TABLE CONTENTS
viii
3. ALLOMETRIC EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
3.1. FOOD INGESTION RATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3
3.1.1. Birds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4
3.1.2. Mammals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5
3.1.3. Reptiles and Amphibians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6
3.2. WATER INTAKE RATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7
3.2.1. Birds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8
3.2.2. Mammals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10
3.2.3. Reptiles and Amphibians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10
3.3. INHALATION RATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11
3.3.1. Birds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11
3.3.2. Mammals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12
3.3.3. Reptiles and Amphibians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12
3.4. SURFACE AREAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13
3.4.1. Birds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13
3.4.2. Mammals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14
3.4.3. Reptiles and Amphibians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14
3.5. ALLOMETRIC EQUATIONS FOR METABOLIC RATE . . . . . . . . . . . . . . . . 3-15
3.5.1. Birds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-18
3.5.1.1 Basal Metabolic Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19
3.5.1.2. Existence Metabolic Rates . . . . . . . . . . . . . . . . . . . . . . . 3-20
3.5.1.3. Free-Living Metabolic Rate . . . . . . . . . . . . . . . . . . . . . . . 3-22
3.5.1.4. Temperature and Metabolic Rate . . . . . . . . . . . . . . . . . . 3-24
3.5.2. Mammals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-26
3.5.2.1. Basal Metablic Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-26
3.5.2.2. Resting Metabolism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-27
3.5.2.3. Field Metabolic Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-27
3.5.2.4. Temperature and Metabolic Rate . . . . . . . . . . . . . . . . . . 3-28
3.5.3. Reptiles and Amphibians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-29
3.5.3.1. Basal and Resting Metabolic Rates . . . . . . . . . . . . . . . . 3-29
TABLE CONTENTS
ix
3.5.3.2. Free-Living Metabolic Rates . . . . . . . . . . . . . . . . . . . . . . 3-30
3.6. MATH PRIMER AND UNIT CONVERSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 3-32
3.6.1. Summary of Operations Involving Logarithms . . . . . . . . . . . . . . 3-32
3.6.2. Summary of Operations Involving Powers . . . . . . . . . . . . . . . . . . 3-32
3.6.3. Unit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33
3.6.3.1. Approximate Factors for Metabolic Equations . . . . . . . 3-33
3.6.3.2. Exact Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33
3.7. ESTIMATING CONFIDENCE INTERVALS . . . . . . . . . . . . . . . . . . . . . . . . . . 3-34
3.8. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-38
LIST OF TABLES
xii
Table 3-1. Metabolizable Energy (ME) of Various Diets for Birds
and Mammals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5
Table 3-2. Allometric Equations for Basal Metabolic Rate (BMR) in Birds . . . . . . . . 3-21
Table 3-3. Regression Statistics for Nagy's (1987) Allometric Equations
for Food Ingestion Rates for Free-Living Animals . . . . . . . . . . . . . . . . . . 3-36
Table 3-4. Regression Statistics for Nagy's (1987) Allometric Equations
for Free-Living (Field) Metabolic Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-37
xiv
LIST OF FIGURES
Figure 3-1. Monthly Variation in Energy Budget Estimated for a
House Sparrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-25
3-1
3. ALLOMETRIC EQUATIONS
Values for key contact rate factors such as food and water ingestion rates have
been measured for few wildlife species. In this section, we describe allometric equations
that can be used to estimate several exposure factors on the basis of animal body weight
using models derived from taxonomically similar species. We emphasize, however, that
measured values from well-conducted studies on the species of concern are likely to be
more accurate and to have narrower confidence limits.
Allometry is defined as the study of the relationships between the growth and size
of one body part to the growth and size of the whole organism; however, allometric
relationships also exist between body size and other biological parameters (e.g., metabolic
rate). The relationship between the physiological and physical parameters and body
weight frequently can be expressed as:
Y = a Wt ± SE of Y, or [3-1]b
log Y = log a + b log Wt ± SE of log Y [3-2]
where Y is the biological characteristic to be predicted, Wt is the animal's body weight
(mass), a and b are empirically derived constants, and SE is the standard error of the mean
value of the parameter.
Equation 3-2 is the log transformation of Equation 3-1. Equation 3-2 represents a
straight line, with b equal to the slope of the line and log a equal to the Y-intercept of the
line. Values for a and b usually are determined empirically from measured values using
linear regression analysis. Once values are determined for a and b, Equation 3-1 can be
used to predict a value of Y from the body weight of the animal. The SE of Y is the
standard error of the mean Y estimated for the mean of the Wt values; the SE of log Y is
the standard error of the mean log Y estimated for the mean of the log Wt values.
3-2
Allometric equations can be used to estimate parameter values for species for
which measured values are not available. The equations presented in this chapter,
however, should not be used for taxonomic categories other than the category for which
each was developed. For example, equations developed for iguanid lizards cannot be used
for amphibians and should not be used for other groups of reptiles without careful
evaluation of likely differences between the groups. It also is important to remember that
the allometric equations presented in this chapter have been developed using mean values
for a number of species within a taxonomic category. Individual species usually exhibit
values somewhat different from those predicted by an allometric model based on several
species. Furthermore, different-sized individuals within a species and individuals at
varying stages of maturation are likely to exhibit a different allometric relationship between
body weight and the dependent variable. For further discussion of within-species
allometric equations related to growth and reproduction, see Reiss (1989).
In the next five sections, we describe empirically derived allometric equations that
relate food ingestion rates (Section 3.1), water intake rates (Section 3.2), inhalation rates
(Section 3.3), surface area (Section 3.4), and metabolic rate (Section 3.5) to body weight.
As discussed above, most of the allometric models differ for birds, mammals, reptiles, and
amphibians, and many also vary within these taxonomic groups. In Section 3.6, we provide
a summary of operations involving logarithms and powers and unit conversion factors for
those persons who may want to modify allometric equations found in the literature.
Finally, in Section 3.7 we describe how to estimate 95-percent confidence intervals for food
ingestion rates and free-living metabolic rates predicted on the basis of allometric
equations presented in this chapter. We present most equations in the untransformed
form only. For equations for which an investigator reported standard errors for the log
transformation of the relationship, we present the equation both ways. For those persons
interested in estimating confidence intervals for other allometric equations, Peters (1983)
provides a simple review of how to estimate regression statistics for equations of the form
of Equation 3-2. Section 3.8 contains the references for this chapter.
3-3
3.1. FOOD INGESTION RATES
Food ingestion rates vary with many factors, including metabolic rate, the energy
devoted to growth and reproduction, and composition of the diet. The metabolic rate of
free-ranging animals is a function of several factors, including ambient temperature,
activity levels, and body weight. In birds and mammals, thermoregulation can
considerably increase an animal's metabolic requirements during the winter, whereas
reproductive efforts can replace thermoregulation as the predominant extra metabolic
expenditure in the spring and summer. Many reptiles and amphibians, on the other hand,
drop their activity levels and metabolic rates in the winter.
For homeotherms (i.e., animals that maintain a relatively constant body temperature
such as most birds and mammals), metabolic rate generally decreases with increasing
body mass (see Section 3.5). The smallest birds and mammals must consume quantities
of food equal to their body weight or more daily; in contrast, the larger homeotherms may
consume only a small fraction of their body weight in food daily. Herbivores tend to
consume larger quantities of food than carnivores because of the lower energy content of
their food. Ingestion rates, expressed in units of food energy normalized to body size (e.g.,
kcal/kg-day), are not significantly different for herbivores and carnivores (Peters, 1983).
Four-legged poikilotherms (those animals whose usual body temperatures are the same as
that of their environment, such as reptiles and amphibians) exhibit the same slope of
decreasing ingestion rates per unit body weight with increasing body size but show a lower
intercept (i.e., lower ingestion rate for a given body weight) than homeotherms (Nagy,
1987).
The rate of food consumption that an animal must achieve to meet its metabolic
needs can be calculated by dividing its free-living (or field) metabolic rate (FMR) (see
Section 3.5) by the metabolizable energy in its food (Nagy, 1987). Metabolizable energy
(ME) is the gross energy (GE) in a unit of food consumed minus the energy lost in feces
and urine. Assimilation efficiency (AE) equals the ratio ME/GE, or the fraction of GE that is
metabolizable. AE is relatively constant among different groups of consumer species of
mammals and birds that are all either carnivorous, insectivorous, herbivorous, or
3-4
granivorous (Hume, 1982; Peters, 1983; Nagy, 1987; Robbins, 1983). Nagy (1987)
calculated the mean ME (i.e., kilojoules of ME per gram of dry matter) of various diets for
birds and mammals from average values of AE for birds and mammals and typical GE
contents of those diets as reported by Golley (1961) and Robbins (1983). These values are
presented in Table 3-1. (For more information on ME and AE, see Section 4.1.2.) Using the
values presented in Table 3-1, Nagy (1987) developed allometric equations for food
ingestion (FI) rates as a function of body weight (Wt) for birds, mammals, and lizards using
estimated FMRs and general dietary composition. In the remainder of this section, we
present these equations for birds (Section 3.1.1) and mammals (Section 3.1.2). Section
3.1.3 summarizes Nagy's food ingestion allometric equations for iguanid lizards. We report
this information even though no iguanid lizards were among our selected species because
it is the only information of this type we identified for any amphibian or reptile.
Nagy's (1987) estimates of FMR are based on doubly labeled water measurements
of CO production in free-living animals. When performed correctly, this method is more2
accurate for estimating the metabolic rate of free-living animals than other methods
commonly used (King, 1974). Other allometric equations for food ingestion rates that we
identified in the open literature are based largely on captive animals without corrections for
the additional energy requirements of free-living animals. For more accurate estimates of
food ingestion rates by type of diet, we recommend following the procedures outlined in
Section 4.1.2 instead of using these generic equations.
3.1.1. Birds
For birds, Nagy (1987) calculated FI rates (in grams dry matter per day) from ME and
FMR and developed the following equations:
FI (g/day) = 0.648 Wt (g), or all birds [3-3]0.651
FI (kg/day) = 0.0582 Wt (kg)0.651
FI (g/day) = 0.398 Wt (g) passerines [3-4]0.850
3-5
Diet
Metabolizable Energy
(kJ/g) (kcal/g)a a
Animal Group
insects 18.7 = 4.47 mammals
18.0 = 4.30 birds
fish 18.7 = 4.47 mammals
16.2 = 3.87 birds
vegetation 10.3 = 2.26 mammals
seeds 18.4 = 4.92 mammals
nectar 20.6 = 4.92 hummingbirds
omnivory 14 = 3.35 mammals and birds
g = grams dry weight.a
Source: Nagy, 1987.
Table 3-1. Metabolizable Energy (ME) of Various Diets for Birds and Mammals
FI (g/day) = 0.301 Wt (g) non-passerines [3-5]0.751
FI (g/day) = 0.495 Wt (g) seabirds [3-6]0.704
where Wt equals the body weight (wet) of the animal in grams (g) or kilograms (kg) as
indicated. We provide the regression statistics for these equations (including sample size
and regression coefficient) and information required to estimate a 95-percent confidence
interval for an FI rate predicted for a specified body weight in Section 3.7. More accurate
estimates of food requirements can be made from estimates of FMR (Section 3.5), dietary
composition, and AE for the species of interest, as outlined in Section 4.1.2.
3.1.2. Mammals
For placental mammals, Nagy (1987) calculated FI rates (in grams dry matter per
day) from ME and FMR values and developed the following equations:
3-6
FI (g/day) = 0.235 Wt (g), or all mammals [3-7]0.822
FI (kg/day) = 0.0687 Wt (kg)0.822
FI (g/day) = 0.621 Wt (g) rodents [3-8]0.564
FI (g/day) = 0.577 Wt (g) herbivores [3-9]0.727
We provide the regression statistics for these equations (including sample size and
regression coefficient) and information required to estimate a 95-percent confidence
interval for an FI rate predicted for a specified body weight in Section 3.7. More accurate
estimates of food requirements can be made from estimates of FMR (Section 3.5), dietary
composition, and AE for the species of interest, as outlined in Section 4.1.2.
Herbivores tend to consume more food than carnivores or omnivores on a dryweight
basis because of the lower energy content of the herbivores' diets. On an energy
basis (e.g., kilocalories), the ingestion rates of carnivores and herbivores are not
significantly different (Farlow, 1976):
FI (kjoule/day) = 971 Wt (kg) (r = 0.942), or herbivores [3-10]0.73 2
FI (kcal/day) = 1.518 Wt (g)0.73
FI (kjoule/day) = 975 Wt (kg) (r = 0.968), or carnivores [3-11]0.70 2
FI (kcal/day) = 1.894 Wt (g)0.70
3.1.3. Reptiles and Amphibians
This section summarizes food ingestion allometric equations for iguanid lizards,
which is the only information of this type we identified for any amphibian or reptile. Nagy
(1987) calculated FI rates (in grams dry matter per day) from ME and FMR values on spring
and summer days and developed the following equations:
3-7
FI (g/day) = 0.019 Wt (g) herbivores [3-12]0.841
FI (g/day) = 0.013 Wt (g) insectivores [3-13]0.773
Again, on an energy basis, carnivores and herbivores are not significantly different and can
be represented by a single relationship:
FI (kjoule/day) = 0.224 Wt (g), or all iguanids [3-14]0.799
FI (kcal/day) = 0.054 Wt (g)0.799
We provide the regression statistics for these equations (including sample size and
regression coefficient) and information required to estimate a 95-percent confidence
interval for an FI rate predicted for a specified body weight in Section 3.7. More accurate
estimates of food requirements for these and other groups of reptiles and amphibians can
be made from estimates of FMR (Section 3.5), dietary composition, and AE for the species
of interest, as outlined in Section 4.1.2.
Allometric equations for FI rates for other groups of reptiles and amphibians were
not found. For other groups, we recommend estimating FI rates from FMR and diet, as
described in Section 4.1.2.
3.2. WATER INTAKE RATES
Daily water requirements depend on the rate at which animals lose water to the
environment due to evaporation and excretion. Loss rates depend on several factors,
including body size, ambient temperature, and physiological adaptations for conserving
water. Drinking water is only one way in which animals may meet their water
requirements. All animals produce some water as a product of their metabolism. The
degree to which metabolic water production and dietary water content can satisfy an
animal's water requirements varies from species to species and with environmental
conditions. Extensive literature describes the allometry of total water flux for various
3-8
groups of animals. Allometric models to predict drinking water intake, on the other hand,
are limited.
3.2.1. Birds
Based on measured body weights and drinking water values from Calder (1981) and
Skadhauge (1975), Calder and Braun (1983) developed an equation for drinking water
ingestion (WI) for birds:
WI (L/day) = 0.059 Wt (kg) all birds [3-15]0.67
where Wt equals the average body weight in kilograms (kg) of the bird species. This
equation is based on data from 21 species of 11 to 3,150 g body weight. Total water
turnover should be proportional to metabolic rate (body weight to the 3/4 power, see
Section 3.5.2.1). The exponent for Equation 3-15 is not significantly different from 0.75
(Calder and Braun, 1983). Additional sources of water not accounted for in this equation
(metabolic water and water contained in food) also help to balance the animals' daily water
losses. For allometric equations for total water flux (including water obtained from food)
for birds, see Nagy and Peterson (1988).
To estimate daily drinking water intake as a proportion of an animal's body weight
(e.g., as g/g-day), the WI rate estimated above is divided by the animal's body weight in kg:
WI (g/g-day) = WI (kg/kg-day), or [3-16]
= WI (L/day) / Wt (kg)
In general, birds drink less water than do mammals of equivalent body weights.
Because of their relatively high metabolic rates, the quantity of metabolic water produced
by birds is greater in relationship to body size than that produced by other vertebrates
(Bartholomew and Cade, 1963). In addition, birds are able to conserve water by excreting
nitrogen as uric acid instead of urea (as excreted by mammals); uric acid can be excreted
3-9
in a semi-solid suspension, whereas urea must be excreted in aqueous solution. On the
other hand, birds exhibit a high rate of water loss from the respiratory system and use
panting and evaporative water loss to prevent overheating at high ambient temperatures.
For example, Dawson (1954) found evaporative losses in two species of towhees to
increase fourfold between 30 and 40C.
Although birds may satisfy some of their water needs by oxidative food metabolism,
it has not been demonstrated that any normally active bird can satisfy its water
requirements with metabolic water alone (Bartholomew and Cade, 1963). The balance
must be obtained from water contained in foods such as insects or succulent plant
material and from drinking water.
As would be expected, birds drink more water at warmer temperatures to make up
for evaporative losses. Seibert (1949) found that juncos (weighing 16 to 18 g) consumed
an average of 11 percent of their body weight in water daily at an ambient temperature of
0C, 16 percent at 23C, and 21 percent at 37C. The white-throated sparrow increased
water consumption from 18 percent of its body weight at 0C to 27 percent at 23C and 44
percent at 37C.
Water consumption rates per unit body weight also tend to decrease with increasing
body weight within a species. For example, in white leghorn chickens, water intake per
gram of body weight is highest in the youngest chicks (45 percent of the body weight at 1
week when chicks average 62 g) and decreases with age thereafter (13 percent of the body
weight at 16 weeks when chicks average 2.0 kg) until egg-laying, when water consumption
increases for the production of eggs (24 percent of the body weight for laying hens)
(Medway and Kare, 1959).
Some species obtain more of their daily water needs from their diet and therefore
drink less water than others; therefore, measured water ingestion values from wellconducted
studies should be used when available. In the absence of measured values,
Equation 3-15 should provide a reasonable central value. Additional information required
to estimate a 95-percent confidence interval was not provided along with this equation.
3-10
3.2.2. Mammals
Based on measured body weights and drinking water values from Calder (1981) and
Skadhauge (1975), Calder and Braun (1983) developed an allometric equation for drinking
water ingestion (WI) for mammals:
WI (L/day) = 0.099 Wt (kg) all mammals [3-17]0.90
where Wt equals the average body weight in kilograms (kg). Additional sources of water
not accounted for in this equation (i.e., metabolic water and water contained in food) help
to balance the animals' daily water losses. The empirically determined exponent of 0.90
does not suggest a simple physiological explanation. If total water turnover (metabolic
water combined with water obtained from food) is proportional to metabolic rate (body
weight to the 3/4 power, see Section 3.5.2.1), then drinking water ingestion would be
expected to scale similarly, as was the case for birds (see Section 3.2.1). For allometric
equations relating body weight to total water flux (including water obtained from food) for
mammals, see Nagy and Peterson (1988).
To normalize drinking water intake to body weight (e.g., as g/g-day; see Chapter 4,
Equation 4-4), the WI rate estimated above is divided by the animal's body weight in kg:
NWI (g/g-day) = WI (kg/kg-day), or [3-18]
= WI (L/day) / Wt (kg)
We present normalized drinking water intakes in the species profiles.
3.2.3. Reptiles and Amphibians
Allometric equations relating body weight to drinking water ingestion rates were not
identified for reptiles and amphibians. The water balance of these groups is complex, in
part because they can absorb water through their skin as well as drink water and extract
water from their food (Duellman and Trueb, 1986; Minnich, 1982). The relative
3-11
contribution of these three routes of water intake depends on the species, habitat,
temperature, and body surface area. In general, the skin of reptiles is less permeable than
that of amphibians. Aquatic turtles (e.g., snapping turtle, painted turtle) also may ingest
large amounts of water when feeding on aquatic plants and animals; however, the
magnitude of such ingestion has not been quantified (Mahmoud and Klicka, 1979). For
further discussion of water balance for these groups, see Duellman and Trueb (1986),
Feder and Burggren (1992), Minnich (1982), and Nagy and Peterson (1988).
3.3. INHALATION RATES
Inhalation rate is one of the respiratory parameters needed to estimate potential
exposure of wildlife to airborne contaminants. Inhalation rates vary with species, body
size, body temperature, ambient temperature, and activity levels. When inhalation rate is
increased, either because of increased activity levels or to promote evaporative cooling,
exposure to airborne contaminants may be increased. As discussed in Section 4.1.4, an
inhalation toxicologist should be consulted when assessing this pathway because
additional respiratory parameters also must be considered (see U.S. EPA, 1990).
3.3.1. Birds
Lasiewski and Calder (1971) developed an allometric relationship for inhalation rate
(IR) associated with standard metabolism (i.e., post-digestive, at rest) for non-passerine
birds (N = 6 species ranging in weight from 43 to 88,000 grams). They excluded
passerines, which have a somewhat higher metabolic rate than non-passerines (see
Section 3.5):
IR (ml/min) = 284 Wt (kg), or all non-passerines [3-19]0.77
IR (m /day) = 0.4089 Wt (kg), or3 0.77
IR (m /day) = 0.002002 Wt (g)3 0.77
As noted above, these inhalation rates were associated with standard metabolic rates.
Free-living metabolic rates are likely to be higher by a factor of at least 2 or 3 (see Section
3-12
3.5); therefore, IRs estimated from these equations should be adjusted accordingly (e.g.,
multiplied by 2 or 3) although IRs might not be directly proportional to metabolic rate.
3.3.2. Mammals
Using measured values from several reports of respiration rates in mammals
(covering 691 data points), Stahl (1967) developed an allometric relationship for inhalation
rate with body size for mammals (N = 691, r = 0.98, SE Y = 45):
IR (ml/min) = 379 Wt (kg), or all mammals [3-20]0.80
IR (m /day) = 0.5458 Wt (kg), or3 0.80
IR (m /day) = 0.002173 Wt (g)3 0.80
As for the equations given for birds, these IRs were associated with standard metabolic
rates. Field metabolic rates are likely to be higher by a factor of at least 2 or 3 (see Section
3.5); therefore, IRs determined from these equations should be adjusted accordingly (e.g.,
multiplied by 2 or 3, although IRs may not be directly proportional to metabolic rate).
3.3.3. Reptiles and Amphibians
In contrast to the fairly regular breathing patterns of most birds and mammals, most
reptiles breath air in distinct episodes. They may take single breaths, or exhibit an episode
of several breaths, and then hold their breath for varying lengths of time (Milsom and Chan,
1986). Inhalation rate varies for reptiles and amphibians not only with body size and
activity level, as for birds and mammals, but also with body temperature. Some gas
exchange occurs normally through the integument of both reptiles and amphibians
(Duellman and Trueb, 1986; Lillywhite and Maderson, 1982). Moreover, for semiaquatic
species, a significant proportion of gas exchange can occur underwater through the skin,
reducing the need to inspire air (Seymour, 1982). For example, in adult bullfrogs, gas
exchange through the skin can account for 18 percent of total oxygen uptake (Burggren
and West, 1982). Given the complexity of the subject, we refer those interested in
3-13
inhalation exposures for reptiles or amphibians to more specific treatments of these topics
(e.g., Duellman and Trueb, 1986; Feder and Burggren, 1992; Gans and Dawson, 1976;
Jackson, 1979; Hutchinson et al., 1968; Lillywhite and Maderson, 1982).
3.4. SURFACE AREAS
The degree to which an animal may absorb contaminants through direct contact
with its skin depends on many factors, including the surface area of the skin available for
contact. Summarizing measured surface areas for more than 100 animals reported by
Hemmingsen (1960), Schmidt-Nielsen (1970, 1972) determined that animals have surface
areas that usually are approximately twice that of a sphere of the same weight (assuming a
specific gravity of 1 for both the sphere and the animal). The permeability of an animal's
skin to contaminants, however, depends on characteristics of the skin (e.g., presence of
keratinized scales) as well as the contaminant (e.g., molecule size, lipophilicity). This
section presents allometric equations for estimating skin surface area; characteristics
affecting skin permeability are not discussed.
3.4.1. Birds
In studies of avian thermal biology, skin surface area is commonly estimated using
Meeh's (1879, cited in Walsberg and King, 1978) formula with Rubner's (1883, cited in
Walsberg and King, 1978) constant of 10:
SA (cm ) = 10 Wt (g) all birds [3-21]skin
2 0.667
where SA is the skin surface area beneath the feathers and Wt is body weight (Walsbergskin
and King, 1978). Although Rubner's constant of 10 was derived originally from domestic
fowl, Drent and Stonehouse (1971) have verified the formula for birds in a variety of taxa
and of weights spanning three orders of magnitude. For passerines, beak surface area
tends to be about 1 percent (range 0.7 percent to 1.6 percent of 10 passerine species) of
skin surface area, and leg surface area about 7 percent (range 5.9 percent to
3-14
7.9 percent of 10 passerine species) (Walsberg and King, 1978). These ratios would be
expected to vary for many non-passerines (e.g., herons, woodcock).
3.4.2. Mammals
Summarizing data from more than 100 mammals, Stahl (1967) developed a
relationship between surface and body weight:
SA (m )= 0.11 Wt (kg), or all mammals [3-22]skin
2 0.65
SA (cm ) = 12.3 Wt (g)skin
2 0.65
This relationship is very similar to that developed for birds (Equation 3-21).
3.4.3. Reptiles and Amphibians
Surface area has been found to be a different function of body weight for adult
amphibians than for birds or mammals (Hutchinson et al., 1968; Whitford and Hutchinson,
1967):
SA (cm ) = 1.131 Wt (g) all frogs [3-23]skin
2 0.579
SA (cm ) = 0.953 Wt (g) bullfrog [3-24]skin
2 0.725
SA (cm ) = 0.997 Wt (g) green frog [3-25]skin
2 0.712
SA (cm ) = 8.42 Wt (g) salamanders [3-26]skin
2 0.694
Models by which to estimate surface areas for turtles (exclusive of the shell and
plastron) and snakes were not found. The general formula for the surface area of a
cylinder can be used to approximate the surface area of a snake if the length and girth are
known or estimated.
3-15
3.5. ALLOMETRIC EQUATIONS FOR METABOLIC RATE
The allometric equations for estimating food ingestion rates provided in Section 3.1
were derived using very simple assumptions about the energetic content and digestibility
of the diet for the species included in the regression equations. