  over  time,  and this variation can
 Influence  the mixture's toxicity.  Thorsland and
 Charnley  (1986) show the temporal Importance of
 cigarette  smoking in a mixture with another
 carcinogen.  In reality, human exposure patterns
 are  even  more complex, and it will be necessary
 to estimate and characterize this time
 variability and to determine its influence.

              2.   OBJECTIVES OF RESEARCH

     Another  problem associated with current
  toxicity  assessment approaches  is that  many
  questions  are  asked  of mixtures,  and  different
  approaches  are  appropriate for  different
  questions.  The questions  will  dictate  the
  research objectives  and corresponding  statistical
  tools.  .
     The most commonly asked questions  probably
  relate to  the  three  given  below.

      1.  Under ambient conditions,  is  the mixture
  hazardous?
        a.   Is interaction likely to occur?  How is
         it  defined?
        b.   What is  the dose-response surface for
         the mixture?
      2.   How toxic is the mixture compared to other
   mixtures?  Other substances?  Are similar
   mixtures equally toxic?
      3.   What is  (are)  the  toxic component(s) of
   the mixture?

      The papers  at this  conference  address the
    first question with most  of  them  focusing on
    question  la,  although the specific questions
    addressed  are  variations  of  the  question.
    Several  such  as Weber et  al.  (1986)  ask whether  a
    given joint action  model  fits  a  data  set.
    Machado  (1986) and  Chen et  al.  (1986)  explicitly
    ask question  la as  to whether  interaction  and
    dose additivlty exist.  Other  papers  examine the
    presence of interaction over a broader range of
    dose-exposure levels and hence try  to describe a
    dose-response surface.  Thorsland and Charnley
                                                       84

-------
(1986) address tht toxicity of mixtures  over  dose
ranges where extrapolation Models er«  required.
Chrlstensen (1986) considers  the  Issue but  his
objective Is different.  Very low doses  (and
hence extrapolation) are of lesser  concern  for
fish than for humans, where risks to Individuals
of 10   or IMS ere of policy concern.
   The collection of papers suggests that
question  la can be answered Cor simple mixtures
of two to three substances.   A response  to  this
question  for more complex sdxtures  Is  hampered by
vnwleldy  experimental designs and unrealistic
data requirements.  This situation  can be alle-
viated somewhat by frsctlonal factorial  designs
although  these were not explicitly  discussed  at
the conference.  Question la  Is  Important  for
screening purposes; answers to  it can  suggest
where  "interaction" is likely to  be present.  A
caution,  however,  Is  that the presence or  absence
of "Interaction" at one set of dose levels  need
not generally"imply the same  result for  other
dose levels.  Thorsland and Charnley  (1986),  for
example,  show  that conclusions  derived at  "high"
dose levels may not be equally  true at "low"  dose
levels.   Experimental results suggest  this  as
well.   In a series of fire toxicology  experi-
ments,  tevin  and coworkers  (1986) demonstrate a
relatively complex "interactive"  effect  of  CO and
CO, on the mortality  of rats. Over a  part  of the
dose  range, mortality response  appears to
increase  with increasing 2  concentrations for  a
fixed  CO  level.  The  very opposite  appears  to
occur  at  other  CO  levels.  Hence,  a conclusion
based  on  experiments  over a limited dose range
could  not be  generalized correctly.
   Most  of  the  historical terminology  problems  so
well  described  by  Kodell  (1986)  relate to
question  la because definitions  of  "Interaction"
were  tied to  specific models.  As we progress
beyond this  screening question  towards
questions Ib,  2,  and  3, much  of  this confusion
will  be resolved.

      3.   GENERAL PROBLEMS  IN  ASSESSING
          MIXTURE  TOXICITY

 3.1   Information  Availability

    The nature of  available  information will
 obviously Influence  the  approach for addressing
 the  mixture's toxiclty.  Most of the conference
 papers assumed that  it  was  possible to identify
 the components of  the mixture.   If  the components
 are'unknown,  the approaches  discussed here  must
 be modified or replaced.  This  will be discussed
 in the next section.

