Quantification of Toxic Effects
for Water Concentration-Based Aquatic Life Criteria
Part A
Section 1: Background - Aquatic Life Criteria Limitations and Needs
Section 2: Toxicity Model Formulations - Binary Endpoints
Section 3: Short-Term Copper Lethality To Juvenile Fathead Minnows
Russell J. Erickson
Mid-Continent Ecology Division
National Health and Environmental Effects Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Duluth, Minnesota
FINAL
April 15,2007
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Executive Summary
This is the first of a series of reports that will present and evaluate methods for improving
how toxic effect levels in aquatic organisms are addressed in the formulation and application of
U.S. Environmental Protection Agency (U.S. EPA) water quality criteria for the protection of
aquatic life. This work is being conducted in support of efforts by the Aquatic Life Criteria
Guidelines Committee of the U.S. EPA Office of Water to develop new guidelines for derivation
of aquatic life criteria.
Section 1 summarizes the current formulation of aquatic life criteria and identifies certain
limitations regarding how well toxic effects on aquatic organisms are quantified in these criteria
as a function of the magnitude and time-variability of exposures. It then broadly describes how
better quantification of toxic effects could address these limitations and improve criteria utility.
Section 2 describes various models for the assessment of binary toxicity endpoints
(yes/no responses such as death) that could be useful in better describing such effects in aquatic
life criteria. It then broadly describes how these models can be parameterized based on standard
toxicity test data and what considerations should go into selecting a model for actual use in
criteria.
Section 3 presents a case study that establishes the feasibility of model parameterization
and demonstrates that these models can adequately describe the observed time-variability of
copper lethality to juvenile fathead minnows for relatively short (<8 d), constant and intermittent
exposures. This section also demonstrates how these models can provide information useful to
criteria and that this information might be adequate for criteria applications using relatively
simple models rather than more complicated models that would be difficult to implements.
Subsequent reports will address model formulations for other endpoints; other case
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studies regarding both acute and chronic exposures and both lethal and sublethal endpoints; and
recommendations regarding the application of these models to aquatic life criteria, including
minimum data requirements for model parameterization. These reports will provide the
technical basis for developing the guidance for using these models to criteria, but are not
intended to provide the actual guidance.
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Section 1: Background - Aquatic Life Criteria Limitations and Needs
1.1 Introduction
This is the first of a series of reports that will present and evaluate methods for improving
how toxic effect levels to aquatic organisms are addressed in the formulation and application of
U.S. Environmental Protection Agency (U.S. EPA) water quality criteria for the protection of
aquatic life ("aquatic life criteria" or "ALC"). This work is being conducted in support of efforts
by the Aquatic Life Criteria Guidelines Committee of the U.S. EPA Office of Water to develop
new guidelines for derivation of aquatic life criteria.
This background section summarizes the current formulation of aquatic life criteria and
identifies certain limitations regarding how well toxic effects on aquatic organisms are quantified
in these criteria as a function of the magnitude and time-variability of exposures. It then broadly
describes how better quantification of toxic effects could address these limitations and improve
criteria utility. Section 2 will describe various models for the assessment of binary toxicity
endpoints that could be useful for better describing such effects in criteria as a function of
exposure magnitude and time-variability. Section 3 will present a case study evaluating how
well such models describe the observed time-variability of copper lethality to juvenile fathead
minnows over relatively short (<8 d) exposures.
Subsequent reports will address model formulations for other endpoints; other case
studies regarding both acute and chronic exposures and both lethal and sublethal endpoints; and
recommendations regarding the application of these models to aquatic life criteria, including
minimum data requirements for model parameterization. It should be noted that the case studies
in this series of reports are intended to provide the detailed technical basis for the development of
guidance for the use of these models in ALC, but will provide neither this actual guidance nor
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elementary explanations of statistical and other mathematical procedures that might be needed in
such guidance.
1.2 Overview of Current Criteria Formulation
U.S. EPA aquatic life criteria are usually derived from laboratory toxicity test results
using procedures described in "Guidelines for Deriving Numerical National Water Quality
Criteria for the Protection of Aquatic Organisms and Their Uses" (Stephan et al. 1985), hereafter
referred to as the "Guidelines." These criteria consist of two concentrations - the Criterion
Maximum Concentration (CMC) and the Criterion Continuous Concentration (CCC).
The CMC is determined based on available "acute values" (AVs) - median lethal
concentrations (ZCsos) or median effect concentrations (ECsos) from aquatic animal acute
toxicity tests meeting certain data quality requirements. To compute a CMC, the Guidelines
require that acceptable AVs be available for at least eight genera with a specified taxonomic
diversity. For each genus, a Genus Mean Acute Value (GMAV) is calculated by first taking the
geometric average of the available AVs within each species (Species Mean Acute Value,
SMAV) and then the geometric average across the SMAVs within the genus. The fifth
percentile of the set of GMAVs so obtained is calculated based on a specified estimation
procedure, and designated the Final Acute Value (FAV). The FAV might be lowered to the
SMAV for an important, sensitive species as appropriate. The CMC is set equal to half of the
FAV to represent a low level of effect for the fifth percentile genus, rather than 50% effect. The
CMC is used in criteria to limit peak exposures by requiring that 1-h averages of exposure
concentrations not exceed the CMC more often than once in three years on average. It should be
noted that use of a 1-h averaging period is not equivalent to a 1-h exposure, but rather to a longer
exposure in which the worst hour is equal to the CMC.
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The CCC is generally determined based on available "chronic values" (CVs), which are
either (a) the geometric average of the highest no-observed-effect concentration (NOEC) and
lowest observed effect concentration (LOEC) for effects on survival, growth, or reproduction in
aquatic animal chronic tests or (b) in some recent criteria, the ECw in such tests based on
concentration/effect regression analyses. If CVs are available for at least eight genera with the
required taxonomic diversity, the CCC is set to the fifth percentile of genus mean chronic values
(GMCVs), by the same procedure used to derive an FAV from GMAVs. Otherwise, the CCC is
set to the FAV divided by a "final acute chronic ratio" (FACR) that is based on acute:chronic
ratios (the ratio of the AV to the CV from parallel acute and chronic tests) for at least three
species with a specified taxonomic diversity. The CCC can also be based on plant toxicity data
if aquatic plants are more sensitive than aquatic animals, or on other data as deemed
scientifically justified. The CCC is used in criteria to limit more prolonged exposures by
requiring that 4-d averages of exposure concentrations not exceed the CCC more often than once
in three years on average.
1.3 Limitations of Current Criteria Formulation
Criteria derived as described above are limited in the following ways regarding how well
the likelihood and magnitude of toxic effects are quantified:
(1) Only one level of effect is considered, rather than how levels vary with exposure.
Acute toxicity analyses are based just on 50% effect, with no specific consideration of
greater or lesser effects or of how rapidly the level of effect changes with concentration.
Chronic toxicity analyses likewise consider only a single level of effect, based either on
what is statistically significant or on some specified level of effect (e.g., 20%).
(2) The actual level of effect represented by criteria concentrations is not well defined.
For acute toxicity, the analysis does start with a specific level of effect (50%), but the
division of the FAV by a factor of 2 results in the CMC corresponding to an unquantified
low level of effect for the fifth percentile species, and unspecified levels of effects for
more tolerant and sensitive species. For chronic toxicity, CVs based on statistically
significant effects can represent a wide range of effect levels depending on the design and
quality of the toxicity tests and the variability of responses.
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(3) Whatever effect levels are represented by criteria concentrations, they correspond to
laboratory exposures with roughly constant concentrations for fixed durations, unlike
natural systems, where exposures generally have no specific duration and can vary
markedly with time. For acute toxicity, the toxicity test duration is 48-96 h (depending
on test species), and no assessment is made of how LC50s differ for shorter or longer
durations, or due to concentration variability within these time periods. For chronic
toxicity, durations can vary from several days to several months or more, but effects for
any one test represent a specific exposure regime that is unlikely to be close to those to
which criteria are applied.
(4) Criteria address the issue of duration and concentration variability by requiring
exposure concentrations to be below criteria concentrations when averaged over periods
shorter than the durations of toxicity tests used to derive the criteria concentrations. Such
an averaging period is intended to preclude exposure concentrations from being
substantially higher than criteria concentrations for more than a small fraction of the test
duration, thus ensuring effect levels stay within an acceptable range. However, the level
of effect that criteria will then represent will depend on the magnitude and pattern of
exposure variability, which constitutes another uncertainty regarding the effect levels
actually represented by criteria conditions.
Some of these limitations are
illustrated in Figure 1.1. In this figure, three
exposure time-series are compared to a
hypothetical CCC based on chronic tests with
a 30-d duration and implemented with a 4-d
averaging period. In Figure 1.1 A, a
moderately variable exposure time-series is
shown that satisfies the criterion
concentration because, although
concentrations on some days exceed the
criterion, the maximum 4-d average does not.
In Figure LIB, the exposure time-series also
satisfies the criterion, but has much lower
Figure 1.1. Three hypothetical exposure
concentration time-series ( ) compared to a
criterion concentration ( ).
O 0.5
100
B
Time
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variability. Its peak concentration is slightly lower than the first time-series, but its overall
average is much higher and exposure concentrations are near the criterion concentration almost
all of the time. For most toxicants, these two time-series should have different effect levels, but
criteria treat them as being of equal concern. What level of effect, therefore, does satisfying the
criterion actually represent? In Figure 1.1C, the exposure time-series is much more variable and
violates the criterion, but the average concentration for the overall time-series is lower than in the
two time-series that do not violate the criterion. Given that the peak concentrations in this time-
series are only present for a short portion of the duration of the toxicity tests used to derive the
criterion, does this time-series truly represent more severe effects than the other two?
These issues and questions can cause difficulties and uncertainties in applying criteria to
the management of discharges and the interpretation of ambient monitoring data. Because
criteria represent low, incompletely-defined levels of effect and because the magnitude of effects
expected from concentrations above criterion concentrations are not quantified, it is difficult to
assess the significance of occasional, minor-to-moderate exceedences of the criteria. To satisfy
the requirement that criteria exceedences be rather rare, site assessments might be more
conservative than actually needed to limit effects to acceptable levels. On the other hand, the
prolonged effects of exposures being near, but not exceeding criteria, are also uncertain. In
general, risk management is made more difficult when the risks associated with criteria are not
well-quantified and are not comparable across different exposure scenarios.
1.4 Expanded Criteria Formulation to Better Quantify Toxic Effects
Improving the quantification of effects in aquatic life criteria will involve a variety of
other issues, but it must start with better descriptions of the relationship of toxic effect levels to
concentration and time. Toxicity test analysis often involves assessing just one level of effect at
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Figure 1.2. Relative impacts of a concentration
time-series (panel A) using a time-dependent
toxicity model (panel B) versus exceedence of a
criterion concentration (panel C).
the end of the test (e.g., 96-h ZCso). This should be expanded to include a range of effect levels
as a function of both concentration and time. Furthermore, this toxicity relationship should
accommodate concentrations being time-variable, rather than roughly constant as is the case for
most toxicity tests.
With such a toxicity relationship, the
impact of any concentration time-series (e.g.,
Figure 1.2A) can be expressed as a quantitative
function of time (Figure 1.2B). This is in
contrast to current criterion formulations which
treat any concentration (over a specified
averaging period) below the criterion as being
acceptable and any concentration above the
criterion value being unacceptable (Figure
1.2C).
Furthermore, this time-series of toxicity
levels can be used to specify the frequency of
any given level of effect within an exposure
Time
time-series. This supports risk characterizations
such as depicted in Figure 1.3, which shows the
risk (=probability of occurrence) for a specified
level of effect versus the mean exposure
concentration. Figures 1.2 and 1.3 were
developed for a particular type and level of
Figure 1.3. Risk of a specified type and level of effect
versus mean exposure concentration.
ju 0.03
o
.£ 0.01
5
ro
_Q
0.001
0.5 0.6 0.7
Mean Concentration
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effect and for an exposure time-series with certain variability characteristics, using one of the
toxicity models discussed later in this report. However, the specifics of these calculations are not
important here, because this is intended to just exemplify the type of analyses that are possible,
which could involve a variety of models, types of effects, and exposure scenarios. Such analyses
could also address measures of exposure other than mean concentration and the uncertainty of
such risk estimates.
With information such as that shown in Figure 1.3, a variety of risk assessment and
management actions regarding water quality criteria could be improved. In general, there would
be clearer meaning of the significance of exposures at or near the criteria limits and better
separation of risk assessment and risk management. The level of exposure to be permitted from
point or nonpoint sources could be related to specific levels of risk, and these levels of risk could
be made more comparable across different exposure conditions. Ambient monitoring data could
be assessed on a quantitative scale, rather than simply determining whether a semi-quantitative
risk is exceeded. Such a quantitative scale could also be used in the interpretation of effects
observed in natural and experimental ecosystems, and thus improve understanding of and
decisions about risks represented by criteria. A significant aspect of all these improvements
would be that criteria attainment need not be based on extreme value analysis of rare
exceedences of criteria concentrations; rather, expected effects could be assessed based on
more-easily measured characteristics of the exposure (e.g., the mean as in Figure 1.2), making
implementation easier and more meaningful.
It should finally be noted that this approach eliminates the distinction between "acute"
and "chronic" effects by addressing the effect of exposure time-series for any endpoint of
interest. Therefore, rather than having separate criteria concentrations based on acute and
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chronic tests, criteria would include toxicity relationships addressing different endpoints.
Because the effect of time is included in these relationships, they could be merged into a single
relationships based on whichever endpoint is most affected under particular exposure conditions,
or based on population dynamics models that integrate these endpoints. Incorporating such
expressions of risk into water quality criteria will also involve changes in how exposures are
expressed (in particular, averaging periods would no longer be part of the criterion formulation)
and how toxicity information across species is integrated. However, the work here will not
consider the overall formulation of the criteria, but rather restrict itself to models for describing
toxic effect levels for specific endpoints as a function of concentration and time, which is a
necessary first step in criteria changes.
1.5 References
Stephan CF, Mount DI, Hansen DJ, Gentile JH, Chapman GA, Brungs WA. 1985. Guidelines
For Deriving Numerical National Water Quality Criteria For The Protection Of Aquatic
Organisms And Their Use. NTIS PB 85-227049, U.S. Environmental Protection Agency,
Washington, DC, USA.
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Section 2: Toxicity Model Formulations - Binary Endpoints
This section will review a variety of models used in aquatic toxicology for describing the
relationship of binary toxic effects (yes/no endpoints such as death) to concentration and time.
