EPA/600/R-21/305
ERASC-017F
December 2021
ALLOMETRIC SCALING OF TERRESTRIAL WILDLIFE ORAL TOXICITY MEASUREMENTS AND
COMPARISON OF ECOLOGICAL TO HUMAN HEALTH ASSESSMENT CONTEXTS
Ecological Risk Assessment Support Center
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati, OH
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DISCLAIMER
This document has been reviewed in accordance with U.S. Environmental Protection Agency
policy and approved for publication. Mention of trade names or commercial products does not
constitute endorsement or recommendation for use.
Preferred Citation:
Farrar, D. and M. Kravitz. 2021. Allometric Scaling of Terrestrial Wildlife Oral Toxicity
Measurements and Comparison of Ecological to Human Health Assessment Contexts. U.S.
Environmental Protection Agency, Ecological Risk Assessment Support Center, Cincinnati, OH.
EPA/600/R-21/305.
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TABLE OF CONTENTS
LIST OF ABBREVIATIONS iv
AUTHORS AND REVIEWERS v
QUALITY ASSURANCE vi
PREFACE 1
1. SUMMARY 1
2. BACKGROUND 2
2.1 Allometric Scaling of Toxicity Measurements 4
2.2 Some Aspects of the U.S. EPA Viewpoint on Scaling of Toxicity Measurements for
Human Health Assessments 5
2.3 Ecotoxicological Data and Ecological Assessments 7
3. SCIENTIFIC ARGUMENTS FOR THE MOST APPROPRIATE EXTRAPOLATION METHOD ....8
3.1 Direct Empirical Evidence from Toxicity Data 8
3.2 Mechanistic Arguments 9
4. SENSITIVITY OF HAZARD ASSESSMENT TO BODY WEIGHT SCALING DECISIONS 11
5. RECOMMENDED DEFAULT SCALING PROCEDURES 12
6. DISCUSSION AND FUTURE WORK WITH AN EMPHASIS ON UNCERTAINTIES IN
EXTRAPOLATION 17
7. REFERENCES 29
APPENDIX A. BODY SIZE AND TISSUE STEADY STATE CONCENTRATION: A NUMERICAL
ILLUSTRATION 38
APPENDIX B. ADDITIONAL DETAIL ON SELECTED SOURCES 40
APPENDIX C. BODY WEIGHT SCALING AS AN APPROXIMATION OF PHARMACOKINETIC
MODELING - ADDITIONAL ANALYSIS OF PUBLISHED RESULTS 44
APPENDIX D. PHARMACOKINETIC DERIVATIONS OF THE ALLOMETRIC SCALING
PROCEDURES 46
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LIST OF ABBREVIATIONS
AUC - area under a curve, here a curve relating tissue concentration to time
BW - body weight
CL- clearance (mass of tissue cleared of toxicant per unit time)
Cmax- maximum tissue concentration
LD10, LD50, etc. - dose lethal with probability 10%, 50%, etc. (a parameter in a dose response
curve, e.g., probit), or an estimate of such a parameter based on data from a toxicity study.
(Whether the reference is to actual parameter value or an estimate will be clarified according to
context.)
NOAEL- no observed adverse effect level (of exposure)
LOAEL- low observed adverse effect level (of exposure)
PBPK- physiologically-based pharmacokinetic
PD - pharmacodynamic (equivalent for present purposes to toxicodynamic)
PK - pharmacokinetic (equivalent for present purposes to toxicokinetic)
RAF - Risk Assessment Forum
TD, TK-see PD, PK
U.S. EPA - U.S. Environmental Protection Agency
U.S. FDA - U.S. Food and Drug Administration
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AUTHORS AND REVIEWERS
AUTHORS
David Farrar
U.S. Environmental Protection Agency
Office of Research and Development
Center for Public Health and Environmental Assessment
Cincinnati, OH
Michael Kravitz
U.S. Environmental Protection Agency
Office of Research and Development
Center for Environmental Solutions and Emergency Response
Cincinnati, OH
REVIEWERS
Robert Paul Hunter
One Medicine Consulting
Olathe, Kansas
Brad E Sample
Ecological Risk, Inc.
Rancho Murieta, California
ACKNOWLEDGMENTS
The first draft of this document was internally (within EPA) reviewed by Jeff Swartout (EPA
Office of Research and Development), and Kristina Garber and Melissa Panger (EPA Office of
Pesticide Programs). Programmatic review was conducted by David Charters (EPA Office of
Land and Emergency Management), Andrea LaTier (EPA Region 10) and Marc Greenberg (EPA
Office of Land and Emergency Management). This document was externally peer reviewed
under contract to Versar Inc., 6850 Versar Center, Springfield, VA 22151 (Contract EP-C-17-
023). Paul Schlosser (EPA Office of Research and Development) provided helpful input into
preparation of the final document.
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QUALITY ASSURANCE
This work was conducted under the U.S. EPA Quality Assurance (QA) program to ensure data
are of known and acceptable quality to support their intended use. Surveillance of the work by
the ERASC Director ensured adherence to QA processes and criteria, as well as quick and
effective resolution of any problems. The QA manager and the ERASC Director have determined
under the QA program that this work meets all U.S. EPA quality requirements. This was written
with guidance from the Center for Public Health and Environmental Assessment (CPHEA)
Program Quality Assurance Project Plan (PQAPP), the QAPP titled "PQAPP for the Superfund
Health Risk Technical Support Center (STSC) and Ecological Risk Assessment Support Center
(ERASC)" QAPP ID L-CPAD-0030721-QP, formally QAPP ID NCEA-16-00003. As part of the QA
system, a quality product review is done prior to management clearance. This document
received internal peer review by at least two scientists and an independent external peer
review by at least three scientific experts managed by Versar under contract EP-C-17-023.
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PREFACE
EPA's Ecological Risk Assessment Support Center (ERASC) provides state-of-the-science
technical information relevant to ecological risk assessments and cleanups at hazardous waste
sites (https://www.epa.gov/risk/erasc). Due to uncertainty surrounding the use of allometry in
ecological risk assessments, ERASC was requested by risk assessors in the EPA Office of Land
and Emergency Management (OLEM) program to clarify the appropriate use of allometric
scaling of toxicity measurements in ecological risk assessments.
1. SUMMARY
The possibility of nonlinear effects of body weights should be considered in any analysis of
biological parameters across species with significantly different body sizes. The term
"allometry" is used where there is a possibly non-linear relationship of a toxicologically relevant
parameter to body weight (BW), particularly when the relationship can be described with a
power function B\Nb with exponent b. (See Section 2. We also use the term "scaling".) This
report discusses scaling defaults for terrestrial wildlife oral toxicity measurements in the form
of exposure values (dose or concentration) associated with specified toxicological outcomes.
These may be in "dietary" form (e.g., ppm toxicant in feed)1 and dose form (e.g., mg/kg or
mg/kg-d). Extrapolation of dose-based toxicity measurements on a "simple body weight basis"
(the case b = 1) is held to apply primarily to lethality measurements based on single doses, or
other situations involving appreciable lethality, while in most other situations an allometric
adjustment based on b = % is the recommended default. Exceptions for particular situations
can be based on direct empirical evidence, mechanistic information, or modeling. The
recommendations are consistent with current health assessment policy (U.S. EPA, 2011).
1 Here the term "dietary toxicity" follows U.S. EPA (2015) and does not refer simply to exposure via the diet. The
term means here that the toxicity measurement takes the form of food concentration associated with an effect, not
converted to dose on a body weight basis by combining the dietary toxicity with a food intake rate.
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It is argued that a biologically consistent approach is to apply allometric adjustments to dose-
based measurements, expressed on a body weight basis, but apply no adjustment to "dietary"
measurements (assume the same critical feed concentration for test and assessment species).
Sufficient assumptions for this approach include that uptake and clearance rates scale to the
same power of body weight and that tissue concentrations over time can be appropriately
summarized by averaging or cumulation (e.g., AUC) (O'Flaherty, 1989). The effect of the
generally higher food intake rate for smaller animals is that tissue concentrations may rise more
rapidly initially in small animals, at a given environmental concentration, without necessarily
producing a substantial difference in longer-term average exposure (as illustrated by simulation
in Appendix A).
2. BACKGROUND
Nonlinear effects of body weight (BW) have been documented for many biological processes
(Schmidt-Nielsen, 1984; Mahmood, 2005). "Allometric" curves allow that such effects of BW on
biological processes may be nonlinear, for example if the biologically effective mass of a
toxicant increases smoothly but not proportionally with species body weight.2 The possible
effect of allometric relationships has been considered in the context of cross-species
extrapolations of toxicity for human health and ecological assessments, and also in veterinary
toxicology.
The U.S. EPA Risk Assessment Forum proposed defaulting to allometric scaling with three-
quarters power of bodyweight (BW3/4) for interspecific extrapolation of toxicity data for
deriving human health oral reference doses for both cancer and non-cancer endpoints in the
absence of other information to support a different interspecific relationship (U.S. EPA 2011).
BW3/4 scaling is viewed as relating primarily to pharmacokinetic (PK) concerns but possibly also
to pharmacodynamic concerns. (In addition to literature cited by U.S. EPA to support the PK
arguments, see Boxenbaum, 1980; Boxenbaum and DiLea, 1995). The approach is considered
2 In practice, a nonlinear allometric relationship of a quantity Y to body weight (BW) is ordinarily expressed using a
power function Y = a'BWS where parameter a is termed the "coefficient" and b the "exponent." Thus b = 1
corresponds to proportionality between Y and BW, while b < 1 to decelerating curves and b > 1 to accelerating
curves.