 3.2  Pharmacokinetlcs

    Another important  information question is that
 of pharmacokinetics.   This  issue was addressed by
 Feder (1986) in his discussion of Thorsland and
 Charnley (1986) and to  a  lesser extent by Weber
 et al. (1986) and others who undertook some
 studies of specific organ systems  In  an effort to
 achieve  "better" model fits.   Obviously,
 responses to a dose can be  more accurately
 estimated If the dose Is that at the  site of
 biologic activity.  Unfortunately,  the "effective
 dose" often Is not known and the "administered
dose" la used  In estimating the dose-response
relationship.  This  Is  obviously less than
optimal In  the case  of  a simple toxic, but the
situation becomes even  more complex in the case
of a mixture.  For example, misinterpretation
could arise  If the relative composition of the
mixture ware to change  aa a result of chemical
Interactions or of differential absorption,
distribution,  metabolism, or elimination, which
varied with  dose or  some other factor independent
of the mixture.  Current pharmacoklnetlc models
attempt to describe  the fate of a single chemical
and do not  treat complexities that mixtures can
introduce.   Such complexities can distort the
estimated dose of a  mixture at target sites,
where toxicity effects  are Initiated.  Pharma-
coklnetic assumptions about compositional changes
in the mixture would also have to hold across all
species involved in  any extrapolation across
species, otherwise the  validity of such extrapo-
lation would be  in question.  Given the
importance  of  this issue, more attention to
phannacokinetics is  clearly warranted in
assessing the  toxicity  of mixtures.  The develop-
ment of both pharaacolcinetic data and models for
mixtures is  needed.

            4.  EMPHASIS  ON COMPONENTS

   The papers  at the conference considered
synthetic approaches in which a mixture was
constructed  from limited (two or three) compo-
nents.  As  indicated above, this emphasis
requires that  the mixture be simple and well-
characterized. These  requirements, particularly
the former,  are not  always realistic.  At issue
is whether  and how existing methods can be
adapted to  more  complex and realistic situations.
   The complexity  issue can be addressed by
extending the  methods  used to several variables
beyond the  two or  three considered.   In this
regard, some of  the  methods are more  amenable
than others.   Those  methods that depend upon
experimental designs are hampered by  practical
considerations.  Toxicology experiments can
rarely accept  more than a limited number of
combinations of  substances, otherwise, they
become too  costly and  uniform experimental
conditions  for all combinations become difficult
to maintain.
   Simple factorial  designs clearly limit
consideration  of mixtures more complex than three
or four substances,  but fractional factorial
designs can extend the  complexity of  mixtures
studies considerably.   For example, designs for a
mixture of  15  components could be constructed
which  required only  52  treatment groups (for
combinations of  doses), vet would still allow
estimation  of  the  toxicity of all IS  substances
singly and  of  pairs  of  six of the substances.  A
simple factorial design for this mixture would
require 32,768 treatment groups.
   Another  approach  to  assessing the  toxicity of
more complex mixtures  is given by Thorsland and
Charnley  (1986), namely, the use of a model to
estimate toxicity.   Their results suggest that
for  their model,  toxicity at "low" doses is
additive across  components in the mixture, I.e.,
"interaction"  effects  become negligibly smaller
as the dose level decreases.  Under these
                                                    85