Two broad classes of models will be presented - "deterministic" models for which an individual
organism will or will not respond as a strict function of the exposure, and "stochastic" models for
which an individual organism might or might not respond, the probability of the organism's
response being a function of the exposure. For convenience, these models will be discussed in
terms of mortality, but will apply to other binary endpoints as well.
2.1 Deterministic Models
2.1.1 Single-Compartment, Lethal-Accumulation-Threshold Model
Although toxicity to aquatic organisms is typically referenced to chemical concentrations
in exposure water, this is done with recognition that effects of chemicals nearly always depend
on the chemical being accumulated into an organism. Studies on various chemicals, organisms,
and endpoints have related effects to the extent of accumulation (see review by Jarvinen and
Ankley, 1999). Toxicity models that relate effect levels to water concentrations can be
developed by combining information on the relationship of effect levels to accumulation
(toxicodynamics) with models or evaluations that address the rate and extent of accumulation
(toxicokinetics) for exposures of interest (e.g., McCarty and Mackay 1993). The simplest such
toxicity model for lethality can be formulated as follows:
(1) An organism accumulates chemical by first-order, single-compartment kinetics (i.e.,
the accumulated concentration in the organism is described by a single, whole-body
value, the gross uptake rate is proportional to the exposure concentration in the water, and
the gross elimination rate is proportional to the accumulation):
dA(t)
dt
= kv-C(t)-kE-A(t) (2.1)
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where A(f) is the accumulated concentration in the organism at time t, C(f) is the exposure
concentration in the water, kv is an uptake rate constant, and kE is an elimination rate
constant. For this model, accumulation at time t can be calculated for any exposure
concentration time-series by numerical integration of Equation 2.1, or by evaluation of
the following integral:
k fx=t
= f (C(x)-k e~kE'(t~x)\dx (22)
= BCFss-C(t)
where ^0 is an earlier time at which accumulation is zero or low enough that it contributes
negligibly to any accumulation at time t; BCFSS (=ku/kE) is the steady-state
bioconcentration factor; and C(t) denotes a weighted running average of the water
concentration, the weighting factor for this average decaying exponentially backward in
time in accordance with the constant kE. By integrating effects of exposure from t0 to t,
Equation 2.2 reflects the fact that accumulation depends on both current and past
exposure concentrations, with the relative importance of the concentrations decreasing
the further back they are from the current time. If C is constant with time and exposure
starts at t=Q, the accumulation at time t would be:
(2.3)
(2) An organism dies when A(t) reaches a lethal threshold LA. Toxicity is assessed
simply by tracking whether A (t) exceeds LA:
LA
where F(t) is the fraction of the lethal condition reached; i.e., the exposure is great
enough to cause mortality if and when F(t) reaches 1.
(2.4)
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Figure 2.1 illustrates how this model
describes if and when mortality will occur. Panel
A depicts a high enough exposure concentration
thatA(i) reaches LA quickly. Panel B uses the
same model parameters as Panel A, but has a
lower exposure concentration that requires more
time to reach the lethal condition. Even lower
exposure concentrations would further delay
accumulation reaching the lethal level, and
mortality would never occur if the exposure
concentration is below that needed to reach LA at
steady state (C-BCFSS
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The model described by Equations 2.1 to 2.4 strictly applies just to a single organism, and
model parameters (LA, k\j, and fe) would be expected to differ among organisms. For groups of
organisms, these parameters would vary according to statistical distributions that need to be
addressed to be able to evaluate statistics such as a median lethal concentration (LCso). Thus, to
fit this model to actual toxicity data, parameter estimation involves not just these three
"organism-level" parameters, but rather a greater number of "distributional" parameters (e.g., the
mean and standard deviation of a distribution for each organism-level parameter).
Although this accumulation-based model allows effects to be expressed as a function of
water concentrations, it requires explicit information on toxicokinetics and on the lethal
accumulation threshold, which is often not well established. A study that addressed the effects
of constant- and fluctuating-exposures of pentachloroethane on fathead minnows in relationship
to accumulation will be presented in the second report in this series. However, requiring explicit
information on accumulation and its relationship to toxicity precludes the use of abundant data
that relate toxicity just to water concentration. Fortunately, this model can be adapted to
describe the relationship of toxicity to exposure water concentration without explicitly
quantifying accumulation.
The relationship of LCsoS to exposure duration for aquatic animals often is observed to
follow a shape similar to Figure 2.2, with ZCsoS declining exponentially from high values at short
durations to a steady value ("asymptotic" or "threshold" ZCso) at long durations. This curve is
idealized and, in practice, its shape can be more complicated due to toxicity consisting of
multiple steps or multiple mechanisms that operate on different time scales, and due to
physiological responses to the toxicant which alter susceptibility. Nevertheless, a decline similar
to that in Figure 2.2 is usually somewhat evident, provided measurements are made over a time
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Figure 2.2. Idealized relationship of lethal
concentrations to exposure duration.
5
c
0
'*= 4
£
c
0
o
§ 3
O
"ro
c
ro
T3 "1
0
2
Q
pi
1
1
I
1
- \
\
\
x^
"Threshold" or "Asymptotic" LC50
i i i i i
0 20 40 60 80 100
Exposure Duration
frame suitable for the chemical and test species.
Units for time or concentration are not given in
Figure 2.1 because these will vary among
chemicals and test organism - for some cases, this
curve might span just minutes, and for other cases
it might span months or more.
Zitko (1979), Mancini (1983), Neely
(1984), and Chew and Hamilton (1985) noted that
such an exponential decline is consistent with,
and plausibly attributable to, a model in which chemical is accumulated by first-order kinetics
and in which death occurs when this accumulation reaches a lethal threshold (i.e., the model
described in Equations 2.1 to 2.4). At short durations, high concentrations are needed to
accumulate enough chemical fast enough to reach the lethal accumulation threshold quickly. As
duration increases, lower water concentrations will cause mortality because there is more time
for chemical to accumulate. With even greater duration, accumulation approaches steady state,
and the lethal water concentration will approach an asymptotic value equal to the LA/BCFss (i.e.,
accumulation can never be high enough to elicit mortality if the water concentration is less than
this). The rate at which this asymptotic lethal water concentration is approached will be
equivalent to the rate at which steady-state accumulation is approached.
Explicit information on accumulation in the model represented by Equations 2.1 to 2.4
can be eliminated simply by dividing Equation 2.1 by LA so that it can be expressed in terms of
the fraction of the lethal accumulation (F(f)) that is present:
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at
(25)
dt LA
where LC«, (=LA/BCFss) is the threshold lethal concentration, the water concentration that would
result in accumulation equal to LA at steady state. This equation simply states that the fraction of
the lethal condition increases with time in proportion to water concentration and decreases with
time in proportion to itself.
Whether mortality can be expected for any time-series can be assessed by numerical
integration of Equation 2.5 or by the following general integral for Equation 2.5, analogous to
Equation 2.2 for accumulation:
^V / \s \SlsJ\s
LA )
^'(C(x)-kE-e^E'(t^})dx (2.6)
Jx=tn \ '
VU ' 'VE
LA
C(t)
?
>1
This equation embodies the perspective that toxicity at any time depends on both current and
past exposure concentrations, with the relative importance of the concentrations decreasing the
further back they are from the time in question. This is an intuitively reasonable concept, and
Equation 2.6 simply provides an expression that describes this weighting.
Compared to Equations 2.2 and 2.4, the water concentration-based expressions of
Equations 2.5 and 2.6 have the advantage of having just two parameters (LCm and fe) rather than
three because k\j and LA are not separable parameters when accumulation is not explicitly
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addressed. For a constant exposure water concentration, the relationship between lethal
concentration and time-to-death (fo) is:
LC= " (2.7)
l-Q-^D V '
which provides the form of the curve presented in Figure 2.2. This equation uses both LC and fo
to emphasize that, whether time-to-death is examined as a function of concentration or lethal
concentration as a function of exposure duration, the same relationship applies.
Again, these equations strictly apply just to single organisms, but can be easily extended
to groups of organisms by treating each of the two organism-level parameters as a distribution
rather than a single value. Thus, this model might involve four parameters consisting of means
and standard deviations for both LC«, and kE. These distributional parameters can be estimated
from standard, constant-concentration, fixed-duration toxicity tests using Equation 2.7, provided
that mortality is monitored for a sufficient number and range of observations times and test
concentrations to encompass an adequate range of effect levels. Parameter estimation methods
will be addressed in Section 2.3 and in the case studies presented in this and later reports. Once
these parameters are estimated, they can be used to predict effect levels for any exposure time-
series using Equation 2.5 or 2.6.
This toxicity model is very simplistic considering the variety of processes and
compartments involved in the accumulation of chemical and the elicitation of toxic effects.
However, for many organisms, chemicals, and exposure conditions of interest, this simple model
might still be adequate for describing toxicity relationships to some acceptable approximation.
For aquatic life criteria, even such an approximate model will provide valuable information on
issues that are currently not well addressed. Nonetheless, consideration should be given to when
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and how additional complexities might be appropriate. The following subsections will discuss
some features that might be part of more complicated deterministic models.
2.1.2 Damage-Repair Models
One simplistic assumption in the single-compartment, lethal-accumulation-threshold
model discussed above is that an organism will die immediately upon reaching a lethal
accumulation threshold, but survive indefinite exposures just below the threshold. More
realistically, once chemical is accumulated, any overt expression of toxicity involves a series of
biochemical reactions with kinetic constraints that might affect time-to-death as much as, if not
more than, accumulation kinetics. To address this issue, some toxicity models relate mortality to
reaching a threshold level of biological damage rather than a threshold of chemical accumulation
(e.g., Connolly 1987, Breck 1989, Ankley et al. 1995, Landrum et al. 2004).
A simple model for damage as a function of chemical accumulation is:
dD(t]
dt
= kD-A(t)-kR-D(t) (2.8)
where D(f) is the accumulated damage to the organism at time t, kv is a damage accrual rate
constant, and &R is a damage repair rate constant. Consideration of damage repair is necessary
because otherwise damage incurred at some past time is considered to persist undiminished and
be as important to effects as damage occurring more recently. Without assuming such repair in
Equation 2.8, damage would increase indefinitely even at low exposures, resulting in no lower
limit on effect concentrations as duration increases (i.e., no threshold effects concentration).
Such zero-threshold models might be appropriate for some applications, and would be a subset
of the more general treatment discussed here. An even more general treatment would be to
assume that damage accrues only from accumulation that exceeds some threshold level; in such
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case, consideration of repair might be less important, but an additional parameter specifying this
threshold accumulation would be needed.
If death occurs upon reaching a lethal damage threshold LD, then this model can be
expressed in terms of the fraction of the lethal condition reached as follows:
dt LD
> \dx (2.9)
\LAx )
where LA^ (=LD-k^/k^) is the minimum accumulation for which the lethal damage can be
reached at indefinite time. To estimate values for L^ and &R requires information on effects and
accumulation over a range of exposure conditions, analogous to parameter estimation for the
relationship of effects to exposure concentration described in Equations 2.5-2.7. Toxicity can
then be related to water concentration by combining Equation 2.9 with Equation 2.2. However,
this again requires considerable information on accumulation and the relationship of effects to
accumulation, and cannot be applied to information that just relates toxicity to water
concentration.
As for the lethal-accumulation-threshold model, this damage-repair model can be adapted
to eliminate the need for explicit information on accumulation, by incorporation of the
expression for accumulation from Equation 2.2 into Equation 2.9:
(2.10)
c(t)>,
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where C(Y) denotes a weighted running average of the water concentration that reflects both kE
and &R, and the threshold lethal concentration LC now equals LAJBCFss-
This model can also be expressed in terms of water concentrations by combining the first-
order differential equations of Equations 2.1 and 2.9 into the second order differential equation:
(2 n)
[kE.kj dt2 u-O dt LC*
For a constant exposure concentration, this differential equation has the following general
solution:
_t, C
F(t) = Z-e-"*
- (2.12)
-k't
Where PI and PI are integration constants depending on initial conditions. For zero accumulation
and damage at t=0, Pi=-C-k^/(k^-kE) and P2=-C-kE/(kE-k^) when &E^£R, andPi=-l and P2=-k
when kE=kx=k. Substituting into Equation 2.12 the lethal condition F=l, C=LC, and t=tD, the
relationship between lethal concentration and time-to-death for this model is:
LC = -
if
""R
IT If- If- If-
11 KE - KR - K
The formulation on the second lines of Equations 2.12 and 2.13, for &E=&R, is needed not only
when this equality is exactly true, but also when these parameters are approximately equal,
which causes parameter estimates to become uncertain, so that equating these parameters
improves error estimates.
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Although Equations 2.7 and 2.13 both involve exponential declines to a threshold lethal
concentration, the equations represent different shapes (Figure 2.3) which can be discriminated
with suitable data. The solid line in Figure 2.3 denotes the single-compartment, threshold-lethal-
accumulation model of Equation 2.7 with LCm=\
and &E=0.05. This is the same curve as Figure
2.2, except plotted on a log/log scale, on which
the slope for this model at early times approaches
-1 for all values of kE.
The dashed line denotes the damage-repair
model of Equation 2.13 withZCco=l and
&E=&R=0.2. These values for k^ and &R were
arbitrarily set equal in this example and their
shared value was selected so that the average LC
over the time range shown was approximately the
same for the solid and dashed lines. The
Figure 2.3. Effects of damage-repair kinetics on
the relationship of lethal concentration to
constant concentration exposure durations. The
single-compartment, threshold-lethal
accumulation toxicity model ( ) (no
damage/repair component), is contrasted with
damage/repair models with kR equal to ( )
and ten-fold greater than ( ) kE.
50
20
10
O
-I 5
2
1
10 20
50 100
relationship for the dashed line is much steeper than for the solid line, approaching a log-log
slope of-2 at early times, because it combines the kinetic constraints of both accumulation and
damage-repair. At short exposure durations, concentrations must be especially high to cause
both substantial chemical accumulation and quick accrual of lethal damage. As exposure
duration increases, the LC drops more quickly than the single-compartment, threshold-
accumulation model because damage accrual is accelerating as accumulation increases.
The dashed-dotted line denotes the model of Equation 2.13 withZCoo=l, &E=0.05, and
&R=5. This much larger value for &R results in little shift from the solid line except at very early
23
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times, although the slope for the dash-dotted line still approaches -2 at sufficiently small times
off the scale of this graph. This simply indicates that very rapid damage-repair kinetics relative
to accumulation kinetics (kR>kE) will cause Equations 2.10 and 2.13 to be approximately
equivalent to Equations 2.6 and 2.7, respectively, except at very early times. Furthermore, if the
accumulation kinetics are much faster than the damage-repair kinetics (k£>kR), Equations 2.10
and 2.13 also become approximately equivalent to Equations 2.6 and 2.7 except at very early
times, with &R substituted for k^.
As such, Equations 2.6 and 2.7 do not just represent a toxicity model based on single-
compartment toxicokinetics and lethal-accumulation-threshold toxicodynamics. Rather, they can
be considered to represent a broader set of models in which the kinetic constant can represent
any process regulating the effect of time on toxicity, not just accumulation (Connolly 1987,
Breck 1989), or can represent the combined effect of multiple processes is this is approximately
first-order. Subsequent use of Equations 2.6 and 2.7 will thus use a kinetic constant k without a
subscript, indicating that the nature of the kinetic process(es) contributing to k are not necessarily
known, and do not actually need to be known if this single constant provides a reasonable
approximation for the toxicity relationships of interest. Similarly, Equations 2.10 and 2.11 can
be more broadly interpreted as describing toxicity for which the kinetics can be reasonably
approximated by two sequential processes, which do not need to be completely characterized for
this model to be useful.
2.1.3 Multiple Mechanisms of Action
Another way in which the relationship of mortality to exposure can be more complicated
than the simple model illustrated in Figure 2.2 is the existence of multiple mechanisms by which
toxicity is elicited. Toxicants can act on multiple biochemical systems, and these actions can
24
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differ with regard to both toxicokinetics and toxicodynamics, so that one mechanism might be
most important for determining lethal concentrations within certain ranges of concentration and
time, whereas other mechanisms would be important for other circumstances.
To extend the model described in Section 2.1.1 to two mechanisms simply requires
applying Equations 2.5 and 2.6 to each mechanism as follows:
where A and B refer to the two mechanisms, and, per previous discussion, k no longer has the
subscript denoting elimination because it is being treated as a more general kinetic coefficient
encompassing the entire toxicity process.
FA(f) and FB(f) must be combined to specify how the two mechanisms jointly contribute
to the overall toxic condition F(f). One possibility for this is that the two mechanisms are
completely independent, so that death occurs when either FA(f) and FB(t) exceeds 1.0, and thus
F(f) will be equal to the larger of these two fractions. For constant concentration toxicity tests,
the relationship of lethal concentration to time-of-death would therefore be:
-] (2.15)
» '\-e-
25
-------
Another possibility is that the two mechanisms additively contribute to damage, so that F(f) is
the sum of FA.(i) and FB(f). For a constant concentration toxicity test, the following relationship
would therefore apply:
1
LC =
1-e"
(2.16)
£C
Figure 2.4 illustrates how such models
would deviate from the simple model of Equation
2.7. The bold solid line in Figure 2.4 again
denotes the model of Equation 2.7 with LCm=\
and A=0.05. For the two-mechanism models, it is
assumed that LC^B is 4-fold higher than LC^A
and that £3 is 40-fold faster than &A, and all
parameters are scaled to produce LCm=l and
approximately the same average LC as the solid
line. The dashed line denotes the model of
Equation 2.15, for which the mechanisms are
Figure 2.4. Effects of multiple mechanisms of
toxicity on the relationship of lethal
concentration to time-to-death. A single-
mechanism, single-compartment, threshold-
lethal accumulation toxicity model ( ), is
contrasted with two-mechanism models with
independent ( ) and additive ( )
mechanisms.
50
20
10
O
-I 5
2
1
5 10 20
^D
50 100
independent. The portion of the dashed line below the solid line indicates how mechanism B
results in lower lethal concentrations at short durations because of its faster kinetics. However,
the high LC^B results in the dashed line crossing the solid line and causing higher ZCs until
intersecting the relationship for mechanism A, which controls toxicity at longer durations
because of the low LC^A- The dash-dotted line denotes the model of Equation 2.16, for which
the mechanisms are additive. The assumed additivity results in a gradual transition between the
26
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two mechanisms, resulting in a smoother curve, but still with two phases reflecting the different
kinetics of the two mechanisms.
2.1.4 Multicompartment Toxicokinetics
The first-order, single-compartment toxicokinetics model described by Equations 2.1 and
2.2 is a highly simplified approximation for chemical accumulation, which might or might not be
an adequate approximation in a toxicity model for a specific chemical, organism, endpoint, and
exposure scenario. An organism consists of various morphological compartments that
accumulate and process chemicals at different rates, such that the concentrations in each
department will have different time-dependencies. Such differences might be large enough to be
important for the time-dependence of toxicity. Physiologically-based toxicokinetic (PBTK)
models have been developed to describe the accumulation and speciation of chemicals in various
compartments in aquatic organisms (e.g., Nichols et al. 1990), and can be a part of accumulation-
based toxicity models. However, PBTK models require considerable physiological,
morphological, and chemical partitioning information for parameterization, and their application
to toxicity predictions requires relating effects to accumulation in a specific compartment. Their
use in aquatic life criteria will be for specific circumstances and require special considerations,
and they will not be addressed in this report.
However, to some approximation, multicompartment kinetics can also be addressed more
empirically and more simply by extending Equation 2.1 to describe multiple compartments,
which is well-established practice in toxicokinetics and pharmacokinetics research (e.g., Gibaldi
and Perrier 1982). The simplest such modification is to treat an organism as consisting of two
compartments, with first-order chemical exchange between the external environment
(compartment 0) and compartment 1 and between compartments 1 and 2:
27
-------
(2.17)
nl2l2
where M;(Y) is the mass of chemical in compartment /' and ky is a transfer rate coefficient from
compartment /' toy (e.g., km is the coefficient for transfer from the external compartment to
compartment 1).
For C constant with time, Equation 2.17 can be integrated to produce:
- -
P-a a-p
(2.18)
A(t\ = M^+M^
V ' W
where Wis the organism weight and a and flare functions of #10, ku, and k2i such that cfrf3=
kw+ku+kn and a-f3= kw-k2\ (Gibaldi and Perrier 1982). With this empirical approach, the
identity of the compartments is typically not completely characterized, so thatMi(^) andM2(i)
are not directly measured. Rather, Equation 2.18 (or comparable equations for other exposure
scenarios) is used to analyze A(f) from accumulation and elimination experiments to estimate the
four model parameters (transfer rate coefficients), which in turn can be used to estimate M\(i)
andM2(f).
To apply such a multicompartment toxicokinetics model to toxicity assessments would
require information on the relationship of effects to accumulation that would allow specification
of the lethal accumulation threshold in terms of either or both compartments. While such an
application is plausible, and desirable when appropriate data are available, it would not address
typical toxicity evaluations based just on water concentrations. As for the single compartment
28
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model, these multicompartment models can be applied to interpretations of water-based toxicity
evaluations without needing to quantify accumulation or the relationship of effects to
accumulation, provided that the relative contributions of the two compartments to the lethal
condition can be inferred. For a lethal accumulation threshold in compartment 1, the relationship
of lethal concentration to time-to-death for a constant exposure concentration is:
LC =
(2.19)
For a lethal accumulation threshold in compartment 2, this relationship is:
LC =
-1
-
_
fl-a
(2.20)
Equations 2.19 and 2.20 provide shapes for the relationship of lethal concentrations to
time that are different from the single-compartment toxicokinetics model of Equation 2.7 and
from each other (Figure 2.5). The solid line in
Figure 2.5 again denotes the model of Equation
2.7 with ZCoo=l and #E=0.05. The dashed line
denotes the model of Equation 2.20 (lethal
accumulation threshold in compartment 2) with
ZCx=l, &io=0.2, #12=0.02, and #2i=0.2. The
relative values for these parameters represent a
situation for which the kinetics of exchange
between the water and compartment 1 and
between compartments 1 and 2 both are important
to the kinetics of mortality in the time frame of
Figure 2.5. Effects of multicompartment
kinetics on the relationship of lethal
concentration to constant concentration exposure
duration. A single-compartment, threshold-
lethal accumulation toxicity model ( ), is
contrasted with two-compartment models with
lethal accumulation threshold in inner ( )
and outer ( ) compartment.
50
20
10
O
-I 5
2
1
10 20
^D
50 100
interest, and the absolute values of these parameters are again scaled so that the average LC is
29
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similar for the different lines. This produces a biexponential decline similar to the damage/repair
model depicted in Figure 2.3; in fact, the model of Equation 2.20 cannot be distinguished from
that of Equation 2.13, which reflects the fact that they both represent two sequential processes
leading to toxicity (single-compartment accumulation followed by accumulation of damage,
versus accumulation in an outer compartment followed by accumulation into an inner
compartment). This reemphasizes the merits, when applying these models to water
concentration-based toxicity data, of not attributing specific mechanisms to the processes causing
toxicity and to recognize that various mechanisms might be responsible for data relationships.
In contrast, the model of 2.19 (lethal accumulation threshold in compartment 1) provides
a two-phase relationship shown by the dash-dotted line in Figure 2.5, for which LC«,=l, kw=0.2,
&i2=0.5, and #12=0.5 . At early times, the LC decreases with time because accumulation in
compartment 1 is increasing and the influence of uptake into compartment 2 is not yet
significant. As time increases, the decrease in LC slows down because uptake into compartment
1 is largely offset by loss of chemical to compartment 2. At even greater times, compartments 1
and 2 approach steady-state with respect to each other, so that more of the uptake from water is
retained in compartment 1, and the log-log slope of the LC thus becomes steeper again, until the
asymptotic LC is approached. This biphasic relationship is similar to the two-mechanism model
of Figure 2.4 and Equation 2.16, and, in practice, distinguishing these models would be difficult,
and probably not important because they would produce similar results. However, Equation 2.19
would be more difficult and uncertain to parameterize because it does include one more
parameter than Equation 2.16.
As for the models in Sections 2.1.1, 2.1.2, and 2.13, the parameters in Equations 2.19 or
2.20 can be estimated from the results of constant concentration toxicity tests provided that the
30
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multicompartment kinetics are important enough to exert appreciable effects over the time-frame
of the data. For the model represented by Equation 2.19, the parameters so estimated can be
used to derive the kinetic constants of Equation 2.17 and 2.18, and then used to calculate toxicity
under time-variable exposures. However, this is not true for the model represented by Equation
2.20, for which parameters estimated from constant concentration toxicity tests are not sufficient
to uniquely specify all the kinetic constants, thus requiring additional assumptions or information
to address time-variable exposures. Therefore, because the multiple compartment of Equations
2.19 and 2.20 provide relationships that (a) are not substantially different from the models of
Equations 2.13 and 2.16 and (b) present more difficulties in parameterization and predictions of
fluctuating exposure effects, they will not be used further in the efforts described in this series of
reports.
2.2 Stochastic Models
2.2.1 Distribution of Time-to-Death and Hazard Rate
The models in section 2.1 are referred to as deterministic because they are premised on
the assumption that any individual organism will either die or not die for any specified exposure
conditions based on a fixed relationship for that individual. Variation between organisms arises
from them having different values for the model parameters. Even if each parameter varies
among different individuals in accordance with a statistical distribution, the response of any
individual is still deterministic. An alternative approach for modeling mortality has its origin in
the statistical analyses of time-to-event such as component failure, life expectancy, etc. Such
statistical tools have been used in environmental risk assessment and toxicology for describing
survival times under toxic chemical exposures (e.g., Dixon and Newman 1991, Newman 1995,
Crane et al. 2002), and these sources provide the basis for the discussion here.
31
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A basic variable of interest in an analysis of survival versus time is the "survivor
function" S, an expression of the statistical distribution of the survival times of the organisms:
S (t, CJ = Probability of test organism survival to time / when exposed to concentration C (2.21)
For a stochastic approach, the survivor function is a cumulative function of the "hazard rate"
h(t,C), the probability of death per unit time per surviving individual:
*(,.c)=' . (2.22)
V ' S(t,C) dt ^ }
The hazard rate represents a stochastic perspective because it specifies a probability that an
organism will die in a given time interval, not a certainty based on the specifics of the exposure
and the model parameters for the organism. Different organisms will die at different times, or
not at all, based partly on random chance, not because of inherent differences between the
organisms (although the hazard rates could be specified to depend on organism attributes if
appropriate). The hazard rate also provides an effective basis for addressing time-variable
exposures because it specifies the instantaneous risk to survivors at any time and therefore can be
used to integrate this variability to estimate the survivor function, by the relationship:
-[* h(t,C(t)\dt
S(t,C(t)} = e Jo l ; (2.23)
2.2.2 Specifying the Hazard Rate
If the hazard rate is constant with time for specific exposure conditions, the survivor
function would have the simple exponential form e~at , analogous to radioactive decay.
However, the survivor function is usually not so simple because, even for a constant exposure
concentration, the hazard rate can change with time. Surviving organisms might be more likely
to succumb upon longer exposure because of increasing chemical accumulation, cumulative
damage, etc., or be less likely to succumb because of compensatory mechanisms, greater
32
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resistance in survivors, etc. The dependence of hazard rate on time must therefore be addressed.
For constant exposure tests, this issue has been addressed by specifying a statistical distribution
for the survivor function (i.e., a statistical distribution for survival times), which can then be
related to the hazard rate using Equations 2.22.
The relationship of hazard rate to exposure concentration can be determined based on the
differences in survivor functions and hazard rates across different exposures. The combined
effect of time and exposure concentration (and other factors such as organism attributes or
physicochemical test conditions) sometimes has been described using a "proportional hazards"
model:
h(t,C) = eg(c)-h0(t) (2.24)
which multiplies a baseline (control) hazard rate ho(f), which incorporates the basic form for the
time-dependence of hazard, by a factor eg^'that is a function of the chemical exposure
concentration. Another model commonly used is the "accelerated failure time" model,
£ (2.25)
where the function g(C) describes the relationship of the median log time-to-death to exposure
concentration and the random variable (£) describes the variability of log time-to-death around
the median. This accelerated failure time model specifies how the survivor function distribution
would vary with exposure concentration, so Equation 2.22 can be used to specify how the hazard
rate varies with exposure.
Although Equations 2.24 and 2.25 incorporate an effect of time and thus might appear to
be applicable to the time-variable calculations indicated in Equation 2.23, this actually is not the
case. The basic difficulty regarding this is that the effect of time in Equations 2.24 and 2.25 is
not some independent function of time, but depends on the exposure history and thus applies
33
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only to the constant exposure in question. If the concentration changes with time, these
equations do not reflect cumulative effects that occurred from the old concentration(s), but rather
cumulative effects that would have occurred if the exposure had been at the new concentration
all along. Equations 2.24 and 2.25 can be applied to time-variable exposures only if the hazard
rate is time-invariant, which implies an instantaneous achievement of the hazard for any
concentration. For any case in which hazard is time-dependent at a constant concentration,
Equations 2.24 and 2.25 do not conceptually address the effects of time-variable concentrations,
so that additional model specifications regarding the effect of time on hazard are needed.