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most appropriate for oral exposure, toxicity caused by the parent compound or a stable
metabolite, clearance through first-order biological (metabolic) processes, and chronic
exposures and effects. It is considered by USEPA (2011) "not generally appropriate "in the case
of a single exposure eliciting sudden and severe toxicity resulting from immediate and
intolerable damage to some critical biological pathway, and where repair processes (i.e., TD)
would be overwhelmed." Nonetheless, U.S. EPA (ibid.) suggests BW3/4scaling may still be useful
for acute exposures with non-lethal effects "in which the functional status of physiological
processes are comparable to the chronic scenario." Allometric scaling is considered (by U.S.
EPA, ibid.) inappropriate when toxicity is attributed to formation of a reactive metabolite or for
very high exposures that saturate the relevant metabolic processes.
In contrast, Allard, et al. (2009) proposed that ecological risk assessors should not use
allometric dose-scaling with body mass when assessing chronic toxicity between species,
stating that "allometric scaling models developed for both human and wildlife risk assessment
are all based on acute toxicity data." (An extended quote from this source can be found in
Appendix B.) However, Allard et al. did not discuss the pharmacokinetic basis for allometric
scaling of chronic toxicity in U.S. EPA (2011).
Due to the uncertainty surrounding the use of allometry in ecological risk assessments, risk
assessors in EPA OLEM submitted a request to ERASC for clarification of the appropriate use of
allometric scaling of toxicity measurements in ecological risk assessments. Specifically, the
problem statement was:
What is the appropriate use of allometric scaling for characterizing toxicity in
ecological risk assessments? 1) What is the default methodology? 2) Can you
develop scientifically justified deviations from the default? If so, how?
This document attempts to answer these questions by reviewing the general types of
arguments used to support science-based policy in scaling decisions as those arguments may
apply to wildlife assessments, leading to recommended defaults, while recognizing that
exceptions may be justified in particular situations. Allometric scaling would be considered in
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Step 3 (baseline risk assessment problem formulation) of the Ecological Risk Assessment
Guidance for Superfund (U.S. EPA, 1997). (It would not be used in the screening steps of an
assessment.) It is important to note that while the information provided in this document may
be useful to a number of programs, particularly Superfund, it is not meant to be prescriptive.
2.1 Allometric Scaling of Toxicity Measurements
U.S. EPA (1993) documents many allometric relationships involving biological parameters for
wildlife. While the primary concern here is the direct scaling of toxicity values, any biological
parameter determining toxicity may itself scale allometrically. Toxicity is influenced by a
balance of uptake processes with processes of elimination or recovery. Internal distribution of
toxicant is also important, but information on distribution is rarely available in an
ecotoxicological setting.
For human health assessments, the U.S. EPA (2011) has reviewed the scientific basis for a
current approach — "BW3/4 scaling" — and recommends that approach as a default for oral
reference doses. Keeping in mind that the emphasis in the U.S. EPA (2011) review is human
health assessment methodology based on mammalian data, that review may be considered for
general toxicological insights on scaling, in combination with analyses more specific to
ecological risk assessment.
A formula for BW3/4adjustment is:
T2 = Ti*(BWi/BW2)1/4
where T2 is the extrapolated toxicity value for Species 2, based on toxicity measurement Ti in
Species 1, both expressed on a BW basis such as mg/kg-d, and BW is indexed 1 or 2 according to
species.3
3 Some confusion seems inevitable from the use of a 1/4 power in a formula said to represent a 3/4-power method.
The 3/4 power appears in the expression for scaling a critical dose (e.g., mg/d) that has not been normalized relative
to BW (see Appendix D). In any case the expressions given can be rationalized pharmacokinetically by assuming
that clearance of a chemical (CL) is proportional to the 3/4 power of BW. A quarter power relates to the fraction of
tissue clear per unit time to BW, because BW3'4 / BW = 1/BW1'4. See U.S. EPA (2011) for additional discussion of
powers of BW in expressions for different types of biological quantities.
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More generally, BW3/4 can be considered one case of a "B\Nb" framework, where b is the
allometric exponent. The general formula is similar to the one just given, but the power in
general is 1 - b. If we choose b = 1 the formula becomes simply T2 = Ti, i.e., the extrapolated
toxicity value equals the measured toxicity value when both are expressed on a body-weight
basis. This "BW1" option can also be termed "extrapolation on a simple body-weight basis."
Various choices of b * 1 represent nonlinearity. The term "surface area correction" refers most
appropriately to the specific choice b = 2/3, a choice that, as a default value, has been replaced
by 3/4 in U.S. EPA (2011). The qualitative effect of using an inappropriate value for the
exponent is to underestimate risk (overestimate effective doses) when extrapolating from a
smaller to a larger species when the assumed value of b is too large, or when extrapolating
from larger to smaller species when the value assumed for b is too small.
For the present document there will be a general preference for a few particular values of b
that seem to have considerable precedent, especially the values 3/4 and 1. Attempts to further
refine allometric methodology by recognizing more context-specific values cannot be dismissed
and are viewed as areas for possible further study. We note for example a recent analysis of
basal metabolic rate (White et al., 2009) suggesting variation of the allometric exponent among
evolutionary lineages of mammals. At the same time, at least one serious theory (the West-
Brown-Enquist theory, West 2017; West et al., 2002; c.f., Savage et al., 2008) attempts a
rationale for an exact value 3/4 for the allometric exponent.
2.2 Some Aspects of the U.S. EPA Viewpoint on Scaling of Toxicity Measurements for
Human Health Assessments
The U.S. EPA (2011) review is directly concerned with human health assessments but can
provide a departure for more general discussions. Notable features of the review include:
• The scientific rationale, summarized in Section 4 of the document, for moving to BW3/4
from "surface area" adjustment (BW2/3) is a combination of empirical evidence from
comparisons of toxic dose values, general biological considerations, and modeling.
General biological considerations involve the scaling of basal metabolic rate and other
physiological rates including glomerular filtration as BW3/4 across species (U.S. EPA, ibid.,
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Table 4.1). A particularly notable analysis of empirical data cited is that of Travis and
White (1988), based on toxicity of 27 chemotherapy agents (see Section 3.1).
• BW3/4 scaling is viewed as relating primarily to PK concerns but possibly also to PD
concerns to some degree (see Rhomberg and Lewandowski, 2006, for further
discussion).
• PBPK modeling is viewed as the preferred approach for addressing the role of PK in
cross-species extrapolation (for human health assessments). However, the possibility of
addressing PK considerations to some degree without such modeling is recognized.
PBPK models are mechanistic biological models defined by rates of movement of
substances between physiological compartments.
• BW3/4 scaling is recommended as the usual default for sublethal oral toxicity
measurements including many acute toxicity measurements. BW3/4 scaling is
considered appropriate particularly when area under a concentration curve (AUC) is an
appropriate summary of tissue concentrations over time. It is suggested that BW3/4
scaling is most appropriate when exposures are in a range in which critical physiological
processes operate in about the same way as without exposure.
• "... BW3/4 scaling would apply most appropriately to those exogenous substances for
which the unmetabolized parent or a stable metabolite is the relevant toxic species and
clearance is according to first-order processes". Under these conditions elimination of
the toxic moiety is expected to vary as BW3/4, which leads to the conclusion that the
concentration of the moiety in the body varies likewise. Conversely, "the applicability of
BW3/4 scaling is less well supported when toxicity is a consequence of exposure to a very
reactive parent compound or metabolite that is not removed from the site of formation
by biological processes (e.g., subsequent metabolism) but chemically reacts with cellular
constituents." In this case total elimination of the toxic moiety is expected to vary as
BW1. (see references in U.S. EPA 2011 for support of these conclusions.)
• Possible exceptions to the proposed default are discussed such as lethal effects and
portal-of-entry effects.
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• It is allowed that specific assessments may differ from the general recommended
defaults for "policy" as well as scientific reasons.
2.3 Ecotoxicological Data and Ecological Assessments
Scaling procedures must be considered for two types of ecotoxicological measurements,
namely dose-based and dietary (or food-based) (see U.S. EPA, 2015). For a dose-based toxicity
measurement, the measurement units are mass toxicant over body weight (e.g., mg/kg) or
dosing rate (e.g., mg/kg-d). For a dietary measurement, the measurement units are
represented as toxicant concentration in feed. Dietary measurements can be converted to
dose rate using information on feeding rate and body weight. The dietary approach has been
used, for example, to report the results of avian reproduction studies. (The terminology is
somewhat confusing: in this context "dietary" means something more specific than exposure
via the diet.)
Additional relevant features of ecological assessment include:
• Ecological assessments may need to address effects on a species category (e.g., birds in
general) or multiple exposed species. The variety of receptors exposed, and differences
in terms of taxonomy, behavior, and physiology, is expected to pose a challenge for any
effort to implement detailed mechanistic (e.g., PBPK) models, as sometimes used in
human health assessments.BW3/4 is protective in extrapolating from small to large
species, the usual situation in human health assessments. For ecological assessments,
in contrast, species tested are not necessarily small compared to species exposed, e.g.,
laboratory rats are large relative to small mammal species of concern in many ecological
assessments.
• Lethality data are ordinarily not considered directly in human health assessments, but
commonly considered in ecological assessments (see particularly U.S. EPA, 2011,
"executive summary").
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3. SCIENTIFIC ARGUMENTS FOR THE MOST APPROPRIATE EXTRAPOLATION METHOD
Different conceptual approaches have been used to justify decisions on BW scaling of toxicity
measurements. Two general types of evidence have been most important (Rhomberg and
Lewandowski, 2006): A "direct empirical" approach uses collections of toxicity measurements
where the same substance is evaluated for multiple species. A second approach uses
mechanistic arguments, particularly relating to 1) the most appropriate summary of
concentration over time and 2) possible body-weight dependency of physiological rates.