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results, mixtures of sevaral known components can
be eaaily addressed toy Adding  the  toxicity of the
components-.  This requires, however, that the
toxicity of  the  components  be  known.
   When toe  components of * mixture are unknown,
two approaches are possible.   A mixture with
SLoSn co-potent, can be fraction. ted Into
•utually exclusive «lxturaa;  tha resulting
mixture,  than can be analysed aa If thay ware
single aub.tances to estimate any "Intaractlon"
of  tha raaulting mixtures.  Such an approach has
bean  considered  to address the toxicity of
unleaded gasoline,  a very complex mixture whose
constituents are not completely specified.
 (?ader at al. 1984).
    Whan the components of the mixture  are not
known, a second approach Is to study the  mixture
 directly as has been done with cigarette  snoke.
 If the mixture  toxicity  is of  interest,  it  may
 not matter whether or not  there  la  Interaction
 among the mixture's components.   The  toxicity
 cou^d be assessed for several  mixtures in the
 same  class  (e.g., different  brands  of  cigarettes,
 vapors  from  different gasolines  or exhaust  fumes
 from  different  diesel engines)  to determine If
 the  toxicity  is relatively robust across the
 class of mixtures.   Feder  et  al.  (1984) discuss
 this  approach as well.
    Extrapolation of toxicity from high to low
 doses could Introduce a problem with  the latter
 approach.   Extrapolation models have  been dev-
 eloped for single substances, and  their appli-
  cation to a mixture could cause problems.
  Consider the example given by Thorsland and
  Charnley (1986).  They  give In Table  1 a bioassay
  design which gives an Interaction  tens  large
  enough to double the risk over that predicted  by
  additlvlty at  low doses.  Under  that  design,  for
  x  at a level  of 1.05415 x  10°  the risk is about
  iSc  10"5.  For x, at 2.10828 x  10'3,  the  risk  Is
  also about  1 x 10'5.  Under  an  additive  model,
  the  risk of  a  mixture of  xx  and  x2 would be
  2 x  10   ,  whereas  the  true  model gives the risk
            "
        ,
of 4 x 10" •  Now if only a mixture of
                                             an
-------
where 0^  Is the  true  carclnogenlclty  of  the
mixture,  e«  Is  some error  associated  with  the
•aaaur* of 7i such as the  error  due to
extrapolating from rat*  to humans  or  the error
aaaociatad with • abort-term test. The f  could
b*  a vector  of the toxlcltle* of the  major
components  of gasoline and X^ Is the  vector of
concentrations of the components.   For example,
f(XifH) - X.B, the model could be Interpreted as
an  additive model under which the toxicity of the
mixture Is the sum of the toxicitles  of Its
constituents.  In this case, 6^ is a measure of
 Interactions among the mixture constituents.
 Given prior  distributions on 6j,  e^,  and  B,  one
 can estimate  the  posterior distribution of the
 mixture  toxicity  given  the data y.   One can
 extend the  context here by defining  £,  such  that
 y.  « 0^  + ej,  where  8£  represents another mixture
 for which no observed data are  available.  Other
 extensions  are  possible.   See Harris (1983)  and
 DuMouchel and Harris (1983).

                   6.   CONCLUSIONS

     The papers at this conference  suggest  that
  statistics  to date  has  concentrated  upon  the
  problem of whether  "Interaction" exists and how
  it can be characterized.   In this area, we have
  made considerable progress.  We now  have  valuable
  screening tools  that indicate when  interactions
  may be important.   Now, we need  to ask more
  specific questions  such as how Important the
  interactions are at doses that nay  be different
  from  those  in  the experiments where  "interaction"
  is measured.   Interpolation and  extrapolation are
  required.   These are roles  for models  that
  attempt to  describe quantitatively  the complex
  biology or toxicology  of mixtures.
      Models  provide a means  to describe  and
  summarize  experimental results  and  to  relate  them
   to underlying biology, but  models for  mixtures
   are In their Infancy.   A research priority  Is  the
   development of improved  models to address  mixture
   toxicity.   Thorsland and Charnley  (1986) provide
   an important example of the direction that  such
   models can take.
      Models  are imperfect tools.  As  such, they
   have limits.  At best, they reflect the  limits of
   biological knowledge.  Models also delve into the
   unknown and unknowable when addressing  such
    issues as  high-to-low dose extrapolation.  In
    these  areas,  models may be the  only available
    tool,  but  their results are subject to  consider-
    able uncertainty,  a greater uncertainty than they
    may  Imply.  The limits  and uncertainties of
    models need to be  stated aa part of their  use.
       In reality,  modeling efforts often lag  behind
    biological developments.   Hence, one  way to
    improve models is  to achieve  greater  under-
    standing of biological  mechanisms.  Biological
    Intuition also can help direct  modeling and
    statistical approaches.  Ueber et  al.  (1985),  for
    example,  help  identify greater needs by following
    their Intuition to illustrate the  poor  behavior
    of  xinc-nickel interactions in the context of
    simple models.
       Models and  statistics  support the major
    strategies  to  assess mixture toxicity,  but models
and statistics are only ana criterion for
development of strategies.  Pragmatism and
biology are foremost considerations.  Pragmati-
cally, it Is not possible  to  test every combi-
nation of substances In every mixture.  The
challenge before us is to  uae statistics to move
•way from this approach towards on* that Is
consistent with biology.

                    INFERENCES •

BRODERIUS, STEVEN  3.  (1986),  "Joint Aquatic
   Toxicity of  Chemical  Mixtures and Structure-
   Toxicity Relationships," presented  at  the
   ASA/EPA Conference  on Current Assessment  of
   Combined Toxicant Effects,  May  5-6,  1986,
   Washington,  D.C.

 CHEN,  JA.1ES  J. ,  HEFLICH,  ROBERT  N. ,  and  HAS3,
   BRUCE S.,  (1986), "A Response-Additive Model
   for Assessing the Joint Action  of fixtures,"
   presented  at the ASA/EPA Conference on Current
   Assessment of Combined Toxicant Effects,
   May 5-6, 1986, Washington,  D.C.