One approach for applying the hazard rate concept to time-variable exposures was
developed by Kooijman and coworkers (e.g., Kooijman and Bedaux 1996a, 1996b). Their model
is based on chemical accumulation, using the same first-order, single-compartment kinetics
model discussed in Section 2.1.1 (Equations 2.1 to 2.4). The hazard rate is assumed to be
linearly proportional to the degree to which chemical accumulation exceeds a threshold
accumulation value, and to change instantaneously as accumulation changes:
h(t) = max(o,d-(A(t)-A0)) (2.26)
where Aft) is measured or estimated as in Equations 2.2 and 2.3, Jis a proportionality constant
called the "killing rate" (Kooijman and Bedaux 1996a, 1996b), and^o is an accumulation
threshold for effects. It is thus the kinetics of accumulation that produce the time-variability of
the hazard rate for constant exposures and allow extrapolation to time-variable exposures.
This model has four parameters [fe and BCFss (or k\j and fe) for the toxicokinetics
(Equations 2.1 to 2.3) and A0 and d for the toxicodynamics] if accumulation and the relationship
of effect levels to accumulation are explicitly addressed. As was the case for the deterministic
model, this stochastic model can also be formulated to be referenced to water concentrations
34
-------
without explicitly including accumulation. When the water concentration (C) is constant,
(2.27)
where Co is a threshold water concentration for effects (=Ao/BCFss) and d' (=d/BCFss) is a
killing rate referenced to water concentrations rather than accumulation. For this formulation,
the three parameters are fe, Co, and d', one fewer than the corresponding deterministic model of
Equation 2.6. For any arbitrary time-series C(t), the hazard rate would be:
h(t) = max(o,d'-(C(t)-C0)) (2.28)
where C(t) is computed as in Equation 2.6.
Unlike the deterministic model examples
shown on Figures 2.2 to 2.5, an individual
organism does not have a fixed relationship for
lethal concentration versus time using this
stochastic model. However, a model comparison
similar to those in Figures 2.3 to 2.5 can be made
based on LCp, the concentration lethal top percent
of a group of organisms. In Figure 2.6, the bold
solid lines are LCpS for the single-compartment,
lethal-accumulation-threshold deterministic model
(Equations 2.7), with median LC=\ and median
£=0.05, and both parameters log-normally
distribution with a logio standard deviation of 0.2.
This line was computed based on Monte Carlo
Figure 2.6. Comparison of LC50 and LCW
versus time of the single-compartment,
threshold-accumulation deterministic model
( ) with median kE=0.05 and median
LC<*=\ ( ) to stochastic models with kE and
LC°° equal to these median values, but with
either a slow ( ) or fast ( ) killing rate.
50
20
10
cf
^l
2
1
0.5
1
10 20 50 100
LT,
50
20
10
o
o" 5
^l
2
1
0.5
50
1
10 20 50 100
LT.
10
35
-------
simulation, and for the ZCso is very close, but not identical, to the line on Figures 2.3 to 2.5 for
the LC of an individual organism with LCm=\ and A=0.05. The dashed lines are the LCp for the
stochastic model of equation 2.28, integrated to predict survival using equation 2.23, with C0=l,
&E=0.05, and a slow killing rate, d'=\. The dash-dotted lines in Figure 2.6 represents the
stochastic model when d'=WO.
For the ZCsoS, the stochastic model with the slow killing rate shows a steeper relationship
than the simple deterministic model, similar to the damage/repair model of Equation 2.13 (Figure
2.3) or the two-compartment model of Equation 2.20 (Figure 2.4). This is understandable
because, like these other models, this stochastic model includes two kinetic processes - the rate
of accumulation and the rate of mortality for a given level accumulation. The similarity of the
LC50s for the deterministic model and the stochastic model with the fast killing rate simply
demonstrates that this stochastic model is roughly equivalent to the single-compartment, lethal-
accumulation-threshold deterministic model when the (deterministic) toxicokinetics are much
slower than the (stochastic) toxicodynamics. Therefore, based on ZCsos, this stochastic model
would not be readily distinguishable from deterministic models discussed previously.
However, some notable differences between the deterministic and stochastic models are
evident when other LCps are examined. For the stochastic model, LCios are lower than the LCsos
at early times, but by a limited extent that depends on the killing rate. At longer exposure
durations, the LCios approach the same value as the LCsos because this stochastic model assumes
that all the organisms have the same sensitivity, so that their effect concentrations become the
same at durations long enough that stochastic differences have diminished. In contrast, for the
deterministic model, LCios can be lower than the LCsos both at early and later times, and the
extent of this difference is independently determined by the standard deviations of the parameter
36
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distributions. Therefore, the variation of LCpS at long versus short durations and the variation of
ZCps at short duration relative to apparent killing rates can provide a basis for determining
whether a deterministic or stochastic model best describes a data set. Of course, the stochastic
model could be expanded to include differences among organisms (e.g., Co could be a
distribution that varies among organisms, rather than a constant), but this would reduce the
importance of the stochasticity and increase the number of parameters.
As with the deterministic model, this stochastic model also can be extended to address
multicompartment kinetics, multiple mechanisms, and damage-repair, and could also include
different concepts regarding the relationship of
hazard rate to accumulation. A simple
modification could be to repeat Equation 2.27 for
two different mechanisms - perhaps one with
high k, d, and Co, so that organisms die quickly
but only at high exposure concentrations, whereas
another mechanism could have low k, d, and Co,
so that organisms continue to die at low exposures
at extended durations. Figure 2.7 illustrates this
Figure 2.7. LC50 vs time relationship for
stochastic model with multiple mechanisms.
50
20
10
o
in
O 5
^l
2
1
5 10 20 50 100
Time
possibility, and shows how the stochastic model can produce results similar to the multiple-
mechanism deterministic models in Figure 2.4.
2.3 Model Parameterization and Selection
2.3.1 Model Parameterization
This section will discuss general principles and approaches for parameterizing either the
deterministic or stochastic binary endpoint models. This general information applies to all the
37
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case studies that will be examined, and further details will be provided as needed in the specific
sections on each study.
For mortality in any toxicity test, the fundamental observation is usually how many
organisms subject to an experimental treatment die between one observation time and the next.
A specific time-to-death for an individual organism is rarely determined; rather, it is only known
that death occurred within the time interval between observations. Similarly, the concentration
needed to kill an individual organism at a specific exposure duration is also not measured; rather,
it is only known that this lethal concentration is between two treatment concentrations. Such a
data set can be perceived as a matrix of treatment concentrations and observation intervals, with
each element of the matrix containing the number of organisms dying during a specific interval
in a specific treatment. If J is the number of experiment treatments (concentrations) and / is the
number of observation times, let N be an 7+1 by /matrix for which the element Ntj (/'
-------
/ (2.29)
where:
^dt
For the deterministic mortality model described by Equation 2.7, an individual organism
will still be alive for a given Q and t if its LCX is greater than Cf(l-e~kt). Thus, for a group of
organisms for which f(k) andffLCaq) are the density functions for the statistical distributions of &
and LCX:
;
where :
(2.30)
^ '
where max and min refer to the maximum and minimum variable values in the distributions of &
Whichever type of model is used, estimates of the model parameters can be obtained
using computerized search algorithms to find the parameter values which maximize the
likelihood L of the observed mortality observations N as a function of the parameter set 0
(Breiman 1973):
j 1+1
(2.31)
For the stochastic model of Equation 2.29, the parameter set @ is Co, fe, and d. For the
deterministic model of Equation 2.30, the parameter set @ would be the parameters forf(k) and
39
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f(LCao), which will typically be a mean and standard deviation for each distribution.
For the case studies presented in this series of reports, custom software developed with
Intel Visual Fortran (Version 9.1, Intel Corporation) using EVISL library (Version 5.0, Visual
Numerics Incorporated) routines were used for this likelihood maximization and for other data
analysis. The search algorithm used was the Box Complex method (Box 1965). However, other
commercial mathematical and statistical software and other algorithms are also suitable for such
analyses. More computational details are provided as needed in the sections for each case study.
For the stochastic model, survivor functions for any time-variable exposure scenario of
interest can be predicted using Equations 2.23. For the deterministic model, predictions are best
conducted by Monte Carlo analysis. In such an analysis, a large number of organism-level
parameter sets (e.g., LCX, k) would be randomly generated from the distributions [e.g.,f(k) and
f(LCao)~\ defined by the maximum likelihood parameter estimates. The model would then be
applied to the exposure scenario of interest using each of these parameter sets to define if and
when each organism was expected to die. These mortality estimates then would be aggregated to
estimate the survivor function or other statistics of interest (e.g., LCsos at a specified time).
Model uncertainty can be addressed in various ways. Standard errors for the maximum-
likelihood parameter values can be obtained from the parameter variance/covariance matrix
estimated by inverting the information matrix for the maximum likelihood analysis (Breiman
1973). However, such standard errors generally underestimate actual uncertainties, especially
for pooled analyses of multiple data sets, because structural model error and variation of
parameters among data sets is not addressed. When data from multiple toxicity tests is available,
an improved assessment of parameter uncertainty can be obtained by estimating separate
parameters for each test, the variation of parameters among tests including both within- and
40
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between-test sources of error. Another alternative is to conduct a pooled analysis of the data
from all the tests, but to include between-test variability as part of the model and estimation
procedure. For example, for the deterministic model, it might be assumed that the means for the
distributions/^ and/fZCoo) for individual tests are in turn distributed with some overall mean
and variance. Case studies later in this series of reports will illustrate how such a broader
analysis might be conducted.
When appropriate estimates of model parameter uncertainty are available, the
uncertainties for model predictions can be obtained by Monte Carlo analysis in which a large
number of sets of parameter values are randomly generated from the parameter uncertainty
distributions. For each of these sets of randomly-generated parameter values, the model
calculations described previously would be conducted, providing a distribution of model results
for any statistic of interest, from which confidence limits can be obtained. For example, for an
LCso at a specified time, if this process produced 999 estimates, the 25th highest value and 25th
lowest value would provide approximate 95% confidence limits.
2.3.2 Model Selection
G.E.P. Box's dictum that "all models are wrong, some models are useful" is a useful
perspective in selecting a toxicity model for use in aquatic life criteria. It must be recognized
that any model provides just an approximation to the toxicity relationships of interest, and that
the selected model need not be perfect to serve some need. Furthermore, the model need not
even be the most complete and accurate available; rather, the most appropriate model would be
one that provides acceptable performance with the lowest complexity and data requirements.
The basic need of aquatic life criteria is for toxicity models which quantify the level of effect as a
function of exposure concentration and duration, including exposures which have a certain
41
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degree of time-variability. Any such quantification represents a fundamental improvement in
defining the risks associated with criteria compared to current practice, and there will be a need
to define specific performance criteria to determine model acceptability. What is the range of
exposure time-series that the model will address? What uncertainty measures are desired and
what level of uncertainty is adequate enough that more complex models are not needed? What is
the minimal amount of data upon which models must be parameterized for a given species, and
will this require that certain model parameters be based on default values derived from analyses
on other species and/or chemicals? This report will not specify such requirements, but will
present approaches and analyses that will assist in their development and application.
In choosing models for consideration, it is useful to examine how mortality varies with
exposure concentration and time (e.g., Figures 2.3 to 2.7), in order to determine what properties
the model should have. To serve this purpose, the case studies will include calculation of LCps
and LTps (the time to kill a percentage p of the organisms at a specified concentration). These
LCps or LTps can be used in simple figures and tables to illustrate data trends and to guide model
formulation more effectively than the matrix of observed deaths within each time interval and
treatment. However, LCps at different times are not statistically independent and do not
represent the fundamental data upon which parameter estimation and model fit must be assessed.
All parameter estimation will be done directly on the basic mortality observation matrix, using
LCps or LTps only for exploratory data analysis or to illustrate model fit.
Once models are selected for consideration and are parameterized with the data, their
relative goodness of fit will be based on the computed likelihood (L) using the Akaike
Information Criterion (AIC):
(2.32)
42
-------
where np is the number of model parameters and a lower AIC indicates a better model. This
formula recognizes that higher values for the likelihood statistic indicate better model fit, and
result in a lower AIC. However, models with more parameters (degrees of freedom) would be
expected to have a better likelihood statistic, so the AIC is increased by the number of
parameters to compensate for the effect of np. Therefore, a model with the lowest AIC would be
considered superior, regardless of the number of parameters.
However, the AIC is not informative regarding the magnitude of deviations of model
predictions from observations or the amount of data variability that is accounted for by the
model, such as the R2 statistic (the fraction of the variance of the dependent variable explained
by the regression) commonly used in regression analysis. Statistics such as R2 also can be used
to illustrate how well data or statistics derived from the data are described by a model, although
using R2 to address statistics such as LCps at different times must be done with recognition of the
lack of statistical independence among observations. Various other measures of the deviation of
observed and predicted ZCps or mortality levels (e.g., mean deviation, mean of the absolute
deviations, standard deviation of the difference) can also be used to describe model uncertainty.
However, whatever measures of fit are used, they should not be the sole basis for
selecting a toxicity model for use in aquatic life criteria. A simpler, but less accurate, model
might be preferred if the level of accuracy still satisfies specified performance goals for exposure
regimes of interst. In general, model selection will involve various statistical and nonstatistical
factors, and must be conducted in an ad hoc fashion.
2.5 References
Ankley GT, Erickson RJ, Phipps GL, Mattson VR, Kosian PA, Sheedy BR, Cox JS. 1995.
Effects of light intensity on the phototoxicity of fluoranthene to a benthic macroinvertebrate.
Environ. Sci. Technol. 29:2828-2833.
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Box MJ. 1965. A new method of constrained optimization and a comparison with other
methods. Computer J. 8:42-52.
Breck JE. 1988. Relationships among models for acute toxic effects: Applications to fluctuating
concentrations. Environ. Toxicol. Chem. 7:775-778.
BreimanL. 1973. Statistics: With a View Toward Applications. Houghton Mifflin, Boston,
MA, USA.
Chew, R.D. and M.A. Hamilton. 1985. Toxicity curve estimation: Fitting a compartment model
to median survival times. Trans. Amer. Fish. Soc. 114:403-412.
Connolly, J.P. 1985. Predicting single-species toxicity in natural water systems. Environ.
Toxicol. Chem. 4:573-582.
Crane M, Newman MC, Chapman PF, Fenlon J. 2002. Risk Assessment with Time to Event
Models. Lewis Publishers, Boca Raton, FL, USA.