3.1 Direct Empirical Evidence from Toxicity Data
The direct empirical approach relies on toxicity measurements of given substances for multiple
species with a range of body weights. Analyses of such data for single-dose LD50s have been
carried out by various authors (Mineau et al., 19964; Sample & Arenal, 19995; Rhomberg and
Wolff, 19986; Burzala-Kowalczyk and Jongbloed, 20117) (additional details for several reviews
provided in Appendix B). For the most part these analyses favor BW1 scaling (i.e., extrapolation
on a simple BW basis) over BW3/4 scaling. However, for avian LD50 data for some pesticides,
there is support for allometric scaling with a coefficient greater than unity (Mineau et al., 1996,
2001; Sample and Arenal, 1999). We are somewhat uncertain of the set of chemicals and avian
taxa to which these results should be held to apply. The database relied upon is said to be
weighted towards cholinesterase inhibitors.
It is not clear how successful the empirical approach, based on toxicity measurements, will be
for sublethal toxicity measurements. However, Travis and White (1988), invoked in U.S. EPA
4 For 37 pesticides contributing avian LD50 data, percentiles calculated here, for allometric exponents tabled are 0.8
(5%), 1.15 (50% - median), and 1.4 (95%).
5 For a subset of 122 substances with an LD50 for 5 or more bird or mammal species the following percentiles of
allometric slope distribution have been calculated: 0.7 (5%), 1.1 (50% - median), 1.7 (95%).
6 The article is used by U.S. EPA (2011) for essentially the same conclusion as stated here. It is based on over 3,000
mammalian single dose toxicity values (which have not been obtained for inspection). The analysis was based on
ratios for species. Data have been re-analyzed using regression methodology by Burzala-Kawalczyk and Jongbloed
(2011) with essentially similar results. Both analyses concluded that the best single value for allometric exponent is
about 1.
7 The article presents a reanalysis of data assembled by Rhomberg and Wolff (ibid.) using regression methods, again
concluding that an allometric exponent of 1 is a reasonable central tendency, while also reporting wide variation.
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(2011) to support BW3/4 scaling, applied the direct empirical approach for 27 chemotherapy
agents by combining a dataset compiled from maximum tolerated doses for 3 species (human,
monkey, dog) with LD10 data from mice and rats.8 For chronic toxicity, possible obstacles
include that the data may be summarized by a NOAEL, which may represent different endpoints
in different studies (as discussed in Allard et al., 2009). For example, an avian reproduction
study may produce measurements of reproductive output (eggs), and growth and survival of
offspring through a series of developmental stages. If feasible, it seems preferable to compare
the same endpoint across species, based on effective doses estimated using a statistical curve-
fitting approach (e.g., nonlinear regression).
3.2 Mechanistic Arguments
Decisions on scaling depend on the most appropriate summary of internal concentration over
time (O'Flaherty, 1989). Subject to various exceptions and qualifications BW3/4 scaling is
supported by various analyses in situations where the concentrations are appropriately
summarized in some type of average or cumulative exposure (O'Flaherty, 1989; Sharma and
McNeill, 2009; Rhomberg and Lewandowski, 2006; U.S. EPA, 2011). Particular cases are the
AUC concentration from a single dose (emphasized particularly in U.S. EPA, 2011), steady-state
concentrations from chronic exposure, and time-weighted average exposures.
The strongest PK arguments for BW3/4 scaling of toxicity measurements relate to situations
where rates for elimination processes are proportional to BW3/4 across species. From a PK
perspective, species BW3/4 ratios are in effect surrogates for corresponding ratios of species
typical clearance rates.9 This conclusion can be derived in a framework of classical
8 Two sets of estimates of the allometric exponent were combined. For one set of 14 substances, estimates ranged
from 0.53 to 0.87; for a second set of 13 substances estimates ranged from 0.53 to 0.96. The authors calculated a
95% confidence interval (0.69, 0.77) for a single estimate.
9 Clearances have units of volume [https://www.sciencedirect.com/topics/immunologv-and-microbiology/clearancel
of tissue cleared of chemical per unit time and as a default may be assumed to scale to the 3/4 power of body weight
(U.S. EPA, 2011). Glomerular filtration (GF) in particular, possibly an important mechanism of elimination for
some substances, scales to the 3/4 power of body weight in mammals and in birds as well (Edwards, 1975; Schmidt-
Nielsen, 1984).
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pharmacokinetics, assuming that AUC is the appropriate internal dose summary and is inversely
proportional to systemic clearance (CL), in turn proportional to BW3/4.
In contrast to AUC or average concentrations, peak tissue concentrations (e.g., Cmax associated
with a single exposure event) appear not to have been associated with a simple approach to
allometric scaling (U.S. EPA, 2011). If allometric scaling of doses is to be used in a situation
where Cmax is considered the most appropriate basis for extrapolation, the best allometric
exponent has not been identified (it may be 1 so far as is known). Modeling (e.g., Fischer, 2005)
may be needed if Cmax is considered the most appropriate internal dose summary.
The analysis of Kirman et al. (2003) has been used by U.S. EPA (2011) as support of BW3/4
scaling on pharmacokinetic grounds. The analysis supports that BW3/4 scaling tends to
approximate the use of PBPK models in extrapolation, assuming that AUC is the best summary
of internal dose. Results from BW3/4 scaling were compared to model-predicted AUC
concentrations for mice, rats, and humans, for 12 "predominantly volatile and lipophilic"
chemicals.10 A likely basis for including these chemicals in this analysis was availability of PBPK
models for mammals involved in health assessments. It may be noted that some of the
substances evaluated are encountered in ecological assessments, but the substances studied
were not selected to represent those of interest for ecological assessors. The analysis indicates
variation across chemicals in how well scaling would approximate the use of current PBPK
models to predict AUC concentrations. For example, methylmercury is an example where the
approximation is comparatively poor. This analysis suggests that as a rule BW3/4 scaling
approximates the use of PBPK models which, it should be noted, may loosely follow allometric
relationships. Additional analysis is desirable to explore what PK properties of the chemicals
reflect greater or lesser success in use of allometric scaling.
Kirman et al. (2003) considered whether the PBPK-based ratios of AUCs comparing pairs of
species were consistent with expectations based on BW3/4 scaling. Another way to express the
10 Benzene, ethanol, styrene, carbon tetrachloride, ethylene oxide, tetrachloroethylene, chloroform, methylene
chloride, trichloroethylene, diisopropyl fluorophosphates, methylmercury, and vinyl chloride.
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results is to calculate an "effective allometric exponent" b such that ratios of body weights,
scaled using the exponent computed, are equal to PBPK-predicted AUC (see Appendix C). Both
sets of computations - those of Kirman et al. along with the effective exponents - are displayed
in Appendix C and appear to support default BW3/4scaling as a PK adjustment, particularly
preferred over extrapolation on a simple body weight basis (under assumptions of the analysis).
Some support for allometric scaling in human health assessments comes from data for humans
and animals on acute toxicity of anti-neoplastic drugs. Allometrically-scaled rodent LD10
measurements are found to be supportable as estimates of human sublethal effects (Travis and
White, 1988). A point of interest for purposes of this section is that tissue AUCs have been held
to be useful dose summaries for these chemicals. It has been observed, however, that such
chemicals are not typical of wildlife assessments - see Allard et al. (2009).
4. SENSITIVITY OF HAZARD ASSESSMENT TO BODY WEIGHT SCALING DECISIONS
To provide some sense of how much difference BW3/4 scaling could make, Table 1 displays
information on sensitivity to relative body weights (test species versus assessment species),
assuming that toxicity and exposure information are combined into a hazard ratio11. The table
gives examples of ratios of species body weights, associated with different multiplicative factors
for adjustment of toxicity measurements when using the BW3/4 approach.
The first example illustrates extrapolation from of a larger, roughly rat-sized species to a smaller
species, roughly mouse-sized. With BW1 scaling we would assume the same toxicity value (on a
body weight basis) in the assessment species as in the test species. With BW3/4 scaling the
toxicity value from the test species is multiplied by a factor of 2, i.e., allometric scaling results in
the assessment species being judged less sensitive than if extrapolation had been based on
BW1. The second example illustrates extrapolation from a smaller (mouse-sized) to larger
(skunk-sized) species. Now, the assessment species is judged more sensitive (lower toxicity
11 Hazard ratio, or HR, is the estimated environmental or tissue concentration divided by the toxic concentration.
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value after scaling) than with BW1 scaling. Indeed, the toxicity value from the test species is
divided by 3.
(Of course, "smaller-to-larger" extrapolation is the general rule in extrapolating toxicity for
human health assessment, so that BW3/4 scaling is generally more health-protective than BW1
scaling in that situation. Ecotoxicological extrapolations can be smaller-to-larger or larger-to-
smaller.)
More precisely, a toxicity value (e.g., a dose estimated to have no detectable effect, a
stipulated magnitude of mortality, etc.) would be "adjusted" (multiplied or divided) by a factor
of 2.0 (2-digit accuracy) if the ratio of species body weights (larger / smaller) is in the range
14.5-17.7, or a factor of 3.0 if such a ratio is in the range 75.7-86.5. (Multiply the toxicity value
by such a factor when extrapolation is larger-to-smaller, divide by the factor when the
extrapolation is smaller-to-larger.)
Table 1. Sensitivity of Hazard Assessment to Allometric Scaling of Toxicity Values3
Adjustment Factor
Ratio of Species
Example Extrapolation
Applied to
Body Weights
Test
Assessment
Measurement
(T est/Assessment)
Species
Species
2.0
15
300-g rat
20-g deer mouse
1/3.0
1/86
35-g mouse
3,000-g striped skunk
a Example (Row 1): The body weight ratio is 15 = 300/20, and the adjustment factor applied to test-
species measurement (on a BW basis) based on BW3/4 scaling is 151/4or approximately 2.
5. RECOMMENDED DEFAULT SCALING PROCEDURES
The evidence available is held to support the following recommendations.