 CHRISTENSEN, ERIK  R. (1986),  "Development of
   Models for Combined Toxicant  Effects,"
   presented at the ASA/EPA Conference on Current
   Assessment  of Combined  Toxicant Effects,
   May  5-6,  1986,  Washington, D.C.

 DUMOUCHEL, WILLIAM H., and HARRIS, JEFFREY £.
    (1983), "Bayes  Methods  for Combining  the
    Results of .Cancer  Studies  in Humans and Other
    Species,"  Journal  of the  American  Statistical
    Association,  78, 293-315.

  ?EDER, PAUL J.  (1986), Discussant's  comments on
    paper by  Thorsland and Charnley, presented to
    ASA/EPA Conference on Current  Assessment  of
    Combined  Toxicant Effects, May 5-6,  1986,
    Washington, D.C.

  FEDER, PAUL J., MARGOSCHES,  Elizabeth,  and
    BAILAR, John.  (f984),  "A  Strategy  for
    Evaluating the  Toxicity of Chemical Mixtures,"
    Draft Report to U.S. Environmental Protection
    Agency, EPA Contract No.  68-01-6721,
    Washington, D.C.

   HARRIS, JEFFREY  E.  (1983),  "Diesel Emissions and
    Lung  Cancer,"  Risk  Analysis,  3, 83-100 and
     139-U6.

   KODELL,  RALPH  L.  (1986), "Modeling the Joint
     Action of  Toxicants:   Basic Concepts and
     Approaches,"  presented at the ASA/EPA
     Conference on Current Assessment of  Combined
     Toxicant Effects, May 5-6, 1986,
     Washington,  D.C.

   LEVIN, BARBARA C., PAABO, M., GUERMAN, T. L.,
     HARRIS,  S. E., and BRAUN, E.  (1986),  "Evidence
     of Toxlcologlcal Synerglsm Between Carbon
     Monoxide and Carbon Dioxide," submitted  for
     publication.

   MACRADO, STELLA  G. (1986),  "Assessment of
     Interaction in Long-Term Experiments,"
     presented at  the ASA/EPA Conference  on Current
                                                        87

-------
  Assessaent of Combined Toxicant Effects,
  Nay 5-6, Washington, D.C.

SIEMIATtCKI, J. and THOMAS, D.C. (1981),
  "Biological Models and Statistical
  Interactions:  An Example frosj Multistage
  Carcinogens,* latematlonal Journal of
  Epidemiology, 10, 383-387.

TBORSLAHD, TODD V. and CHARNLET, GAIL.  (1986),
  "Use  of the  Multistage Model  to Predict the
  Carcinogenic Response Associated with Tl»e-
  Dependent  Exposures  to Multiple Agents,"
  presented at the ASA/EPA Conference on
  CurrentAsacsaaent of Combined Toxicant Effects,
  May 5-6, Washington, D.C.

WEBER, LEVERM J., MDSKA, CARL, YINGER, ELIZABETH
  rOIKMEISTER, aad SHELTON, DENNIS.  (1986),
  "Concentration aad Response Addition of
  Mixtures of Toxicants Dslng Lethality, Growth,
  and Organ Syste* Stadias," presented at  the
  ASA/EPA Conference on Current Assessaent of
  Combined Toxicant Effects, May  5-6,  1986,
  Washington, D.C.
                                                      88

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A:  ASA/EPA Conference on Current Assessment of Combined Toxicant Effects Program



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-------
                    APPENDIX B:  Conference Participants

                           ASA/EPA Conference on
            The Current Assessment of Combined Toxicant Effects
                               May 5-6,  1986
                             Washington,  D.  C.
John C. Bailar
468 N Street, S. W.
Washington, D. C.  20A60

R. Clifton Bailey
WH 586, EPA
401 M Street, S. W.
Washington, D. C.  20460

Steven P. Bayard
CAG   RD-689'
Environmental Protection Agency
401 M Street, S. W.
Washington, D. C.  20460

Jeff Beaubier
TS-798, EED, OTS
USEPA
401 M  Street, S. W.
Washington, D. C.  20460

Judith S.  Bellin
ORB RD 681
Environmental Protection Agency
401 M  Street, S. W.
Washington,  D.  C.  20460