Dixon PM, Newman MC. 1991. Analyzing toxicity data using statistical models for time-to-
death: An introduction. In: Newman MC, Mclntosh AW (eds). MetalEcotoxicology:
Concepts and Applications. Lewis Publishers, Boca Raton, FL, USA.
Gibaldi M, Perrier D. 1982. Pharmacokinetics. Marcel Dekker, New York, NY, USA.
Jarvinen AW, Ankley GT. 1999. Linkage of effects to tissue residues: Development of a
comprehensive database for aquatic organisms exposed to inorganic and organic chemicals.
SETAC Press, Pensacola, FL, USA.
Kooijman SALM, Bedaux JJM. 1996a. The Analysis of Aquatic Toxicity Data. VU University
Press, Amsterdam, The Netherlands.
Kooijman SALM, Bedaux JJM. 1996b. Analysis of toxicity tests onDaphnia survival and
reproduction. Water Res. 7:1711 -1723.
Landrum PF, Steevens JA, Gozziaux DC, McElroy M, Robinson S, Begnoche L, Chernak A,
Hickey J. 2004. Time-dependent lethal body residues for the toxicity of
pentachlorobenzene toHyalella azteca. Environ. Toxicol. Chem. 23:1335-1343.
Mancini JL. 1983. A method for calculating effects on aquatic organisms of time-varying
concentrations. Water Res. 17:1355-1361.
44
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McCarty LS and Mackay D. 1993. Enhancing ecotoxicological modeling and assessment.
Environ. Set. Tech. 27:1719-1728.
Neely WB. 1984. An analysis of aquatic toxicity data: Water solubility and acute LC50 fish
data. Chemosphere 7:813-819.
Newman MC. 1995. Quantitative Methods in Aquatic Ecotoxicology. Lewis Publishers, Boca
Raton, FL, USA.
Nichols JW, McKim JM, Andersen ME, Gargas ML, Clewell HJ, Erickso RJ. 1990. A
physiolotically based toxicokinetic model for the uptake and disposition of organic
chemicals in fish. Toxicol. Appl. Pharmacol. 106:433-447.
Zitko V. 1979. An equation of lethality curves in tests with aquatic fauna. Chemosphere 2:41'-
51.
45
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46
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Section 3. Short-Term Copper Lethality To Juvenile Fathead Minnows
3.1 Description of Study and Exploratory Data Analysis
Lindberg and Yurk (1982, 1983 a, 1983b) conducted a series of tests on the toxicity of
copper to juvenile (ca. 30-day-old) fathead minnows that consisted of (a) 31 constant-exposure
toxicity tests with durations ranging from 2.5 h to 192 h (including observations of mortality
after termination of exposure when exposure duration was <24 h) and (b) 6 pulsed-exposure
toxicity tests with pulse durations ranging from 2.5 to 12 h and pulse intervals ranging from 8 to
24 h. Five of the constant-exposure tests showed exposure concentrations which drifted with
time or were otherwise uncertain, and will not be used here. The rest of this data set will be used
here to demonstrate the use of constant-exposure tests for toxicity model parameterization and
selection, to test the validity of model assumptions, and to evaluate model applicability to time-
variable exposures.
Exploratory data analysis of the constant-exposure tests showed three attributes important
to model selection.
(1) For each test, LT50s were estimated for each concentration at which mortality
exceeded 40% by the end of the exposure. All the LT50s so computed are plotted versus
concentration on Figure 3.1 A, and demonstrate a strong relationship to water
concentration (Cw) similar to that depicted in Figure 2-2. The solid line denotes a
regression analysis for the relationship described in Equation 2-7 (conducted using
Sigmaplot, version 9.01, SPSS, Chicago, IL, the graphics software used to create this
figure) to illustrate that these data do approximately follow this relationship.
(2) However, closer inspection of the data shows certain deviations from the simple
exponential relationship. This is more clear in the log/log plot of Figure 3. IB of the same
LT50s. For times <24 h, the LT50s generally fall below the fitted regression line and, for
times between 24 and 96 h, they generally fall above the line. These deviations are even
47
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more clear in Figure 3.1C, which shows
LC50s calculated from just several tests for
which LC50s could be calculated for times
from 6 h or less to at least 96 h. In contrast
to a smooth single-phase exponential decline,
Figure 3.1C suggests two phases to the
response curve - a rapid mortality that is
starting to level off at about 300 |ig/L by 24
h, followed by a second phase in which LC50s
drop again to level off near 100 |ig/L. This is
the sort of behavior exhibited by the models
presented earlier which either have two
mechanisms (Figures 2-4, 2-7) or two-
compartment kinetics in which toxicity
reflects accumulation in an outer
compartment (Figure 2-5).
(3) Several of the constant-exposure tests in
this study had durations <24 h, with
monitoring of mortality after the cessation of
exposure. Figure 3.2 displays LC50s plotted
at the exposure duration, but either calculated
(a) based just on mortality that had occurred
by the cessation of exposure (filled symbols,
arrows denoting LC50s greater than the
indicated values) or (b) based also on any
mortality that occurred after the exposure
(open symbols). Virtually no delayed
mortality was present for 24 h exposures and
the difference between LC50s including and
excluding delayed mortality was less than a
factor of 1.1 for 12 h exposures; however,
this difference was about a factor of 1.25 for
8 h exposures, a factor of at least 1.4 for 4 h
exposures, and even greater for 2.5 h
exposures. The smaller LC50s when delayed
Figure 3.1 Panels A, B: Exposure concentration
versus LT50 ( ) for pooled tests of acute copper
lethality to juvenile fathead minnows, with fitted
exponential relationship ( ). Panel C: LC50
versus time for selected tests ( T A
Data not adjusted for delayed mortality.
Tlme(h)
Figure 3.2 LC50s at the end of short-duration
toxicity tests based on mortality during exposure
() or on mortality both during and after
exposure (O).
O
D)
' 500-
O 400-
O
O
O
O
O
I 8
Time (h)
48
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mortality is included further accentuate the two-phase behavior noted in Figure 3.1.
Deterministic models that have both an accumulation and a damage-repair component, or
that have effects related to accumulation in an internal compartment, could account for
mortality persisting beyond the end of exposure. Such models are also suggested by the
log/log slopes being greater than 1.0 at short durations for the solid symbols in Figure 3.2
(although the indeterminate LC50s make the exact slopes uncertain). Stochastic models in
which the hazard rate depends on past exposure (e.g., being a function of accumulation
that persists for some time after exposure stops) could also explain the delayed mortality
and steeper slopes.
These data therefore suggest a complicated toxicity relationship with more than one of
the mechanistic possibilities discussed in Section 2. This therefore provides a good case study
for not only discussing how models can be formulated and parameterized, but also for addressing
the level of model complexity that can be justified and supported by constant concentration
toxicity tests. Equally important, this is a good case study for exploring whether certain model
complexities, even if they are supported by calibration data sets, are actually important for
making model predictions for the time-variable exposures to which aquatic life criteria are
applied.
3.2 Evaluation of Deterministic Models
3.2.1 Models Evaluated
Based on the exploratory data analyses discussed in Section 3.1, three deterministic
models were evaluated for this data set.
(1) Model Dl was the one-compartment, lethal-accumulation-threshold model of
Equation 2.5-2.7. This is the baseline model that might typically be used in the absence
of a demonstrated need for more complex models. The evaluation here will emphasize
how much error is introduced by using such a model and not addressing the complexities
of the toxicity relationship discussed in Section 3.1. The organism-level parameters for
this model are designated as ZC» A and kA, to denote a single-mechanism "A" with one
kinetic-constant and a threshold lethal accumulation.
49
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(2) Model D2 was the two-mechanism, independent-action model of Equations 2.14-2.15.
This model was included because it exhibits biphasic behavior (Figure 2-4) similar to that
observed in the data (Figure 3.1). The organism-level parameters for this model are
designated LC^A, LC«,?B , kA, and kB, "B" denoting the mechanism for the faster toxicity
phase and "A" the slower phase.
(3) Model D2X extended Model D2 by applying the damage/repair model of Equations
2.10 and 2.13 to the fast toxicity phase, to address the delayed mortality associated with
this phase (Figure 3.2). The organism-level parameters for this model are designated
ZC» A, LC^B, kA, k-Bi, and kB2. The two kinetic constants for the fast phase are
differentiated by numbers to emphasize that, although one parameter might reflect
chemical accumulation and the other damage/repair, it is uncertain what kinetic processes
are associated with each constant. When an analysis indicated that km and kB2 had the
same or very similar values, the analysis was redone with the assumption that these
values were the same, as described in Equation 2.12 and 2.13.
The multicompartment models of Section 2.1.4 were not part of this evaluation. As noted
earlier, these models produce relationships that are not readily distinguishable from those of the
multiple-mechanism and damage/repair models. In fact, as also noted earlier, the models used
here should not be treated as definitely describing specific toxicity processes, but as general
kinetic formulations that provide useful approximations to a variety of possible processes.
Additionally, it is not possible to conceptually reconcile the toxicity relationships illustrated in
Figures 3.1 and 3.2 with the two-compartment models discussed in Section 2.1.4. Because the
first phase of toxicity involves delayed mortality and a steep log/log slope at earlier times, it
must reflect toxicity occurring due to accumulation in an inner compartment (Equation 2.20), but
the biphasic behavior would then require a third compartment as a sink for the inner
compartment or as an additional site of toxicity. This creates complexities which are not needed
to address the behaviors in Figures 3.1 and 3.2, and adds to the difficulties regarding
parameterization and prediction already noted in Section 2.1.4 for the multicompartment models.
50
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3.2.2 Model Parameter Distributions
The default assumption regarding the variation of organism-level parameters among
different organisms was that they were log-triangular and independently distributed. Thus,
fitting the models to toxicity data required estimating a mean and standard deviation for each
organismal-level parameter, which would require four distributional parameters for Model Dl,
eight for Models D2, and ten for Model D2X (eight if the two kinetic constants for the fast
toxicity phase are equal to each other). Two issues were addressed regarding this default
assumption:
(1) Because estimation of a large number of parameters might be problematic for
many toxicity data sets, some analyses using Models D2 and D2X assumed that
the distributions for the organism-level parameters shared the same relative
standard deviation, resulting in fewer distributional parameters and allowing an
evaluation of whether individual standard deviations were important for model
performance.
(2) Consideration was also given to the uncertainty that might be introduced by
assuming independence among the organism-level parameters, when, in fact,
there might be some correlation among these parameters (i.e., an organism with a
higher-than-average LCm might tend to have higher-than-average k, or vice-versa).
Fully evaluating the extent and nature of the various correlations would be very
difficult, if not infeasible, but some analyses with Models Dl and D2 assumed a
correlation coefficient of either -1 or +1 between k and LCm. to determine
whether the default assumption of a correlation coefficient of 0 might cause
substantial errors.
3.2.3 Pooled Data Set Analyses
Analyses were conducted on the pooled data of all 26 constant concentration tests to
provide the most complete information for assessing model attributes and overall parameter
values. These pooled data include some tests (those with exposures < 24 h) in which delayed
mortality was monitored. Three versions of this pooled data set were used in the analyses to
51
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variously address this delayed mortality information:
(1) Pooled data set "PE" (delayed mortality "excluded") consisted of all the
observations up to the end of the exposure in each toxicity test, and thus did not
include information on the observed delayed mortality. This dataset was analyzed
using all three models and the various options concerning the parameter
distributions, allowing the relative merits of the model formulations to be assessed
using the type of data typically available from toxicity tests. Fitting Model D2X
to such data still supports prediction of delayed mortality, and allowed evaluation
of how well delayed mortality was predicted even when it was not included in the
model parameterization.
(2) Pooled data set "PI" (delayed mortality "included") consisted of all the
observations in each toxicity test, including those made of mortality after the end
of exposure. This dataset was only used to parameterize Model D2X because
Models Dl and D2 do not explicitly address such delayed mortality. This allowed
comparison of how parameter values and model fit differed for Model D2X when
delayed mortality was included or excluded in parameter estimation, and thus how
well Model D2X actually describes the processes responsible for mortality both
during and after exposure.
(3) Pooled data set "PA" ("adjusted" for delayed mortality) addressed the delayed
mortality information so that Models Dl and D2 could be applied. This was
accomplished by relating observed mortality to the exposure period needed to
elicit the mortality, not to the time at which the actual mortality was observed.
For exposures <24 h, for which delayed mortality was assessed, this required
combining all the observations for each test and concentration (including the
delayed mortality) into a single observation which specified the mortality
resulting from that exposure duration and concentration. For exposures >24 h,
observed mortality up to 24 h was combined, so that the first observation was the
cumulative mortality at 24 h, thereby eliminating any appreciable effect of
delayed mortality on the time course of toxicity; later observations were not
modified. This still allows a meaningful expression of risks, because the
mortality resulting from a particular exposure is addressed, albeit not the exact
time sequence of the mortality. It also allows Models Dl and D2 to be used to
address delayed mortality, because these models can be considered to describe the
attainment of a lethal condition, with the delay in the observed mortality not being
52
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explicitly modeled. This data set was not used to parameterize Model D2X
because adjusting the data for delayed mortality already served the purpose of the
second kinetic constant in the first phase of toxicity of this model; however, how
well Model D2X predicted these adjusted data was still evaluated.
Table 3.1 summarizes the distributional parameter estimates for each model without
(pooled data set PE) and with (pooled data set PI or PA) consideration of delayed mortality.
Parameter estimation assumed that the logarithms of the organism-level parameters have a
triangular distribution, so the table includes the estimated mean and standard deviation of the
base 10 logarithm of the parameters. For example, for model Dl parameterized using pooled
data set PE, the logio(£Cco;A) is estimated to be distributed with mean 2.042 and standard
deviation 0.152. The table gives the standard error for each such distributional parameter
estimate, this uncertainty being very small because of the large amount of data in the pooled dat
sets. The table also lists the antilog of the mean estimates for the log parameters, which provides
a median estimate for each organism-level parameter on its original scale. For model Dl
parameterized using pooled data set PE these median estimates are 110 ug Cu/L for LC«, and
0.0284/hfor£.
This table provides three columns for the AIC, corresponding to the three different
pooled data sets used in the analyses. The AIC in bold text indicates the data set to which the
likelihood was maximized for each analysis; however, AICs for the other data sets also can be
computed, and are provided where useful for discussing model performance below. For
example, if Model D2X is parameterized using data set PE, how well does it predict the delayed
mortality data in set PI, compared to when it is parameterized using set PI? If the model is
sufficient, such cross predictions should be good, and discrepancies can help identify model
limitations. However, AICs can only be compared within columns because the different
53
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]
fable 3.1. Deterministic model parameter estimates based on pooled constant-concentration toxicity tests (26 tests, 2.5-192 hr).