A general recommendation is to recognize that nonlinear body-size effects are likely in
biological data, toxicological or otherwise, based on species with widely different typical body
weights. Examples that are likely to be of toxicological significance are allometry in rates of
uptake or clearance of toxicants, with rates tending to be slower relative to body weight in
larger species. Scaling relationships for toxicity measurements reflect the combined effect of
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scaling relationships involving multiple biological processes that are expected to determine
species toxicity.
For purposes of development of specific recommendations for terrestrial wildlife, most toxicity
measurements can be classified as single-dose measurements designed to measure lethality
(generally as an LD50), repeated-dose, or "dietary" dose levels from longer-term experiments.
(Repeated dose and dietary dose levels may be interconverted with assumptions particularly
regarding ingestion rate, but the distinction is considered important at least in pesticide
assessment.) Repeated-dose studies may be lethal by design (e.g., a feed study used to
estimate the LC50). Dose-related lethality may sometimes be observed in studies not designed
to be lethal.
The proposed default allometric exponent for single dose LD50s is b = 1 (extrapolation on a
simple body weight basis). The primary support for this recommendation is the analyses of
Rhomberg and Wolff (1998) and Sample and Arenal (1999), using single-dose LD50 studies. The
same is proposed for any dose-based study (single or repeated dose) with substantial lethality
(about 50% or greater).
The proposed default for toxicity extrapolation for repeated-dose studies with limited or no
lethality is b = 3/4. Similar assumptions lead to a default no allometric scaling for "dietary" (or
food based") measurements (i.e., measurements reported as toxicant concentrations in food).
It may be noted that toxicity results reported as a food dose rate (e.g., mg/kg-d) will generally
derive from a food concentration, and so there is an argument for avoiding explicit allometric
scaling, but to use the food concentration without conversion and directly assume that equal
concentrations are equipotent across species (Sample et al., 2014). Exceptions would include
where an extrapolation would be between species not considered to be comparable with any
available adjustment. A possible example would be if a test species is monogastric, and the
receptor species is a ruminant that consumes and ferments a high volume of plant material.
The U.S. EPA (2011) proposes that the rationale for 3/4 scaling applies when physiological
processes function in a similar way with exposure as without exposure. This suggests that the
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rationale for such scaling is inapplicable at some level of lethality. (In particular we do not
propose b = 0.75 for LD50s.) However, limited lethality does not seem to disqualify doses from
scaling with b = 0.75: The latter has in fact been supported by U.S. EPA (2011) based on
correlation of rodent LD10 measurements to human doses that were chosen to be generally
non-lethal (Travis and White, 1988).
When low levels of lethality are observed only after a week or more in a repeated-exposure
scenario, this may be taken as an indication that clearance mechanisms have had enough time
to act to reduce toxic effects to some degree, and that BW3/4 adjustment or use of a dietary
approach may be appropriate. (In discussing single- versus repeated-dosing, Rhomberg and
Wolff (1998) discuss the concepts of standing levels and rates of regeneration of toxicological
defenses.) Simple PK models are potentially of use in interpreting acute and subacute response
data. Fischer (2005), in an ecological context, extrapolated a half-life across species; however
the U.S. EPA (2014) discussion of the extrapolation of PK parameters, in a human health
context, needs to be taken into account.
It is possible that the biological basis for BW1 extrapolation of highly lethal doses is applicable
as well to some sublethal effects associated with single doses. However, U.S. EPA (2011)
generally favors BW3/4 for sublethal acute effects.
For dietary sublethal effects it may be unnecessary to account for allometry explicitly (e.g., by
converting measurements to dose rates such as mg/kg-day and then applying allometric
scaling). No theoretical basis was encountered for a general body-weight dependency of
average tissue concentrations on body weight for species exposed to similar concentrations in
feed. Thus, as a rule, a critical dietary concentration determined for a test species can be
assumed for the assessment species as well. A set of assumptions that appear to support this
approach include that internal doses are appropriately summarized by cumulation or averaging,
and that rates of assimilation (ingestion plus absorption) and clearance scale in the same way
relative to body weight (e.g., as BW3/4). Then average internal dose does not change
systematically with body size because the allometric effects on assimilation and clearance
effectively cancel (Appendices A & D; O'Flaherty, 1989; U.S. EPA, 2005, 3-7). Note that this
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argument would not justify extrapolations between monogastric species and species that
consume and ferment plant material. These may generally ingest at a high rate relative to body
size, and the effect of fermentation on the toxicant would also need to be considered. Probably
no simple extrapolations to such species from monogastric species will be biologically
defensible.
See Box 1 for a summary of recommended defaults for use of allometric scaling for
characterizing toxicity in ecological risk assessments.
Box 1. Recommended Defaults for Use of Allometric Scaling for Characterizing Toxicity in
Ecological Risk Assessments. Note: Recommendations are subject to a principle of
using as much of the available science as possible in a given situation.
• The default for single dose LD50, or when the duration of dosing before toxicity is
observed is less than a chemical's half-life, is b = 1 (extrapolation on a simple body
weight basis). The same is proposed for any dose-based study (single or repeated
dose) with substantial lethality (about 50% or greater).
• The default for toxicity extrapolation for repeated-dose studies, where toxicity is only
observed after five half-lives, with units reported as a dose rate such as mg/kg-day,
and limited or no lethality is b = 3/4.
• The default for a "dietary" or ("food based") toxicity, reported and applied as
toxicant concentration in feed, is no allometric scaling.
Arguments for BW3/4 scaling as the most common default for sublethal doses include the
following: 1) validation with data on sublethal acute toxicity of anticancer drugs (Travis and
White, 1988; U.S. EPA, 2011); 2) comparability of BW3/4 scaling to application of PBPK-based
AUC estimates (Kirman et al., 2003; Appendix C); 3) a tendency - with various exceptions - for
allometric exponents to be less than 1 relating clearance to body weight across species, in
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compilations such as those of Chiou et al. (1998), Huang et al., (2015), and Tang and Mayersohn
(2005); and 4) an apparent, current viewpoint in veterinary medicine that a BW1 default is
particularly unsafe as a general rule (Hunter and Isaza, 2008). (With regard to the final point,
however, neither BW3/4 scaling nor any other point prediction method is assumed to be safe, in
view of the uncertainties. The best that can be done is to attempt to use all available scientific
information and recognize the remaining uncertainty. Limitations of these lines of evidence are
explored further in the discussion.) These arguments rely to some degree on data involving
substances not likely to be subjects of ecological risk assessment; however, some consideration
may be given to use of information on such substances for purposes of recognizing general
biological patterns.
Risk assessment computations are somewhat simplified if toxicity measurements can be used in
a sense "as reported" (without transformations). However, the implications are different for
different types of toxicity measurements, depending in particular on whether or not the toxicity
is in the form of dose relative to body weight. An "as-reported" use of a dose-based
measurement would be to assume that the same dose can apply to test and assessment species
if both are expressed on a BW basis. An as-reported use of dietary toxicity can assume that the
same critical concentration in feed applies to both test and assessment species. These
approaches are similarly appealing with respect to simplicity but are supported by different
assumptions and have different implications in practice.
Acceptable implementations of allometric scaling could involve triggers based on relative body
weights, comparing test and assessment species. If allometric scaling is considered
burdensome relative to value added, such scaling could be implemented when the ratio of body
weights (assessment species/test species) exceeds a specified threshold. This approach is not
pursued further here, and specific cutoff values based on ratios of body weights are not
proposed, but sensitivity analyses like those in the preceding section would be helpful if such an
approach is pursued.
Cross-species toxicity extrapolation depends on the identification of groups of species and
substances such that extrapolation is to be allowed within but not among groups.
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Extrapolation between conventional vertebrate classes (amphibian, reptiles, birds, mammals)
has been discouraged (Aliard et al., 2009). Here, no effort has been made to develop guidelines
on whether or not species are too different for extrapolations to be allowed. Additional criteria
for appropriate classification may be based on classification or phylogeny, or on physiological or
behavioral traits, such as homeo-/heterothermy, carnivory/herbivory, ingestion of infrequent,
large meals by snakes, concentration of urine in arid conditions, seasonal dormancy, long-
distance migration of birds, differences in renal physiology, employment of fermentation of
plant material by some species but not others, and so on. Further elaboration of allometric
methods to provide distinct criteria for more and smaller taxa, or other species groups, would
depend on having enough data for each group to support a specific scaling rule. Obtaining
toxicity data representing vertebrate classes may be difficult enough, and there could only be
greater difficulty in obtaining adequate data for more and smaller groups of species.
6. DISCUSSION AND FUTURE WORK WITH AN EMPHASIS ON UNCERTAINTIES IN
EXTRAPOLATION
Assessment methodological development seeks to optimize the use of scientific information,
with recognition of uncertainties, allowing that the timely development of assessments
requires precise, practical guidance. There is much that can be done to further the
development of methods that meet such objectives related to body weight scaling. In the
remainder of this section, we recognize some important themes related to these broad
objectives.
Using as Much Scientific Information as Practical, and Allowing for Policy. Recommendations
presented here on scaling toxicity measurements are subordinate to a broader principle of
attempting to fully utilize the science available in a given situation. The applicable science may
suggest a specific approach for scaling toxicity measurements, or an approach that does not
involve scaling. In some situations, the information might support a simple modeling approach.
The information needed will relate to combinations of species and substances; however, it is
not feasible here to identify information likely to be available for every important combination.
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An important objective has been to clarify the PK basis of BW3/4 scaling as promulgated by U.S.
EPA (2011) for human health assessments. A central idea is the use of BW3/4 as a surrogate for
systemic clearance (CL). (Technically, ratios of BW3/4serve as surrogates for ratios of
corresponding CL.) The available science could suggest other surrogates. A ratio of glomerular
filtration rates might be considered as a surrogate for the ratio of species CL (Lin, 1995).