Hiranmay  Biswas
WH-553
 7702 Middle  Valley Drive
 Springfield,  Virginia  22153

 Steven J.  Broderius
 Energy Research Laboratory
 Environmental Protection Agency
 Duluth,  Minnesota   55804

 Mary J.  Camp
 USDA
 5800 Eastpine Drive
 Riverdale, Maryland  20737

 Richard A. Carchman
 Medical College of Virginia
 Box 613
 Richmond, Virginia  23298

 Joseph S. Carra
 Environmental Protection Agency
 401 M Street, S.W.  TS298
 Washington, D. C.  20460
Hans Carter
Medical College  of Virginia
Box 32, MCV  Station
Richmond, Virginia   23298

Eric Y. Chai
Shell Development Co.
   PO Box  1380
Houston, Texas 772251

Chao W. Chen
Carcinogen Assessment  Group
Environmental Protection Agency
2313'Falling Creek  Road
Silver Spring, Maryland  20904

James J. Chen
Biometry Division
National Center  for
Toxicological Research
Jefferson, Arkansas   72079

Jean Chesson
Battelle Washington Operations
2030 M  Street, N.  W.,  Suite  800
Washington,  D.  C.   20036

Eric Christensen
Department of  Civil Engineering
University of  Wisconsin
Milwaukee,  Wisconsin  53201

Vincent James  Cogliano
ORD, OHEA,  CAG
Environmental  Protection Agency
 401 M  Street,  S. W.
•Washington,  D.  C.   20460

James  M.  Daley
 PM223
 Environmental Protection Agency
 401 M Street,  S. W.
 Washington,  D.  C.   20460

 Kurt Enslein
 President
 Health Designs,  Inc.
 183 Main Street
 East Rochester,  New York   14604
                                    91

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'i.V.C./H'i.'djil:
  »
                 .:.
Linda S. Erdrelch
OHEA-ECAO  (ORD)
26 West St. Glair Street
Cincinnati, Ohio 45268

Thomas R. Fears
National Cancer Institute
7910 Woodmont Avenue
Landov Building, Km. 3B04
Bethesda, Maryland 20892

Paul I. Feder
Battelle Columbus Laboratories
Applied Statistics and
 Computer Application Section
505 King Avenue, Room 11-9082
Columbus, Ohio  ^3201

Bernice T. Fisher
Environmental Protection Agency
401 M Street, S. W.
Washington, D. C.  20460

Mary J. Frankenberry
TS-798
Environmental Protection Agency
401 M Street, S.W.
Washington, D. C.  20460

Paul H. Friedman
OSW - WH-562B
Environmental Protection Agency
401 H Street, S. W.
Washington, D. C.  20460

David Gosslee
 Statistical Group Leader
Math. & Statistical Research
Martin  Marietta Energy  Systems
Building  9207A - PO Box Y
Oak  Ridge,  Tennessee  37830

G. Jay  Graepel
E. I. du Pont de Nemours & Co.
 Haskell Laboratory
 PO Box  50 - Elkton Road
 Newark, Delaware   19714

 Gary Forrest Grindstaff
 E329 (JS-798),  401 M  Street,  S.  W.
 Environmental Protection Agency
 Washington,  D.  C.   20460

 James Leonard Hansen
 TJnion Carbide Corporation
 PO Box 8361,  770-203A
 South Charleston,  West  Virginia  25526

                                     92
                                                     Richard C* Hertzberg
                                                     Environmental Criteria and
                                                      Assessment Office
                                                     Environmental Protection Agency
                                                     Cincinnati, Ohio  45268

                                                     Joseph F. Heyse
                                                     Merck Sharp & Dohme Res. Labs
                                                     WP16-100
                                                     West Point, Pennsylvania  19486

                                                     Stephanie Irene
                                                     Environmental Protection Agency
                                                     401 M Street, S. W.
                                                     Washington, D. C.  20460

                                                     W. Barnes Johnson
                                                     Stat. ?ol. Branch, PM 222, USEPA
                                                     401 M Street, S. W.
                                                     Washington, D. C.  20460

                                                     Henry D. Kahn, Chief
                                                     Statistics-Section - WH 586
                                                     Environmental Protection Agency
                                                     401 M Street, S. W.
                                                     Washington, D. C.  20460

                                                     Ralph L. Kodell
                                                     Division of Biometry
                                                     National Center  for
                                                      Toxicological Research
                                                     Jefferson, Arkansas  /2079