Darentheses denote standard error of parameter estimates. Bold AIC values denote AIC for data set used in parameterization.)
Analysis
Options
Model Dl
(Separate SD)
(Pooled Set PE)
Model D2
(Shared SD)
(Pooled Set PE)
Model D2X
(Shared SD)
(Pooled Set PE)
(Unequal k)
Model Dl
(Separate SD)
(Pooled Set PA)
Model D2
(Shared SD)
(Pooled Set PA)
Model D2X
(Shared SD)
(Pooled Set PI)
(Equal k)
Median Value for Parameter
iC.,A
HgCu/L
110
91
87
104
90
88
kA
1/hr
0.0284
0.0136
0.0131
0.0326
0.0137
0.0192
LC..B
HgCu/L
229
263
264
174
ksi
1/hr
0.088
0.224
0.187
£B2
1/hr
0.341
0.165
Mean of Logic Parameter
iC.,A
2.042
(0.007)
1.957
(0.011)
1.941
(0.010)
2.017
(0.007)
1.956
(0.011)
1.943
(0.010)
kA
-1.546
(0.009)
-1.865
(0.019)
-1.884
(0.012)
-1.486
(0.012)
-1.862
(0.018)
-1.716
(0.016)
LC..B
2.368
(0.007)
2.420
(0.004)
2.422
(0.005)
2.240
(0.006)
fel
-1.056
(0.012)
-0.649
(0.082)
-0.726
(0.011)
km
-0.467
(0.099)
-0.783
(0.006)
Standard Deviation of Logic Parameter
iC.,A kA
0.152 0.220
(0.003) (0.003)
LC..B
fei
0.150
(0.002)
fej2
0.163
(0.002)
0.112 0.372
(0.005) (0.008)
0.153
(0.002)
0.194
(0.002)
Akaike Information Criterion
AIC-PE
6896
6597
6437
6793
AIC-PI
11570
9315
8835
AIC-PA
4562
4552
4018
4771
54
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columns have different general magnitudes depending on the amount of data in each data set.
For Models D2 and D2X, the analyses shown in Table 3.1 are those in which the
organism-level model parameters share the same standard deviation. Analyses in which separate
values for these standard deviations were estimated (not shown) provided little or no
improvement in the AIC and produced similar values for the standard deviations of the different
organism-level parameters. For Model Dl, the analyses shown in Table 3.1 are those with
separate values for the standard deviation of k and LCm because this did result in appreciably
better fit than the analyses with a shared standard deviation (not shown). The different values for
the standard deviations of & and LC«, for Model Dl are symptomatic of this model ignoring the
biphasic nature of the model, which causes the standard deviation of the k to be inflated because
it is a compromise value for the different kinetic constants of the two phases.
Analyses in which the organism-level model parameters were assumed to be correlated
rather than independent are also not shown in Table 3.1 because this resulted in poorer model
fits. Such poorer fit provides support for the assumption of parameter independence in the
analyses, although it is still possible that some limited correlation exists.
Figure 3.3 illustrates the relative fits of the analyses shown in Table 3.1 by comparing
predicted ZCsos and ZCios to average observed ZCsos and ZCios at 2.5, 4, 8, 12, 24, 48, 96, and
192 h. The left panels show observed LCps for mortality occurring up to the designated exposure
duration; the model prediction lines are for Models Dl and D2 parameterized with data set PE,
and Model D2X parameterized with both data sets PE and PI. The right panels show observed
ZCpS adjusted for the mortality occurring after the end of exposure; the model prediction lines
are for Models Dl and D2 parameterized with data set PA, and Model D2X parameterized with
both data sets PE and PI. The predicted ZCps were calculated by Monte-Carlo simulations.
55
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Figure 3.3. LC50 andLC10 versus time fortoxicity of copper to juvenile fathead minnows for constant exposures. The left
panels address LCps at the end of the specified exposure periods (not adjusted for mortality after the exposure) while the right
panels address LCps adjusted for mortality occurring after the specifed exposure periods. Circles () denote average
observed LCps from 26 constant-exposure toxicity tests. For model Dl ( ) and model D2 ( ), parameterization
used pooled data set PE for the left panels and pooled data set PA for the right panels, because predictions of LCp with and
without delayed mortality required the different data treatments. For model D2X, parameterization with pooled data set PE
( ) and pooled data set PI ( ) are included on both left and right panels because this model can do predictions of
LCP with and without delayed mortality with either parameterization. The inset boxes provide the relative sum of squared
deviations (R2) of the predicted and observed LCps for each model; the R2 is identified according to the model used for the
prediction and the data set used to parameterize the model.
Not Adjusted for Delayed Mortality
Adjusted for Delayed Mortality
2000 -i
1000 -
o
05
500 -
200-
100-
50-
2000 -I
1000 -
500 -
100-
50-
6 12 24 48 96 192
12 24 48 96 192
2000-1
1000-
500-
o
O)
0° 200 H
100 -
50 -
2000-1
1000-
^ 500-
O
O)
o
, r 200 -
100 -
50 -
3 6 12 24 48 96 192 3 6 12 24 48 96 192
Exposure Duration (h)
56
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9999 sets of organism-level model parameters were generated by random selection from the
maximum likelihood estimates for the distributions of these parameters (Table 3.1). The model
was then applied to each parameter set to estimate LCs for exposure durations from 2 to 200 h
(for Model D2X, both including and excluding mortality expected after exposure). The LCp for
each duration was set to the appropriate percentile within these sets of LCs. The fits of the
model-estimated LCps to the observed LCps were summarized using the R2 statistic (Figure 3.3),
although this not statistically rigorous because theZCps are not statistically independent.
The analyses summarized in Table 3.1 and Figure 3.3 support the following observations
regarding the relative merits of the models which were evaluated:
(1) Adding the second mechanism of toxicity results in appreciably improved fit relative
to a single mechanism. When parameterized using data set PE, the AIC-PE for Model
D2 is 4.5% less than that for Model Dl (Table 3.1). When parameterized using pooled
data set PA, there is even a greater improvement of 13% in AIC-PA (Table 3.1), because
adjusting for delayed mortality further increases the biphasic nature of the toxicity. This
better fit is evident in Figure 3.3, in which Model D2 closely follows the biphasic nature
of the average observed LC50s andZCi0s, whereas the simple exponential decline of
Model Dl shows substantial deviations. This is reflected in the R2 for the deviation of
model predictions from the LC50s and LCws. Although Model D1 has respectable R2s
(86-90%), Model D2 is much better (98-99%).
(2) Adding the second kinetic constant to the fast mechanism of toxicity in Model D2X
results in additional improvement in fit relative to Model D2. When parameterized using
pooled data set PE, the AIC-PE for Model D2X is 2.5% lower than that for Model D2
(Table 3.1). The importance of this second kinetic constant is most evident in the AICs
in Table 3.1 that included the delayed mortality (data set PI). For Model D2
parameterized using data set PE, AIC-PI, which includes the delayed mortality, is 4973
greater than AIC-PE; this increase is the maximum possible because this model predicts
no delayed mortality. For Model D2X parameterized using pooled data set PE, this
increase is only 2878, indicating it predicts a large fraction of the delayed mortality, in
addition to the improved fit during the exposure period. The importance of the second
kinetic constant is also suggested in the left panels of Figure 3.3, where Model D2X
57
-------
shows steeper slopes than Model D2 in the first phase of toxicity, such steeper slopes also
being evident in the observed LC50s and LCws. (This benefit of the second kinetic
constant is not reflected in better R2s for Model D2X than Model D2 because these R2s
do not consider the "greater than" values at 2.5 h and thus do not adequately account for
the steep slopes at short durtions)
(3) Including the delayed mortality in the parameterization of Model D2X has mixed
effects on model performance. There was a 5% decrease in AIC-PI when Model D2X
was parameterized using data set PI compared to the AIC-PI when the model was
parameterized using data set PE. Although this decrease is appreciable, it is not much of
an improvement considering that the better fit was based on a lot of additional
information. Furthermore, improved fit for the delayed mortality was at the expense of a
worse fit to the data within the exposure periods (AIC-PE in Table 3.1) and resulted in a
loss of the biphasic behavior of the model and much poorer R2 for predicting ZCps during
the exposure period (Figure 3.3). These problems with fit suggest that the mechanisms
causing the delayed mortality are not accurately described by the model. This
inadequacy of the model is also indicated by £Bi and km being equal when parameterized
using data set PI. When these constants are equal, the delayed mortality is at its
maximum relative to the mortality within the exposure period, and the fact that the model
parameterization was pushed to this limit indicates an inadequacy for describing the
relationship of mortality during and after exposure.
(4) Model D2 can effectively describe delayed mortality if the data can be adjusted for
these delays as with data set PA. As noted above, Model D2 parameterized to data set
PA results in high prediction R2s for the average observed ZC50s and ZCi0s adjusted for
delayed mortality (right panels of Figure 3.3). In contrast, Model D2X parameterized
based on pooled data set PE resulted in higher AIC-PA than Model D2 (Table 3.1) and
smaller R2s for the delay-adjusted ZC50s and ZC10s in Figure 3.3. Although this poorer fit
for Model D2X is due to the advantage Model D2 has in being parameterized based on
the adjusted data, it still indicates that Model D2X does not completely reflect the
processes producing delayed mortality. Nonetheless, Model D2X parameterized with
data set PE still produced good R2 for predicting the LCps in the right panels of Figure
3.3, and thus would still be useful for data sets without explicit information on delayed
mortality.
The pooled analyses thus showed importance for addressing both the biphasic nature of the toxic
response and the existence of delayed mortality. If appropriate information is available on the delayed
58
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mortality to adjust the data to reflect time-to-lethal-exposure rather than time-to-death, using such
adjusted data with Model D2 provides the best performance (provided the risk predictions do not need to
explicitly address time-to-death). In the absence of such information, Model D2X provides a means to
address much, but not all, of the delayed mortality based on inference from the mortality patterns within
the exposure period. However, although these analyses have shown certain differences in model
performance for describing constant-concentration tests, it is not possible to state how important this will
be for addressing fluctuating exposures. Therefore, all three models will be considered in section 3.2.5
regarding pulsed exposures, using the parameter values from Table 3.1.
3.2.4 Individual Data Set Analyses
The pooled data addressed above has much more information than typically will be
available for model parameterization, and the standard errors of the parameters are consequently
misleadingly small because the analyses assume that whatever model is being evaluated is
absolutely correct and that the true parameter values are the same for all the toxicity tests. Under
such assumptions, this large amount of data results in small standard errors that do not reflect
model formulation error or variations among tests, and thus do not provide a basis for reasonably
assessing the uncertainty of model estimates. To address the issue of parameter differences
among tests, and thus better describe uncertainty, the models were also parameterized based on
individual toxicity tests within the pooled data set which were of sufficient duration to include
both phases of toxicity (seven 96-hr and five 192-hr tests). Because these tests had no delayed
mortality information, this parameterization was analogous to pooled data set PE regarding the
type of information available. Based on the findings of the pooled analyses, Model Dl was
parameterized with separate standard deviations for the two organism-level model parameters
and Models 2 and 3 were parameterized with shared standard deviations.
Table 3.2 provides the averages (across tests) of the maximum likelihood estimates
59
-------
Tal
hr
)le 3.2. Deterministic model parameter estimates based on analysis of individual constant-concentration toxicity tests (seven 96-
tests, five 192-hr tests). (Parentheses denote standard deviation of individual estimates.)
Analysis
Options
Model Dl
(Indiv SD)
(96 hr Tests)
Model Dl
(Indiv SD)
(1 92 hr Tests)
Model D2
(Shared SD)
(96 hr Tests)
Model D2
(Shared SD)
(1 92 hr Tests)
Model D2X
(Shared SD)
(96 hr Tests)
Model D2X
(Shared SD)
(1 92 hr Tests)
Median Value of Parameter
^
104
105
61
95
64
95
*A1
0.0317
0.0205
0.0087
0.0125
0.0095
0.0123
^
211
279
225
303
*B1
0.117
0.091
0.200
0.192
*M
0.703
0.502
Mean of Logic Parameter
^
2.016
(0.029)
[0.083]
2.020
(0.026)
[0.089]
1.784
(0.127)
[0.069]
1.980
(0.030)
[0.107]
1.804
(0.099)
[0.060]
1.981
(0.030)
[0.107]
*A1
-1.499
(0.055)
[0.118]
-1.688
(0.039)
[0.224]
-2.062
(0.171)
[0.224]
-1.903
(0.048)
[0.174]
-2.026
(0.135)
[0.206]
-1.906
(0.055)
[0.170]
^
2.324
(0.021)
[0.085]
2.446
(0.029)
[0.092]
2.350
(0.018)
[0.108]
2.480
(0.022)
[0.089]
*B1
-0.933
(0.039)
[0.079]
-1.039
(0.056)
[0.110]
-0.701
(0.083)
[0.163]
-0.720
(0.076)
[0.112]
&B2
-0.157
(0.237)
[0.291]
-0.304
(0.240)
[0.322]
Standard Deviation of Logic Parameter
LC^ kAl LC^ *BI
0.109 0.278
(0.021) [0.031]
[0.041] (0.055)
0.135 0.175
(0.016) [0.026]
[0.025] (0.040)
0.125
(0.008)
[0.017]
0.128
(0.007)
[0.012]
*B2
0.131
(0.008)
[0.020]
0.131
(0.007)
[0.011]
Akaike
Information
Criterion
194-491
350-446
176-455
337-415
173-456
337-407
60
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and standard errors for the logio distributional parameters (mean and standard deviation) of each
organism-level model parameter, with separate entries for each test duration. This table also
provides the standard deviations of these distributional parameters across the individual tests.
The magnitude of these standard deviations among tests relative to the average of the standard
errors estimated for the individual tests is indicative of whether there are important sources of
uncertainty not included in the model analyses (e.g., variability of organism sensitivity across
tests). These analyses of individual toxicity tests support the following observations regarding
the models:
(1) On average, the AIC was reduced by Model D2 substantially (9%) in comparison to
Model Dl, reinforcing the importance of describing the biphasic nature of the data and
illustrating how this is evident even in the individual data sets. Model D2X provided no
further reduction of the AICs (<0.5%), which was likely due to these individual toxicity
tests not having sufficient information at shorter durations and high concentrations to
discriminate Model D2 and Model D2X, in contrast to the more diverse set of tests used
for the pooled analysis. This inadequate information to parameterize Model D2X
resulted in large standard errors for km-
(2) In general, the average distributional parameter estimates for the individual toxicity
tests were similar to the estimates for pooled data set PE (Table 3.1) and were also
similar between the two different test durations in Table 3.2. A notable exception to this
similarity is that, for Models D2 and D2X, the ZC» A and kA were substantially smaller on
average for the individual 96-h tests than for either the pooled data or the individual 192-
hr tests. This represents a limitation of addressing two phases of toxicity from such short
tests, in which evaluating the second phase of toxicity depends on data from 24 to 96 h, a
limited time range which is unlikely to contain enough information to extrapolate well to
longer durations. Providing the best fit to data at <96 h can thus result in erroneous
extrapolations to longer durations. This indicates the need for integrating analyses of
acute toxicity tests with longer exposures, which was satisfied at least to some degree by
the 192 h exposures.