Some authors report allometric results for blood half-lives instead of CL, considering the former
to scale more predictably with body weight (Riviere et al., 1997; Antonissen et al., 2015).
However, U.S. EPA (2014) states that "Half-life is not an acceptable basis for" calculation of
data-derived extrapolation factors (DDEFs). Therefore, blood half-life should be combined with
the volume of distribution to yield CL before use in scaling. (Volume of distribution may be
represented with an uncertainty distribution, if desired - see discussion of quantitative
uncertainty evaluations.)
Echoing U.S. EPA (2011), it is understood that alternatives to the defaults proposed may be
appropriate for particular regulatory entities, based on science or policy, for specific situations.
For example, an entity that regulates pesticides may determine that there is data of sufficient
quality and quantity to justify routine scaling of toxicity estimates, for some pesticides and
receptors, with an allometric exponent developed for the specific situations. An example
proposed for avian risk assessments is Mineau et al. (1996). Regulatory entities may choose to
develop tiered assessment schemes, in which higher-tier assessments require more data.
Relevant Species and Substances for Developing Ecological Assessment Methods. Species
used to evaluate human health effects are a biased sample for ecological assessment purposes.
For example, these species are generally not large predators. However, we advocate
development of comparative frameworks that address variation across species and substances,
without automatically excluding species commonly involved in human health. FARAD
(FARAD.org) is a source of PK information on domesticated mammal species, some of which
could be relevant to wildlife receptors in a given assessment context (also see Martinez et al.,
2006; Mahmood et al., 2006). We have not encountered arguments that domestication as such
reduces the relevance of a species for wildlife assessments.
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The data on PK is fairly rich for pharmaceuticals, but the relevance to ecological assessments of
such data is open to some question. Data on toxicity of anti-neoplastic drugs has played a
significant role in development of the human health assessment approach of U.S. EPA (2011).
Allometric scaling of toxicity of such chemicals is not necessarily limited to mammals:
Antonissen et al. (2015) is an example where a blood half-life for an anti-neoplastic substance
scales allometrically in birds (based on applications to four species including a passerine). The
questions are, what properties of these substances account for the reported success in scaling
their biological properties? And are those properties also characteristic of some substances of
concern for ecological assessment?
Risk assessors are understood to require guidance that is transparent and practical. A helpful
development might be a list of qualitative "indicators for particular applicability of BW3/4 (or
BW1) scaling," easy for assessors to use. These indicators may be based largely on intrinsic
properties such as lipophilicity that are easily measured in vivo. Criteria based on in vivo PK
parameters would be relevant as well, but data for such parameters may often be more difficult
to obtain than information on intrinsic properties considering the variety of ecological
receptors. It is very important for such information to be brought to bear in the identification
of the most appropriate test species to use in a given assessment. Decision trees seem to be
viewed favorably by assessors and may be useful for this purpose.
We think there would be consensus on the value of organizing comparative information in a
form useful to assessors. This could take the form of groups of species and substances that can
be handled in similar ways in an assessment. PBPK modeling, though perhaps infeasible for
most specific assessment situations, could have a role in combination with classifications of
species and substances if the models suggest that a simpler approach provides an adequate
approximation for some groups defined by the classification. The properties of the chemicals
considered by Kirman et al. (2003) should be correlated to how well BW3/4scaling approximates
PBPK results. For purposes of developing comparative frameworks, we encourage information
exchange between human health assessors, ecological risk assessors, and veterinarians.
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Extreme phylogenetic extrapolations. Additional guidance is desirable on extrapolations that
are to be considered extreme because of involving species with profoundly different biology.
Important biological differences between vertebrate groups include the absence of a renal
portal system in mammals and major differences between groups in the regulation of body
temperature. Birds and mammals are as a rule endotherms, other vertebrates ectotherms. (A
related distinction is between homeothermy, poikilothermy, and heterothermy.)
We do not dispute a recommendation (Allard et al., 2009) to avoid extrapolations across
vertebrate classes. In view of the difficulty of such extrapolations, they should perhaps be
treated as a special topic. However, some preliminary remarks are offered. Criteria for
protection of aquatic life are potentially protective of aquatic life stages of amphibia (a
possibility that is not reviewed here). Special allometric procedures have been advocated for
extrapolating pharmaceutical doses from mammals to reptiles or amphibians (Hunter, 2010).
The methodology involves, in addition to a factor based on body weights, multiplicative factors
for each of several groups based on tendencies for species in some groups to have higher
metabolic rates than similar-size species in other groups (e.g., generally higher metabolic rates
for endotherms than for similar-sized ectotherms, higher metabolic rates for passerines than
non-passerines). We note, however, that a current phylogenetic framework for animals
(tolweb.org) places birds with reptiles. Thus, consideration might be given to a unified PK
framework for reptiles and birds. In fact, some similarities relevant to pharmacokinetics are
discussed in Hunter (2010). PBPK modeling is expected to be relatively well developed for
poultry, at least for pharmaceuticals.
Some Areas of Uncertainty
Metabolic elimination of toxins. Any aspect of absorption, distribution, metabolism or
excretion (ADME) may contribute to uncertainty in a given assessment context. Species
variation in metabolic elimination of toxins is a biological factor that may not be handled well
by allometric methods (e.g., Hutchinson et al., 2014). PBPK modeling as used in human health
assessment frequently assumes BW3/4scaling of liver perfusion and metabolic rates are typically
expressed allometrically (e.g., metabolic Vmax = VmaxC*BW3/4) although the allometric
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coefficient (VmaxC) is not kept constant between species. These assumptions result in some
tendency for systemic clearance to scale in a similar way. A parameter of interest is intrinsic
clearance, which relates the rate of metabolism to the concentration of the parent at the site of
metabolism. To evaluate species variation, efforts should be made to make use of information
on enzyme activity, e.g., P450 (see Head and Kennedy, 2010; Manning et al., 2013). It may be
noted that differences in metabolism related to diet have been reported among populations of
a single species (Malenke et al., 2012).
The role of metabolism in determining toxicity to wildlife is reviewed by Hutchinson et al.
(2014). In particular they state that:
"The essential purpose of xenobiotic metabolism is to convert lipid-soluble, non-polar
and non-excretable chemicals into water soluble, polar molecules that are readily
excreted. ... wildlife species with low metabolic competency may exhibit zero-order
metabolic (pharmacokinetic) profiles and thus high API [Active Pharmaceutical
Ingredient] toxicity, as in the case of diclofenac and the dramatic decline of vulture
populations across the Indian subcontinent. A similar threat looms for African Cape
Griffon vultures exposed to ketoprofen and meloxicam, recent studies indicating toxicity
relates to zero-order metabolism [some technical detail redacted]. While all aspects of
ADMET [ADME + Toxicity] are important in toxicity evaluations, these observations
demonstrate the importance of methods for predicting API comparative metabolism as
a central part of environmental risk assessment." [Italics and bracketed parentheticals
added.]
Trophic ecology as an illustration of uncertainty. Two relatively extreme types of diets seem
illustrative, namely hyper-predation (e.g., cats), and low-nutrient plant diets.
Low nutrient plant diets are often associated with gut fermentation, which occurs in the foregut
for some species and in the hind gut for others. Stevens and Hume (2004) provide a general,
comparative account of vertebrate digestive physiology. Use of the best-justified allometric
scaling of an oral toxicity measurement is not expected to reduce uncertainty appreciably in an
extrapolation between species that do and do not use fermentation. However, we cannot say
that clearance of intravenous doses, if available, could not be extrapolated on a BW3/4 basis. As
usual, we underline that the role of scaling is determined by the scientific information relevant
to the context.
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For mammals with a diet essentially entirely of meat Shrestha et al. (2011) report a tendency
towards genetic loss of capability for metabolizing some toxic substances, consistent with low
dietary exposure to plant secondary compounds in some evolutionary lineages. (Incidentally
we note the use by these authors of reconstructed phylogenies rather than taxonomy in
describing evolutionary loss of metabolic function.)
Body size and gastro-intestinal physiology. In developing allometric expressions, it is helpful
(other things being equal) to use species with a wide range of body sizes. In general, datasets
may be sensitive to the largest-bodied species that have been studied. The largest terrestrial
vertebrates will be mostly ruminants that consume and ferment plant material at a high
rate. The largest monogastric species tested may be carnivores (most often dogs). There is a
substantial amount of veterinary literature on extrapolations of pharmaceutical doses involving
large species (Mahmood et al., 2006; Martinez et al., 2006, 2009; Hunter and Isaza, 2008;
Sedgwick, 1993). FARAD.org may be a source of relevant PK information.
Additional Uncertainties and General Remarks. There are of course many uncertainties in the
use of laboratory toxicity measurements. While a thorough treatment is beyond the scope
here, it is well to keep these in mind. Acute lethality is often evaluated using gavage.
Relevance of the results to exposure by ingestion with food is obviously a difficult issue. For
dietary studies, it may be difficult to state exactly the ingestion rate of toxicant. Some feed will
be spilled, and high feed concentrations may elicit aversive responses.
The best summary of internal dose over time might be the subject of further, useful study. The
strongest argument for BW3/4as a PK adjustment appears to assume that the most appropriate
internal dose metric is inversely proportional to BW3/4. This occurs using AUC as the internal
dose metric when elimination scales as BW3/4. However, Cmax cannot be excluded as the most
biologically relevant summary. The ideal summary of concentration over time could be
something different from either of these.
Some literature (e.g., Huang et al., 2014) is disparaging towards adoption of any default
allometric approach, whether with exponent 3/4 or some other value, on grounds that not
enough biology would be considered and that exponents are variable. Nonetheless,
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extrapolation of toxicity across species is often necessary, and even the use of BW1 involves an
assumption about the role of body weight. A comprehensive account of extrapolation
uncertainty and variation in species sensitivity will not be attempted here.