                                                     Aparna M. Koppikar
                                                     CAG/OHEA/ORD  RD689
                                                     401 M Street, S. W.
                                                     Washington, D. C.  20460

                                                     Daniel R. Krewski
                                                     Chief, Biostatistics &
                                                      Computer Applications
                                                     Environmental Health Directorate
                                                     Health & Welfare Canada
                                                     Ottawa, Ontario K1A OL2
                                                     CANADA

                                                     Herbert Lacayo, Jr.
                                                     PO Box 15521
                                                     Arlington, Virginia  22215

                                                     Emanuel Landau
                                                     Staff Epidemiologist
                                                     American Public Health Association
                                                     1015  15th Street, N. W.
                                                     Washington, D. C.  20005

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Barbara A.  Leczynski
Battelle Washington Operations
2030 M Street, N. W.
Washington, D. C.  20036

Walter S. Liggett, Jr.
National Bureau of Standards
Gaithersburg, Maryland  20899

Bertram D. Litt
Office of Pesticides
Environmental Protection Agency
TS-769C2
1921 Jefferson Davis Highway
Arlington, Virginia  22201

Stella G. Machado
Radiation Epidemiology  Branch
National Cancer Institute, NIH
LAN  3A22
Bethesda, Maryland   20205

Sam  Marcus
National Center  for  Health Statistics
13417 Keating Street
Rockville,  Maryland  20853

Elizabeth  H.  Margosches
Office  of  Toxic  Substances
Environmental Protection Agency
TS-798
401  M Street, S.  W.
Washington, D.  C.   20460

Margaret McCarthy
 Div. of Public Health
University of Massachusetts
 Atnherst, Massachusetts  01003

 Bruce Means
 Environmental Protection Agency
 401 M Street, S. W.
 Washington, D.  C.  20460

 Suresh Moolgavkar
 Fred Hutchinson Cancer Research Ctr.
 1124 Columbia St.
 Seattle, Washington 98104

 Paul D. Mowery
 SCI Data Systems
 530 College Parkway, Suite N
 Annapolis, Maryland  21401
Cornelius J. Nelson
OTS/EED/DDB
Environmental Protection Agency
401 M Street, S. W.
Washington, D. C.  20460

Jerry'L. Oglesby
SCI Data Systems
530 College Parkway,  Suite N
Annapolis, Maryland   21401

G. P. Patil
Center for Statistical Ecology
 and Environmental Statistics
Pennsylvania State University
University Park, Pennsylvania  168C2

Reva Rubenstein
Environmental Protection Agency
401 M Street, S. W.
Washington, D. C.  20460

Frederick H. Rueter
Consad Research  Corporation
121 North Highland Avenue
Pittsburgh, PA  15206

Michael  Samuhel
Battelle Washington Operations
2030 M  Street,  N. W. , Suite  800
Washington, D.  C.  20036

Jitendra Saxena
Environmental Protection Agency
 (WH  550)
401 M Street, S. W.
Washington, D.  C.  20460

Cheryl  Siegel Scott
TS-798
Environmental Protection Agency
401 M Street, S.W.
Washington, D.  C. 20460

Robert  L. Sielken
 Sielken, Inc.
 3833 Texas Avenue
 Bryan,  Texas  77802

 Janet Springer
 FDA
 11204  Schuylkill Road
 Rockville, Maryland   20852
                                     93

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 ^-^^'^^^^^
Ms. Judy A.  Stober
Health Effects Research Lab.
26 West St.  Clair Street
Cincinnati,  Ohio  45268

David J. Svendsgaard
Biometry Div./HERL  MD55
Environmental Protection Agency
Research Triangle Park, NC   27711

Todd W. Thorslund
ICF Clement  Associates
1515 Wilson  Boulevard
Arlington, Virginia  22209

Lavern J. Weber
Oregon State University
Mark 0. Hatfield
 Marine Science Center
2030 South Marine Science Drive
Newport, Oregon   97365-5296

Dorothy Wellington
6402 Middleburg Lane
Bethesda, Maryland  20817

Ronald E. Wyzga
Electric Potter Research Institute
P.O. Box 10412
3412 Hillview Avenue
Palo Alto, California  94303

Grace Yang
Department of Mathematics
University of Maryland
College Park, Maryland  20742

Mr. Robert P.  Zisa
Office  of Compliance Monitoring  EN-342
Environmental Protection Agency
401 M Street,  S.  W.
Washington,  D.  C.   20460

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