(3) As expected, the average standard errors of the distributional parameters from the
analyses of individual toxicity tests (Table 3.2) are much greater than the standard errors
for the pooled data analyses (Table 3.1). This simply reflects the greater uncertainty of
61
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any estimates from more limited data. More importantly, the average standard errors (in
parentheses in Table 3.2) are generally smaller than the standard deviations across
individual tests (in brackets in Table 3.2), indicating the presence of between-test
variability that isn't addressed in the error estimation portion of model parameterization.
An exception to this is again the ZC» A and kA estimates for the 96-hr tests, which had
high standard errors because of the lack of sufficient information from longer times
needed to adequately characterize this phase of the toxicity.
These analyses of individual level tests thus further indicated some importance of
addressing the biphasic nature of the data, indicating the advisability of using Model D2 and
D2X. However, these analyses also showed potential problems in parameterizing either Model
D2 or D2X based just on 96-hr tests and in parameterizing Model D2X (and thus inferring some
of the delayed mortality) without sufficient data on deaths at early times at high concentrations.
These analyses also demonstrated the potential problems with basing model prediction errors on
standard errors estimated as part of model parameterization procedures that do not address
between-test variability. The next subsection will further explore how important these
performance and uncertainty issues are in the prediction of the effects of pulsed exposures,
including consideration of how parameter variability among different toxicity tests can provide
uncertainty information for model predictions.
3.2.5 Predictions of Pulsed Exposures
Although analysis of constant exposure tests provides some basis for evaluating the
appropriateness of different model formulations, the ultimate measure of the utility of a model
will be how well it predicts effect levels for a range of exposure scenarios with fluctuating
concentrations. In this section, such predictions will be examined for some intermittent exposure
toxicity tests.
Lindberg and Yurk (1982, 1983a, 1983b) conducted six intermittent exposures consisting
of 2.5- to 12-h exposures to copper separated by 5.5- to 20-h exposures to control water (8 to 24
62
-------
hr total cycle time for copper plus control exposure periods). Experimental procedures resulted
in a rapid enough transition between the exposure and control periods that these experiments can
be treated as "on-off' or "rectangular" pulses for the purposes of model predictions. Figure 3.4
shows measured ZCsos and ZCios (filled circles) at the end of the control period following each
pulse (thus including any delayed mortality after the pulse), based on the average concentration
over the entire cycle. The average concentration is used here rather than the pulse concentration
because it provides a more useful comparison among different exposure scenarios, including
constant exposures. Figure 3.4 also shows the ZCioS andZCsoS (empty circles) from constant
exposure tests run simultaneously with each pulsed-exposure test.
For the pulsed exposures tested, the measured ZCps (filled circles) show only small
effects of time. Relative to the first pulse, LCsos changed by less than 5% for the once-daily
pulses (24-hr exposure cycle), by about 20% for the twice-daily pulses (12-h exposure cycle),
and 35% for the thrice-daily scenarios (8-h exposure cycle). This is in contrast to the much
greater time-dependence of the constant exposure ZCps (empty circles), where the changes were
60-70%. Such reduced time-dependence of pulsed-exposure LCps versus constant-exposure
ZCpS is expected when constant-exposure ZCps decline less than proportionately with time, so
that high pulses averaged over the exposure cycle are more damaging than a constant exposure
with the same average over that period. The initial fast phase of toxicity and the delayed
mortality also contribute to the limited time-dependence of the pulse LCps.
Another feature of the observed data is that the pulsed exposures ZCps (filled circles) at
the end of the tests are higher than for the companion constant exposure tests (empty circles)
when the pulse period is 50% of the pulse cycle, but lower when the period is only 17-33% of the
cycle. This effect is rather small, but noteworthy because pulsed exposures generally exert
63
-------
Figure 3.4.
observed LC
cycle. Open
predictions a
model D2 (-
predictions o
1000
500
200
100
50
1000
500
200
100
50
1000
.~. 500
_l
D) 200
£= 100
O
=J3 50-
L_
o
C 1000-
o
O 500-
| 200-
O 100-
50
1000
500
200
100
50
1000
500
200
100
50
Poxicity of copper to juvenile
Ps for each pulse based on ave
circles (O)denote observed L
re provided for models Dl (
f Model D2X parameterized i
LC50s
° 0
---.^J3 ° ° o o
24 48 72 96 120 144 168
O
^^^^^*^*^^~~*"*«*^Mita^ft^fc»^4,^ta*i
24 48 72 96 120 144 168
*~*":ii'*^ii iig Q
12 24 36 48 60 72 84
*^*^-«- 0 «
12 24 36 48 60 72 84
6j..
8 16 24 32 40 48 56
2::;:^^^
8 16 24 32 40 48 56
fathead minnows in pulsed exposures. Solid circles ()denote
rage concentration over pulse cycle and mortality at end of pulse
C50s at the same times in companion constant exposure tests. Model
) and D2X ( ) parameterized with pooled data set PE and
ed data set PA. Gray band denotes 10-90th percentile range of
ising individual 192-hour toxicity tests.
1000
500
200
" ^ 100
50
LC10s
o
o
...^ 0 0 0 o 0
192 24 48 72 96 120 144 168
1000
500
200
UAAfe 100
50
0
0
192 24 48 72 96 120 144 168
1000
500
200-
"" 100-
50
<^°_o 0
96 12 24 36 48 60 72 84
1000-
500-
200
""""" 100-
50-
° ° o
^t*^^~^-lwl
96 12 24 36 48 60 72 84
1000
500
200
100
50
H^^^fc*
64 8 16 24 32 40 48 56
000
500
200
100
50
°^. 0 o 0
C^*-.* 8 ft .
4 h Pulse/
24 h Cycle
^
192
12 h Pulse/
24 h Cycle
MMHIt-
192
4 h Pulse/
12 h Cycle
96
6 h Pulse/
12 h Cycle
f+tm
96
2.5 h Pulse/
8 h Cycle
_
64
4 h Pulse/
8 h Cycle
64 8 16 24 32 40 48 56
Time (hr)
64
64
-------
greater effects than constant exposures when compared on the basis of average concentrations,
regardless of the nature of the pulses. This effect is also noteworthy in that it is not predicted by
the models considered here, so it might reflect unidentified processes in the pulsed exposure that
ameliorate effects, such as physiological recovery/adaptation during the control period exposure
between pulses. Such processes would also contribute to the lack of time-dependence in the
observed pulsed ZCps.
Figure 3.4 also includes model predictions of the pulsed-exposure ZCsos and LCws. The
dashed and dashed-dotted lines represent predictions based on Models Dl and D2X,
respectively, parameterized with pooled data set PE. These predictions for Models Dl thus do
not reflect delayed mortality and for Model D2X only reflect expected delayed mortality inferred
from the mortality patterns within the exposure periods of the constant concentration tests. The
bold solid line denotes prediction based on Model D2 parameterized with pooled data set PA,
and thus includes expectations based on more complete information regarding delayed mortality.
For each model, 999 sets of organism-level parameters were randomly generated from the
distribution parameters in Table 3.1 and each such set was used to generate an expected
organism LC (including mortality delays) for each pulse of the intermittent exposures sequences
shown in Figure 3.4. The predicted pulse LCwS and LCsoS were set to the 10th and 50th
percentiles of the resultant LC sets.
The shaded areas on Figure 3.4 denote the 10th and 90th percentile predictions based on
parameters estimated from the individual 192-h toxicity tests. The average and the standard
deviation of the distributional parameters across the 192-h toxicity tests for Model D2X in Table
3.2 were used to randomly generate 999 sets of distributional parameters for this model, which in
turn were each used to generate 999 sets of organism-level parameters. Each of these 999 sets of
65
-------
organism-level parameters was used to predict the pulse LCws and ZCsos, and the 10th and 90th
percentiles of these ZCps bound the shaded areas on Figure 3.4. The width of the shaded area
thus represents the general magnitude of the uncertainty that would arise in LCps due to variation
among tests.
The models predict more time-dependence of the pulsed LCps than actually observed
(Figure 3.4, Table 3.3). The degree of the time dependence in these predictions decreases from
Model Dl to Model D2 to Model D2X when parameterized using pooled data set PE, reflective
of the importance of the initial fast toxicity phase and of delayed mortality for the low ZCps in
the initial pulses. The predictions of Model D2 parameterized using data set PA, for which the
delayed mortality is fully reflected in the initial fast toxicity phase, show even less time-
dependence, the predicted ratio of the first to the last pulse only being 10-20% higher than
observed (Table 3.3). This predicted time dependence indicates the models (as parameterized)
do not fully describe all the processes important for this pulsed toxicity. Nevertheless, the
predictions are still within a factor of 2 of observations and, if the initial fast toxicity phase is
addressed, the limited time-dependence of the observed pulsed LCps is predicted well enough for
these models to have considerable utility.
Table 3.3. Observed and predicted ratios of LC50 of first pulse to LC50 of last pulse .
Pulse Duration/
Recovery (h)
4/20
12/12
4/8
6/6
2.5/5.5
4/4
Ratio of LC50 of first pulse to last pulse
Observed
1.03
1.02
1.2
1.3
1.5
1.6
Model Dl-PE
2.0
1.7
2.7
3.0
3.7
3.7
Model D2-PE
1.6
1.5
2.0
2.3
2.5
2.6
Model D2X-PE
1.5
1.4
1.7
2.0
2.3
2.3
Model D2-PA
1.2
1.2
1.4
1.6
1.7
1.7
66
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3.3 Evaluation of Stochastic Models
3.3.1 Models Evaluated
Based on the exploratory data analyses discussed in Section 3.1, two stochastic models
were evaluated:
(1) Model SI was the single-compartment and single-mechanism model of Equation
2.27-2.29. As for deterministic Model Dl, this is a baseline model that might typically
be used in the absence of a demonstrated need for more complex models. The parameters
for this model are COA, dA, and kA, where the subscript A designates the mechanism.
(2) Model S2 included two independent mechanisms to address the biphasic nature of the
data discussed in Section 3.1. For the stochastic model being used here, multiple,
independent mechanisms simply involve summing the hazard rate expressions (Equation
2.28) for the separate mechanisms. For the second mechanism, the model parameters are
designated as COB, dB, and kB.
3.3.2 Pooled Data Set Analyses
Parameters for Models SI and S2 were estimated using both data set PE (excluding
delayed mortality observations) and pooled data set PI (including delayed mortality) as described
in Section 3.2.3. Table 3.4 provides the maximum likelihood estimates and standard errors for
the log-tranformed parameters, the untransformed parameter values, and the AIC scores for each
analysis. Figure 3.5 shows the model predictions for ZCsoS and LCwS under constant exposure,
both unadjusted and adjusted for delayed mortality, analogous to Figure 3.3.
The analyses summarized in Table 3.4 and Figure 3.5 support the following observations
regarding the relative merits of the models which were evaluated:
(1) As for the deterministic model, adding the second mechanism of toxicity results in
appreciably improved fit relative to a single mechanism because of the biphasic nature of
the data. Including the second mechanism for parameterization using pooled data set PE
resulted in a 6% reduction of AIC-PE and, for parameterization using pooled data set PI,
67
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Table 3.4. Stochastic model parameter estimates based on pooled constant-concentration
toxicity tests (26 tests, 2.5-192 hr). (Parentheses denote standard error of log parameter
estimates.)
Analysis
Options
Model SI
(Date Set PE)
Model SI
(Date Set PI)
Model S2
(Data Set PE)
Model S2
(Data Set PI)
Parameter Values
COA
76
75
69
68
dA
xlO3
0.244
0.150
0.148
0.107
kA
0.356
0.597
0.184
0.489
COB
269
340
dB
xlO3
0.743
0.222
kB
0.393
0.719
Log Parameter Values
COA
1.883
(0.002)
1.876
(0.003)
1.836
(0.003)
1.834
(0.004)
dA
-3.613
(0.011)
-3.824
(0.008)
-3.831
(0.016)
-3.969
(0.013)
kA
-0.448
(0.019)
-0.224
(0.029)
-0.734
(0.024)
-0.311
(0.040)
COB
2.430
(0.009)
2.532
(0.012)
dB
-3.129
(0.022)
-3.653
(0.031)
kB
-0.406
(0.018)
-0.143
(0.051)
AIC
PE
7085
7252
6687
7079
PI
9709
9418
10360
9291
resulted in a 2% reduction in AIC-PI (Table 3.4). For the LCps in Figure 3.5, the R2 for
Model S2 is always higher than for Model SI.
(2) As for the deterministic model, including the delayed mortality in model
parameterization had mixed effects, again indicating that the nature of the relationship of
the delayed mortality to the mortality during the exposure period was somewhat different
than assumed by the models. When the model was parameterized using the delayed
mortality (data set PI), the AIC-PI was, as expected, lower than when the models were
parameterized using data set PE (Table 3.4); however, this was at the expense of a poorer
fit during the exposure period (AIC-PE). This is reflected in Figure 3.5, in which R2s are
worse for the models parameterized based on data set PI.
(3) In general, the stochastic models showed poorer fit to the data than the deterministic
models. Whether parameterized to data set PE or PI, the stochastic models had higher
AICs than the comparable deterministic model analyses (Table 3.4 versus Table 3.1).
Similarly, the R2 for the predicted LC50s and LCws also were slightly to substantially
poorer for the stochastic models (Figure 3.5) than for the deterministic models (Figure
3.3).
68
-------
Figure 3.5. LC50 andLC10 versus time fortoxicity of copper to juvenile fathead minnows for constant exposures. The left
panels address LCps at the end of the specified exposure periods (not adjusted for mortality after the exposure) while the right
panels address LCps adjusted for mortality occurring after the specif ed exposure periods. Circles () denote average
observed LCps from 26 constant-exposure toxicity tests. Model predictions are for model SI parameterized with pooled data
set PE ( ) and pooled data set PI ( ) and model S2 parameterized with pooled data set PE ( ) and pooled data
set PI ( ). The inset boxes provide the relative sum of squared deviations (R2) of the predicted and observed LCps for
each model; the R2 is identified according to the model used for the prediction and the data set used to parameterize the
model.
Not Adjusted for
2000 -
1000 -
^f 500 -
O
D)
o
.j0 200 -
_l
100 -
50 -
f&
\ \ " *
%
XY*
^^ \ " ^
>
Delayed Mortality Adjusted for Delayed Mortality
Model R2
Sl-PE 61%
SI-PI 36%
S2-PE 95%
S2-PI 73%
^N.
v -w
>w*f .
^ Aix
2000 -
1000 -
^f 500 -
O
D)
o
.j0 200 -
_l
100 -
50 -
3 6 12 24 48 96 192
2000
1000 -
2f 500 -
O
%
o° 20°-
_i
100 -
50
\^ *
*$
Model R2
Sl-PE 73%
SI-PI 80%
S2-PE 89%
S2-PI 88%
1
'^****ataB*
2000
1000 -
2f 500 -
O
D)
o° 20°-
_i
100 -
50
3 6 12 24 48 96 192
v\
x\^.
*v*
3 6 12
^
*'\
x'-Vs,
3 6 12
Model R2
Sl-PE 82%
SI-PI 61%
S2-PE 95%
S2-PI 80%
v
%i\
Ns'V
1
24 48 96 192
Model R2
Sl-PE 87%
SI-PI 90%
S2-PE 93%
S2-PI 92%
*" A
**^f-
4
»
24 48 96 192
Exposure Duration (h)
69
-------
(4) One aspect of the poorer fit of stochastic models is that they overestimate the
difference between the LC50s and LCws (Figure 3.5) at short durations and underestimate
the difference for the longer durations, in contrast to the deterministic models (Figure
3.3). This is again due to the difference between LCps being tightly linked to the effect of
time (killing rate) for the stochastic model, whereas for the deterministic model the
difference between LCps is independent of the time effect. If the time course of toxicity
indicates the need for a certain killing rate, this can cause such inappropriate estimates of
the difference between LC50s and LCws. In addition, Model S2 also did not show the
strong observed biphasic behavior for LCws (Figure 3.5), whereas the Models D2 and
D2X did (Figure 3.3).
3.3.3 Predictions of Pulsed Exposures
Figure 3.6 provides predictions for the pulsed exposures using Models SI and S2
parameterized using data set PE, contrasted with the predictions for Model D2 parameterized
with data set PA. Like the deterministic models parameterized without information on delayed
mortality, these stochastic models predict more time dependence of the pulse ZCsos and LCws
than was observed, but stochastic model S2 still provides reasonable predictions. For some
exposure durations, the stochastic models also tend to give poorer predictions of the difference
between the ZCsos and ZCios, as was true for the constant concentration exposures (Figure 3.5).
By assuming that all organisms have identical sensitivities, the stochastic models also
require that, if a certain percentage of organisms are killed in the first pulse, at least this
percentage of the survivors will be killed in the next pulse, and so on until all the organisms have
died after a sufficient number of pulses. The data in these experiments contradict this, with
various exposures showing partial kills in the first pulse with little or no subsequent mortality in
subsequent pulses. This aspect of the stochastic models also would predict a convergence of the
ZCsos and LCws at later pulses, which, while not always evident in the time span of these tests,
is also contradicted by the observed ZCps. This indicates the importance of having some
70
-------
Figure 3.6
observed ]
cycle. Op
prediction
to model I
1000
500
200
100
50
1000
500
200
100
50
1000
500
d
§> 200-
C 100
tS 50-
-1 »
§ .
c 1000-
o
O 500
g- 200-
0 100-
50
1000
500
200
100
50
1000
500
200
100
50
. Toxicity of copper to juvenile fathead
LCpS for each pulse based on average co
en circles (O)denote observed LC50s at
s are provided for models SI ( ) an
LCSOs
1000 -
500 -
0
° 200 -
~ 50 -
24 48 72 96 120 144 168 192
1000
500
0
50
24 48 72 96 120 144 168 192
1000
500
50
12 24 36 48 60 72 84 96
1000
'....o 0
*^-« P f| . . 200-
100 -
50
12 24 36 48 60 72 84 96
1000
O 50° '
100-
50
8 16 24 32 40 48 56 64
000
0, 500'
-^P... o o
" 100-
50
minnows in pulsed exposures. Solid circles ()denote
ncentration over pulse cycle and mortality at end of pulse
the same times in companion constant exposure tests. Model
i S2 ( ) parameterized with pooled data set PE, compared
ita set PA.
LC10s
4 h Pulse/
0 o 24 h Cycle
^ 0 0 0 0 o o
24 48 72 96 120 144 168 192
12 h Pulse/
0 24 h Cycle
0
24 48 72 96 120 144 168 192
4 h Pulse/
° o 1 2 h Cycle
^^ - » 9
12 24 36 48 60 72 84 96
6 h Pulse/
° ° o 12 h Cycle
~^ * j
12 24 36 48 60 72 84 96
2.5 h Pulse/
-^ o o o 0 8 h Cycle
8 16 24 32 40 48 56 64
4 h Pulse/
r--S 1 I 8 8 h Cycle
8 16 24 32 40 48 56 64 8 16 24 32 40 48 56 64
Time (hr)
71
-------
differences in sensitivity among the organisms to explain both the time-dependence of the pulse
ZCpS and differences between the ZCsos and ZCios, which is just not addressed by the stochastic
models as formulated. The stochastic models could be modified to include differences among
the organisms; however, this would increase model complexity and make the stochastic aspect of
these models less important, if not superfluous.
3.4 Summary and Model Application to Aquatic Life Criteria
The above analyses of the data of Lindburg and Yurk (1982, 1983a, 1983b) demonstrated
that the toxicity models discussed in Section 2 can be effectively parameterized and, in some
cases, used to make useful predictions for both constant and pulsed exposures. For this data set,
the stochastic models (Models SI and S2) and the single-mechanism deterministic model (Model
Dl) do not perform well, lacking the ability to describe certain features of the observed
mortality. In particular, Models SI and Dl do not address the biphasic nature of the data and the
stochastic models do not address sensitivity differences among organisms that are evident in the
data. The deterministic models which address the biphasic nature of the data and delayed
mortality, and which include differential sensitivity among the organsisms (Model D2X and
Model D2 parameterized with data adjusted for delayed mortality), provide better performance
that approximates observed mortality reasonably well.
However, the applicability of any of these models to aquatic life criteria is still not
resolved without comparing their predictions to the types of exposure time-series that aquatic life
criteria must handle, and the poorer performing models identified above still might have
acceptable utility. Pulsed-exposures such as those examined above represent an extreme in time-
variability, which would tend to maximize prediction errors for the models used here and
magnify differences between the models. In particular, the low LCps of the initial pulses and the
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limited time-dependence of LCps of later pulses are in large part due to the abrupt, high exposure
of previously unexposed organisms to the first pulse, which increases the importance of the
initial fast toxicity phase and the delayed mortality associated with this phase. This would be
relevant to spill situations or short, intermittent exposures preceded by low concentrations, but
might not be important to more typical exposure time-series of concern to aquatic life criteria.
The sensitivity of predictions to model formulations will be further examined here using
some hypothetical exposure time-series more germane to aquatic life criteria. Ten-year exposure
time-series were generated by randomly selecting base 10 logarithms of mid-day concentrations
from normal distributions with different standard deviations (o=0.1 or 0.3) and auto-correlation
coefficients (p=0.5 or 0.99). Concentrations between these mid-day concentrations were
assigned by linear interpolation. For a median concentration of 100 [ig Cu/L, Figure 3.7 shows
the first year of these time-series, illustrating how they differ both in variability and smoothness.
It should be noted, however, that these exposure time-series are still relatively simple and do not
incorporate seasonality and intermittency that might be important for aquatic life criteria. Other
types of time-series will need to be examined as new criteria procedures are developed, before
conclusions about these models are finalized. The examples given here are intended just to
demonstrate how these models can be applied to exposure scenarios of concern and how the
merits of different model formulations can be assessed.
These time-series cannot be analyzed like the constant and pulsed exposure toxicity tests
discussed previously, which have a definite beginning and end and use a single set of previously-
unexposed organisms. For the deterministic models, once organisms of a given sensitivity die,
the impacts of the rest of the exposure time-series on organisms with that sensitivity are not
assessed. By random chance, the deaths for this sensitivity level can be early or late in the
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Figure 3.7. Portions of time-series used for comparing risk levels predicted by different toxicity models under
fluctuating log-normal concentration scenarios. These panels show the time-series with log mean = 2, log
standard deviation = 0.1 or 0.3, and daily autocorelation coefficient = 0.50 or 0.99.
500 i-
200
100
50
20
100
150
200
250
300
350
20 "-J-1
0
500
200
100
50
20 "-1
50
100
150
200
250
300
350
50 100 150 200 250 300 350
Time (days)
a = 0.1
p = 0.5
a = 0.3
p = 0.5
a = 0.1
p = 0.99
a = 0.3
p = 0.99
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time-series, which does not provide a meaningfully measure of risk; rather, risks can be
meaningfully quantified only by testing multiple sets of organisms with different time-series or
different starting points. For the stochastic models this is not as serious of an issue, because the
assumption that all organisms having the same sensitivity allows risks to be computed for the
surviving organisms; however, these risks will vary depending on the starting point of exposure,
so there is still a need for different sets of organisms with different time-series or exposure
starting points within a series. The use of multiple sets of organisms has the added advantage of
allowing some consideration of the exchange of organisms in natural populations between areas
of low and high exposure.
The analyses conducted for the time-series of Figure 3.7 therefore involved a new cohort
of organisms being introduced each day and evaluating the combined effect on all such cohorts.
To avoid an indefinite buildup of the number of organisms and needing to computationally track
so many organisms, the cohorts were gradually removed over a period of 50 d, emulating
migration out of the exposure area. The effect measure used was the percentage mortality each
day of the combined organisms for all cohorts at the start of the day. This mortality rate was
evaluated for each of the four time-series at 100 median concentrations ranging from 5 to 200 [ig
Cu/L. Six models were used in this analysis, including Models Dl, D2, D2X, SI, and S2
parameterized with data set PE, and Model D2 parameterized using data set PA (which provided
the best predictions for the pulsed exposures and is used here as a reference).
The risks of specific mortality rates were computed as the percentage of days the rate was
exceeded over the ten-year time-series. Figure 3.8 shows the risks for 1% and 10% mortality per
day as a function of mean exposures concentration for each time-series and model. This figure
supports the following observations:
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Figure 3.8. Risk curves for 1% and 10% mortality per day versus average concentration for four exposure
scenarios of log normal concentration distributions with different standard deviations (a) and daily
autocorrelation coefficient (p). Curves are given for model D2 parameterized using pooled data set PA ( ),
for models Dl ( ), D2 ( ), D2X ( ), SI ( ), and S2 ( ) parameterized using pooled data set
PE. The bold and narrow solid lines for model D2 are generally indistinguishable.
Mortality 1%/day
Mortality 10%/day
T3
CD
T3
CD
CD
O
X
LJJ
"CD
CD
O
100
10
0.1
100
10
0.1
LJJ
CD 100
E
i-
^ 10
CD
O)
I 1
CD
0.1
100
10
0.1
a = 0.1
p = 0.5
a = 0.3
p = 0.5
a = 0.1
p = 0.99
a = 0.3
p = 0.99
10 20 50 100 200 5 10 20
Mean Concentration (ng Cu/L)
50
100 200
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(1) These models can be used for providing useful information regarding a range of effect
levels for a variety of exposure scenarios, allowing the risks of aquatic life criteria to be
more meaningfully defined. This includes relating effect levels to more easily monitored
and controlled measures of exposure such as mean concentrations, rather than extreme
values.
(2) Figure 3.8 illustrates how mean concentrations must be lower for more variable
exposures, which is expected because it is necessary to limit the high end of the
concentration distribution responsible for most of the toxicity. In contrast, the different
daily autocorrelations examined have little effect on risk, because the effects for this
endpoint and toxicant do not require high exposures for prolonged times.
(3) For the deterministic models, there is very little effect of model formulation and
parameterization. The risk curves for the two different parameterizations of Model D2
are not distinguishable on Figure 3.8, and differ only slightly from Model D2X. Model
Dl risk curves are also very similar to those for Model D2, predicting at most 20% higher
median concentrations for the same probability of an effect, despite using only a single
toxicity mechanism to describe the biphasic data. The lack of sensitivity to model
formulation is attributable to mortality for these exposure time-series being largely due to
the slower toxicity mechanism, which is nearly identical for Models D2 and D2X and
which is also reasonably approximated by the single mechanism of Model D1 (Table
3.1). In contrast, the pulsed exposures examined earlier were highly affected by the
faster toxicity mechanism.
(4) The stochastic models produce risk curves somewhat different than the deterministic
models, tending to overestimate the risk of the larger effect levels and underestimate the
risk of the smaller effect levels relative to Model D2. This again reflects the lack of
independence in these models between the effect of time and the number of organisms
affected, which contributed to the poorer performance of these models for the pulsed and
constant exposures examined earlier. However, the deviations from the deterministic
models are never large, being at most 50% and usually much smaller. Thus, the
deficiencies of these models do not keep them from having considerable utility for these
types of exposure time-series.
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As noted earlier, the risk curves given here are just one example of how these models can
be applied to aquatic life criteria. Various other possibilities exist for toxicity endpoints,
exposure measures, and structuring the population of organisms being evaluated. More simple
model applications are also possible. For the stochastic models, risk curves similar to Figure 3.8
would result from applying the models to a single cohort of organisms. For the deterministic
models, the lethal condition variable F(f) could be tracked without removing organisms when
F(f) exceeds one to simply provide a "meter" for the severity of the exposure time-series. These
models could also be used merely to select better averaging periods under the current aquatic life
criteria framework, without explicitly addressing the magnitude and time-variability of effects.
Any final analysis needs to be tailored to the total framework being used for criteria (e.g., are
these results to be fed into a population model?) and a more complete description of the exposure
time-series to be addressed. Nonetheless, the analyses presented here do demonstrate the
feasibility of analyzing toxicity data with relatively simple models which then can provide
information on risks useful for aquatic life criteria.
3.5 References
Lindberg C, Yurk J. 1982, 1983a, 1983b. Progress in studies to determine effects of intermittent
dosing of copper on fathead minnows. In: Second-, third-, and fourth-quarter reports to
U.S. EPA, Cooperative Agreement CR809234020, Aquatic Pollutant Hazard Assessments
and Development of a Hazard Prediction Technology by Quantitative Structure-Activity
Relationships. Center for Lake Superior Environmental Studies, University of Wisconsin,
Superior, WI.
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