Differences in species sensitivity may relate to diet (with associated differences in physiology,
especially gastro-intestinal), toxicodynamics, and toxicokinetics. Pharmacokinetic (PK)
considerations are sometimes classified as ADME (absorption, distribution, metabolism, and
excretion). Components of PK that can impact species extrapolation are differences in binding
to molecular components (a factor in distribution), and differences in metabolism (discussed in
more detail below). These may interact in complex ways, e.g., transformation of a substance to
compounds that may or may not be as readily excreted as the parent. Calabrese (1991)
provides a general discussion. However, despite of the variety of ADME factors that may be
important, an analysis of clearance (CL) for 115 xenobiotic substances report generally better
support in mammals for BW3/4than BW1 scaling across four categories of substances, (i)
proteins, (ii) compounds eliminated mainly by renal excretion, (iii) compounds eliminated by
metabolism, or (iv) compounds eliminated by renal excretion and metabolism combined (Hu
and Hayton, 2001). All subgroups except (ii) showed a b value statistically "not different from
0.75." For group (ii) the average coefficient "was 0.65, which differed from 0.75 but not from
0.67". (However, we may note that dependence of allometry on mechanisms of elimination
may be described at different levels of granularity by different biologists. For example, Walton
et al. (2004) summarize allometric effects for subcategories of renal excretion among mammals
for purposes of extrapolation to humans.12; references alluded to in footnote are in Walton et
al. (2004)).
12 "The prediction of kinetics in humans using allometric scaling of data from a range of animal species has been
successful for a number of compounds that are eliminated largely unchanged in the urine (e.g. [references for 5
compounds]), and for the renal clearance aspect of compounds eliminated by both metabolism and excretion
([references for 2 compounds]). However, allometric scaling has been less successful when the compound
undergoes active transport in the kidney (e.g. [reference for napsagatran]) or is extensively bound to plasma proteins
(e.g. [references for 2 compounds]). Mahmood (1998) analysed the results of allometric scaling of renal clearance
for eight drugs and concluded that renal clearance in humans would be under-predicted for drugs cleared largely by
tubular secretion."
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Species variation in sensitivity to a substance is expected to be determined in part by variation
in aspects of species biology that determine pharmacokinetics. Extensive information exists on
species variation in PK parameters for pharmaceuticals (e.g., Mahmood, 2005). Situations
involving interspecies extrapolation of pharmaceutical effects (efficacy or toxicity) include first
uses of a new drug in humans and veterinary pharmacology.
When adequate data is available it is better to estimate an allometric exponent for PK
extrapolation from the data than to use a fixed allometric exponent such as 1 or 3/4. However,
use of the most appropriate allometric approach does not guarantee confident predictions for
every substance. For example, an analysis by Riviere et al. (1997) of comparative
pharmacokinetics of 44 drugs noted a large spread of individual values around the regression
line. (For a statistic to represent this residual spread in regression, a common choice is the R2
statistic.)
We note that in collections of PK data representing multiple substances, estimates of the
exponent are variable and sometimes far from 0.75 but still cluster closer to 0.75 than to 1
(e.g., Tang et al., 2007 Table 2, for i.v. administrations). Huang et al. (2015) found a mean of
0.82 for serum CL for 85 drugs. On the one hand this variation suggest that extrapolated point
estimates are highly uncertain and could pose unacceptable risk, particularly from a standpoint
of first use in humans. At the same time, such results point to the possibility of the use of
uncertainty distributions or uncertainty factors, rather than point estimates only. The central
tendency (e.g., median) of an appropriate distribution might be 0.75,1, or some other value. A
distribution centered on 0.75 could assign appreciable probability to a value of 1 or lower. It is
understood that options for collecting additional data may be limited in a site assessment
context. However, it is not possible to say with confidence what data may be available in the
future.
Qualitative and Quantitative Uncertainty Evaluations. No one expects that every important
biological difference between species will be addressed by scaling of oral toxicities. Where an
extrapolation on a simple body weight basis would not be viewed as plausible because of a
likelihood of profound biological differences between species, the extrapolation based on
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changing the exponent to 3/4 would not, as a rule, be plausible either. Therefore, an initial step
is to decide whether an extrapolation between a given pair of species is plausible at all.
Assuming that an extrapolation is plausible for a given pair of species, but some likely biological
differences are identified, a subsequent question can be, can species differences be quantified
with some proportionality constant (factor) in an expression relating internal dose (e.g., AUC) to
exposure? For example, absorption is customarily quantified as fraction of ingested dose that is
absorbed. For pharmaceuticals, some factors affecting absorption of pharmaceuticals have
been reviewed by Kararli (1989). If the biological factor can be represented this simply, we only
need information on the ratio of the proportionality constant for the two species. (We do not
need the two specific values of the constant for each species.) Consideration might be given to
a quantitative uncertainty approach (uncertainty factor or distribution). The median should be
1 if there is no evidence to suggest which species has the larger value of the constant. A likely
"first cut" for an uncertainty distribution, when the variable is positive and there is no clear
upper bound, is a lognormal, with allometric assumptions readily incorporated (e.g., Chiu and
Slob, 2015). In particular Monte Carlo simulation is not required for uncertainties for products
of random variables modeled as lognormal.
If species sensitivity differences cannot be expressed this simply, then a more complicated
approach might be considered. Covering possible models is beyond the scope of this document
(but see Fischer, 2005). In particular, at this time we have not investigated whether there is
some practical modeling approach for using information on intrinsic clearance (which reflects
the metabolic rate at a cellular level).
Data and Methods for Developing Empirical Scaling Expressions. We have no grounds for
dismissing any analyses cited here, that have estimated allometric exponents by regressing
toxicity measurements on body weight across species. The evidence can be strengthened if a
purported difference among groups can be supported by biological arguments.
We observe that guidance for such analyses could relate to quality of toxicity data, appropriate
body weights (e.g., possible use of species default body weights), handling of cases of multiple
studies per substance and species, number of species, appropriate variety of species (e.g.,
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range of body sizes, phylogenetic representation, diversity of physiology), meaningful groupings
of species based on life history and/or phylogeny, meaningful groupings of substances, criteria
for deciding if allometric differences are similar or different among groups, and statistics (e.g.,
mean or median) for summarizing results for groups of chemicals or species.
The allometric exponent is usually estimated by ordinary bivariate linear regression, relating the
log of the dependent variable to the log of body weight. The assumption of no error in body
weights is probably acceptable for applications with species body weights ranging over several
orders of magnitude (discussed, for example, by Kilmer and Rodriguez, 2017). However, if the
body weights are largely at one extreme (say, most of the species included are small), then
results may be very sensitive to measurements for a few species representing the other
extreme (say, large species). Ideally, both extremes will be represented by biologically and
taxonomically diverse species.
Over-representation of some taxa can in principle be handled with phylogenetic generalized
least squares (e.g., Smears and Rohlf, 2016), an essentially straightforward extension of linear
mixed modeling. (However, this methodology may not yet be very well known among
ecological assessors.) Regression results without such adjustment assume independent
observations and are expected to overstate the information in the data to some degree, so that
it may be too easy to find differences among groups (in statistical terms, to make Type I errors).
Phylogenetic distance may serve as a surrogate for unrecognized physiological differences
among species. Standard taxonomy does not necessarily provide a good reflection of
phylogeny. Several uses of reconstructed phylogenies were encountered in the course of
preparing this review (Bakken et al., 2004; Shrestha et al., 2011; White et al., 2009). Additional
development would be required if such methods are to be practical for ecotoxicologists.
Some investigation may be given to other statistical methods that 1) can address multiple
predictors of a parameter of interest such as CL, not only body weight (i.e., multivariate
prediction methods), or 2) emphasize accurate prediction. In particular, meta-analyses of
allometric exponents computed for multiple species and substances are subject to decisions on
"lumping versus splitting." Groups with too few species may not provide enough data for a
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reliable summary (e.g., median), while groups that are too large may be so heterogeneous that
some species are not accurately represented by a group summary. In the context of predictive
accuracy, this is recognized as a tradeoff between bias (with groups too large) and variance
(with groups too small), or as a problem of underfitting versus overfitting (as discussed, for
example, by Efron and Hastie, 2016). Statistical decision tree methods (e.g., Anderson et al.,
2014) will result in a good balance of parsimony and fit, with an objective of accurate
prediction. To compare a few alternative groupings, the Akaike Information Criterion may be
considered.
The use of quantitative uncertainty distributions (e.g., with Monte Carlo) can assign greater
weight to the exponent (0.75 or 1 etc.) that is considered most plausible. Bayesian methods in
particular can be used for developing uncertainty distributions, allowing rigorously for values
other than a default, where supported by data. A Bayes prior distribution for the exponent may
be centered on the single value considered a priori most plausible. The uncertainty in the
exponent, considering the data, would be expressed in a posterior distribution for the
exponent. Such a specialized approach might need to be restricted to higher-tier assessments.
For data on chronic effects, attention may focus largely on reproductive effects. Multiple
endpoints may be available from a given study. Studies may report a NOAEL, which may be
based on different endpoints in different studies. It may be preferable to focus on a single
endpoint and estimate an effective exposure for each study by statistical curve fitting. Perhaps
the ideal endpoint for such analyses would have high ecological significance, as well as being
amenable to curve fitting.
Model Validation. If current practices for extrapolation of toxicity values are viewed as
validated, then the validation criteria should be stated and used to compare alternative
extrapolation approaches. Assuming that toxicity values will be extrapolated on some basis,
recourse is to the approach with the strongest scientific basis. The current validation status of
BW1 extrapolation seems dubious. In a zoological pharmacological context, Hunter and Isaza
(2008) treat BW1 as a form of extrapolation, in fact a risky one:
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"From practical experience, many drugs do not have a simple linear relationship relative
to weight. At the extremes of the weight range, this method tends to overdose large
animals and underdose small animals, which may be very clinically significant. Again,
the simplicity of the calculation tempts many practitioners to use this potentially
dangerous method of extrapolation without consideration of the consequences. In fact,
many clinicians who read dosage recommendations from formularies are not even
aware that an extrapolation is being made nor the risks of the associated assumptions."
[Italics added.]
A famous example of over-dosing a large species, based on a BW1 extrapolation, is West et al.,
1962).
Ecological assessment, as a discipline, may define validation criteria that are meaningful and
feasible with data likely to be available. Validating that a risk assessment methodology with a
particular approach to toxicity data will protect a species in the field is beyond the scope of our
discussion, considering that the assessment will contain other components, e.g., exposure
assessment, that are uncertain (and outside the scope of our discussion). More focused
validation would relate to the efficacy of scaling as, specifically, a PK adjustment. We suggest
that there may be some role for exposure biomarkers if available, to provide a limited
validation. These could point to species differences in internal exposure not accounted for by
the suggested BW3/4 scaling of toxicity values.
It may be helpful to identify validation exercises based on predictions of PBPK models. An
impressive validation of a mechanistic (e.g., PBPK) model would be if it is shown to provide
accurate predictions when applied in conditions other than those where the model was initially
developed, e.g., a different time pattern of dosing, without re-optimizing parameters or
complicating the model (e.g., adding more compartments). A simple approach such as BW3/4
scaling of toxicity would have a degree of validation, for some combinations of species and
substances, if shown to be consistent with a validated PBPK representing those combinations.
28
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APPENDIX A. BODY SIZE AND TISSUE STEADY STATE CONCENTRATION: A NUMERICAL
ILLUSTRATION
Tissue concentration of a toxicant is determined by relative rates of uptake and elimination.
Both types of processes are subject to allometric trends across species (Schmidt-Nielsen, 1984;
U.S. EPA, 2011). As a rule, larger species will consume less on a weight specific basis, and also
eliminate less rapidly (on a weight specific basis). A bias may result if the risk calculations
address allometry for one type of process - uptake or elimination - but not the other. While
direct measurements of some rates of interest may be available for a species of interest (e.g.,
U.S. EPA, 1993), this section considers the effect of assuming that rates conform to general
expectations based on allometry.
In theory AUC or steady-state concentration should not depend on body weight other things
being equal (O'Flaherty, 1989). The table below illustrates the theoretical lack of BW
dependency, based on comparing tissue concentration trajectories for a 35 g "mouse-sized"
species and a 300 g "rat-sized" species. It is assumed that the environmental concentration is
the same for each species and that differences result from allometry in rates of uptake and
clearance. The rate for each of the two species are assumed to conform to the same
proportionality to BW3/4across species.13
It may be helpful to assume that the elimination process is glomerular filtration, simply
because for that process the pharmacokinetic definition of "clearance," as volume of tissue
cleared per unit time, seems relatively obvious. A general definition of clearance (Toutain and
Bosquet-Melout, 2004) applies to elimination by various mechanisms.
We assume initially (Column 3) that each species ingests a toxicant (in arbitrary mass units)
proportional to whole-organism metabolism. If the toxicant is mixed in tissues before any
elimination, the result (Column 4) is a lower tissue concentration for the larger animal,
13 More technically, if the two species belong to a group with the same allometric coefficient and exponent then the
coefficient can be ignored for purposes of the illustration. Also, the conclusions do not depend on the value of the
allometric exponent b (3/4 or otherwise) so long as the same value is assumed for both uptake and clearance.
38
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reflecting the lower weight-specific food consumption (lower mass food intake per unit body
mass). Next (Column 5), we assume that a volume of body fluid is removed, again typical for
the body weight, and replaced with uncontaminated fluid. We assume that the fraction of
ingested toxicant removed is proportional to the ratio of fluid removed to body weight.
(Technically, for the argument presented we are using body weight as a surrogate for volume of
distribution, which would be used if directly available.) After this elimination event, the tissue
concentrations are more nearly equal in the two species (Column 8). Columns 9 and 10 show
the results of repeating these computations for two additional cycles of ingestion and
elimination, which suffice for the illustration. While the simulation approach is crude, it
suggests tissue concentrations converging to the same value in each species, as expected based
on theory.
Example of uptake and elimination related to body size. Table columns have been numbered to facilitate cross-referencing with explanations
(in text).
1
2
3
4
5
6
7
8
9
10
Species
BW
(g)
Toxicant
Mass
Uptake
BW3/4
Tissue
Cone.
BW 1/4
Glomerular
Filtration
(mass in 1 time
unit)
% Toxicant
Eliminated
% Toxicant
Remaining
Tissue
Cone.
(Add 1
Step)
(Add 1
Step)
Mouse-sized
35
14.39
0.41
14.39
41%
59%
0.24
0.38
0.47
rat-sized
300
72.08
0.24
72.08
24%
76%
0.18
0.32
0.43
39
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APPENDIX B. ADDITIONAL DETAIL ON SELECTED SOURCES
Allard, P., Fairbrother, A., Hope, B., Hull, R., Johnson, M., Kapustka, L., Mann, G., McDonald, B.,
and Sample, B. 2009. Recommendations for the Development and Application of
Wildlife Toxicity Reference Values. Integr Env Assess and Man 6:28-37.
Type of
Source
Assessment Context /
Taxonomic Restriction
Substances of
Special Interest
Ch ronicity
Objectives / Approach
General policy document
Journal
article
Ecological/Birds and
mammals
General
General
addressing multiple
practices in toxicity
reference value derivation
Procedure Described or Recommended: "Don't use allometric dose-scaling with body
mass when assessing chronic/subchronic toxicity between species. ... Allometric scaling
...has been used for wildlife risk evaluations despite its multiple limitations. ... is no
longer recommended for use in wildlife risk assessment (U.S. EPA, 2005). First,
supporting data are limited. Much of the mammalian data are based on anticancer
drugs evaluated in Freireich et al. (1966) rather than contaminants typically evaluated in
wildlife risk assessments. Second, the allometric scaling models developed for both
human and wildlife risk assessment are all based on acute toxicity data. Their
applicability to chronic toxicity data is unknown.... Because modes of action can vary
dramatically for the same chemical over acute and chronic exposures (discussed in more
detail below), it is likely that interspecific scaling factors based on chronic toxicity data
also will differ from those based on acute toxicity data. Additionally, given the variation
in cross-species physiological responses in different organ systems, it is reasonable to
expect multiple chronic scaling factors for a given chemical, depending on the mode of
action considered. In their current forms, neither allometric scaling nor ICE [interspecies
correlation estimation] models represent chronic toxicity, and, therefore, their
application to chronic data is not recommended. In the absence of suitable models, we
favor the use of toxicity information as reported, because it is often unknown whether
target species would be more resistant or more sensitive."
40
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Mineau, P., Collins, B., and Baril, A. 1996. On the use of scaling factors to improve interspecies
extrapolation of acute toxicity in birds. Reg Toxicol Pharmacol 24:24-29.
Type of Source
Assessment
Context/ Substances of
Taxonomic Special Interest
Restriction
Chronicity
Approach
Journal article
presents
analysis of
pesticide
lethality to birds
Ecological/Avian towards
cholinesterase
inhibitors
37 pesticides,
"heavily weighted"
LD50
Estimate
allometric
coefficient using
regression
Procedure Described or Recommended: "We used an avian LD50 database to derive
empirically the appropriate scaling factor for birds. With a subset of 37 pesticides of varying
structures but heavily weighted to cholinesterase inhibitors, we found that the appropriate
scaling factor in birds is usually higher than 1 and can be as high as 1.55. Extrapolations on the
basis of weight alone or, worse, the use of inappropriate mammalian scaling factors could lead
to serious underprotection of small-bodied bird species modeled in the course of risk
assessment procedures."
41
-------
Rhomberg, L., and Wolff, S. 1998. Empirical scaling of single oral lethal closes across mammalian
species based on a large database. Risk Anal 18:741-753.
Type of Source
Assessment
Context /
Taxonomic
Restriction
Substances of
Special Interest
Ch ronicity
Objectives /
Approach
Journal article
presents meta-
analysis of lethal
effects and
discusses policy
implications
Human
health/Data
for mammals
in multiple
orders
Data analysis
based on 135,000
substances,
largely of
occupational
human health
concern.
Single-dose
lethal
(LD50)
Compare alternative
body weight
adjustments based
on pairwise species
comparisons
Procedure Described or Recommended: "We find a good correspondence of LD50 ... across
species when the dose levels are expressed in terms of mg ... per kg of body mass. ... contrast
with earlier analyses that support scaling doses by the 3/4-power of body mass to achieve
equal subacute toxicity of antineoplastic agents. We suggest that, especially for severe toxicity,
single- and repeated-dosing regimes may have different cross-species scaling properties, as
they may depend on standing levels of defenses and rate of regeneration of defenses,
respectively."
42
-------
Sample, B., and Arenal, C. 1999. Allometric models for interspecies extrapolation of wildlife
toxicity data. Bull Environ Contam Toxicol 62:653-663.
Type of Source
Assessment
Context/
Taxonomic
Restriction
Substances
of Special
Interest
Ch ronicity
Objectives /
Approach
Journal article
presents meta-
analysis of lethal
effects and
discusses policy
implications
Ecological/Birds
and mammals
Multiple
classes of
organic and
inorganic
compounds
Single-dose
lethal
(LD50)
Regression analysis
of 2,853 lethal oral
dose measurements.
Objectives were
characterization of
relationship to body
weight and
comparison of birds
to mammals
Procedure Described or Recommended: "Do not extrapolate from birds to mammals or vice
versa. Use a chemical-specific scaling factor or possibly a factor for a chemical group, e.g.,
chlorinated organics ... . Use BWA1-2 (birds) or BWA0-94 (mammals)." [Note that the avian value
suggested by these authors derives from Mineau et al., 1996.]
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APPENDIX C. BODY WEIGHT SCALING AS AN APPROXIMATION OF PHARMACOKINETIC
MODELING - ADDITIONAL ANALYSIS OF PUBLISHED RESULTS
For the human health assessment context, U.S. EPA (2011) indicates PBPK modeling as the
preferred approach for addressing PK considerations. The model-based analysis of Kirman et al.
(2003) is cited by U.S. EPA (ibid.) as support for BW3/4 scaling. The analysis draws on previous
modeling efforts for each of 13 substances. The table below reproduces selected results from
that analysis, along with a re-expression in terms of the best allometric exponent b for
approximation of PBPK modeling, assuming blood AUC to be the appropriate summary of
tissue-level exposure over time.
The authors evaluated BW3/4 scaling by comparison to model-based predictions of blood AUC
for mouse, rat and human for 12 lipophilic, predominantly volatile chemicals. The results were
expressed as equivalent internal doses (EID) comparing two species, which are ratios of AUC
(AUCa I AUC/,) at the same external dose (mg/kg-d). (For various quantities, subscripts h and a
will indicate respectively human and non-human animal.) The table below gives the geometric
mean EID values for the 9 chemicals as reported in their Table 4, based on model blood
concentration of parent.
The approach taken by the authors was to evaluate whether their PBPK-based ratios were
reasonably close to BW3/4 expectations. Here, instead of taking 3/4 as the point of reference, an
effective allometric exponent is computed using the formula
7 * log EID
b = 1
¦0g(BW7BwJ •
Here BWa and BW/, are species body weights. This approach identifies a value of b that is in a
sense best for purposes of approximating the effect of using PBPK modeling (assumed to be the
preferred approach).
Overall, these results seem to provide reasonable support for BW3/4 scaling as a PK-based
adjustment. Extrapolation on a simple body weight basis appears less supported for parent
compound. It should be noted that some geometric mean EIDs (bold) are reported by the authors
to only one digit of precision.
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It might happen that parametrization of a PBPK model makes use of BW3 4 scaling in estimation
of some parameters. Then, there would be a degree of circularity in using the results to claim
that allometric scaling of toxic doses approximates PBPK-based results. Nevertheless, it seems
to be still of interest how well BW3/4 scaling approximates current PBPK-based estimates.
Allometric exponents for approximation of PBPK model-based ratios of
species AUCs for
Zontinuous Oral Exposures.
Dose
(mg/kg-d)
mouse - human
rat - human
EID
effective b
EID
effective b
0.0001
0.13
0.74
0.28
0.77
0.001
0.11
0.72
0.25
0.75
0.01
0.10
0.70
0.23
0.73
0.1
0.10
0.70
0.22
0.72
1
0.09
0.69
0.22
0.72
10
0.08
0.67
0.19
0.70
100
0.04
0.58
0.14
0.64
1000
0.05
0.61
0.29
0.77
10000
0.11
0.72
0.41
0.84
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APPENDIX D. PHARMACOKINETIC DERIVATIONS OF THE ALLOMETRIC SCALING PROCEDURES
U.S. EPA (2011) advises in the context of human health assessments that allometric scaling
serves primarily to address pharmacokinetic (PK) considerations and to some degree
pharmacodynamic (PD) considerations. Two PK derivations are given for BW3/4 scaling of
doses. Each assumes some form of averaging or cumulation of internal toxicant concentrations,
as the toxicologically most appropriate summary of a tissue-time curve. In addition, we show
the theoretical independence of equilibrium tissue concentrations from body size. Results in this
section have been compiled for convenience, with no claims of originality.
Models here are simplistic, first-order and single-compartment. Lin (1995) can be recommended
as an introduction to species PK differences based on simple models (allowing that the source
could have a particular focus on extrapolation among mammals). Such models may be taken as
a reasonable basis for low-tier assessment methodologies, at least.
Two species are indexed 1 (test species with toxicity measured) and 2 (assessment species with
toxicity measurement unavailable). It may be helpful to think of extrapolating from Species 1 to
Species 2. Various quantities are subscripted 1 or 2 particularly W\ and the body weights of
the two species.14
The allometric exponent will be assumed to be 3/4 (however, results are easily generalized).
Scaling based on biological equivalence of AFC from single doses. Here we assume a single
exposure event recorded as mg toxicant per kg body weight. Tissue concentration eventually
diminishes with time as the toxicant is eliminated by some mechanism, as described by a
concentration-time curve. The area under a concentration-time curve (AUC) may be considered
as a basis for dose equivalence results (e.g., U.S. EPA, 2011). AUC is a measure of cumulative
internal exposure, with units concentration*time (say, min*mg/L).
14 This section uses color- a dark green for quantities associated with the test species (Species 1) and red for the
assessment species (Species 2).
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The AUC-based extrapolation of toxic dose (mg/kg) from Species 1 to Species 2,
(wx\°-2S
toxicity2 = toxicity1 X J
can be derived from an expression for AUC (Lin, ibid., Expression 4),
AUC _ Fx dose
clearance
Here dose is mass (mg) of chemical ingested, F (between 0 and 1) is bioavailable fraction, and
"clearance" (units volume/time) denotes the volume of tissue cleared of toxicant per unit time
(units volume/time). Lin (ibid.) expresses Fasa product of factors representing fractions
absorbed and surviving breakdown in the liver and gut wall. In any case, such factors are here
assumed similar between Species 1 and 2, so that they cancel approximately in the derivation.
(However, refinements of extrapolation methodology may be based on known species
differences in such factors.)
Suppose that the biological response will depend on AUC in the same way, for Species 1 and 2,
that is, a given AUC will or will not produce an effect in Species 1 according as it does or does
not produce the effect in Species 2. Differences in AUC for a given dose are assumed to be
based on differences in clearance. A first-order clearance process is assumed (i.e., one that is
independent of dose), with a rate that scales across species with an exponent 3/4, as with
glomerular filtration rate (GFR) or other processes that scale with body weight in the same way
as blood flow (Lin, ibid.; U.S. EPA, 2011 particularly Table 4.1). Then from the AUC
expression we have
dose^, dosej
where dose- is the dose just sufficient for the toxicological effect in Species i (/ = 1 or 2).
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Rearranging this expression leads to the factor for extrapolating a dose in mass units (not BW-
normalized):
/W2\3/4
dose*, = dose, x —
KWj
To derive the factor appropriate for application to a BW-normalized toxicity measurement, first
multiply both sides of the previous equation by 1:
dose, dose, (W2\3^
W2 = Wt ———~ (—-) .
2 W2 1 \wj
Then rearrange to yield the scaling expression:
dose^ dose^ Wx /VK2\3/4
___ _
_ dose^ /lli\1/4
Wj Wj '
In summary, while BW3/4 scaling of a toxic dose that is not BW-normalized (in mass units)
involves a factor (W2/W1)314, the corresponding factor for a BW-normalized toxic dose is
(Wi/W2)m (also see U.S. EPA, 201 1, ix).
Scaling for repeated exposure based on biological equivalence of equilibrium body burden.
Now assume that dosing is repeated, and the results are reported (for the sake of concreteness) in
units mg toxicant per kg body weight, per day (mg/kg-d). An extrapolation expression can be
derived by viewing an average dose as based on a sum of AUCs corresponding to individual
meals. A stochastic version of this idea may be useful for probabilistic risk assessment based on
exposures that vary in space and time, and feeding behavior more or less unpredictable. For a
deterministic approximation of average body burden we may view uptake and elimination as
continuous and use steady-state value, found by equating uptake and elimination rates, as an
approximation of average body burden. For a single species, consider the amount of toxicant in
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any 1-kg volume of tissue and write the steady-state equation for that volume, for a single time
unit, as
mg toxicant taken in = mg toxicant eliminated
= (mg toxicant in tissue) x (fraction eliminated).
Regarding the right-hand side note that the total volume of tissue cleared of toxicant in a unit
time for the whole organism is assumed proportional to W3/4 Assuming the same fraction
eliminated per unit time in each 1 kg portion of tissue, that fraction would be Wil4/W= MW] 4 so,
[toxicant in tissue]
toxicant taken in oc —
VI/1/4
or
toxicant in tissue oc [toxicant taken in] x VK1/4.
(The proportionality signs indicate neglect of constants that are assumed to be similar across
species and cancel in the development of the extrapolation factor.) Finally, to develop the factor
for extrapolation from Species 1 to Species 2 we want to know the Species 2 external exposure
(mg/kg-d) that will yield the steady-state body burden (mg/kg) sufficient for a toxic effect in
Species 1 (in each 1 kg volume of a given species.) Equating body burdens we write:
[Species 2 mg/kg-d] x w21/4 = [Species 1 mg/kg-d] x Wi114
which can be solved to yield the usual extrapolation factor.
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Steady-state body burden does not depend on body weight when uptake and elimination
scale to the same power of body weight (also see O" Flaherty, 1989). The following relates to
species assumed to belong to a group which has the same allometric expressions for toxicant
uptake and clearance rates. Equilibrium tissue concentration can be identified by setting input
rate equal to elimination rate:
toxicant input (mg) = toxicant elimination (mg)
Suppose that metabolic rate and clearance scale to the same power b of body weight (e.g., both
are BW3 4). Then
Cf00dWb oc Ctissue W*
where Cf00d, Ctissue are respectively concentrations of toxicant in food and (at steady state) tissue.
(On the left-hand side, the exponent is based on food ingestion being proportional to metabolic
rate.) Therefore, tissue concentration is simply proportional to feed concentration given these
relationships. If toxicity is measured as a feed concentration associated with a biological effect,
then we know of no PK argument for body weight scaling of food concentrations. With different
allometric exponents for uptake and elimination there may be some body weight dependence of
critical feed concentration but the effect may be small if the exponents are not too different.
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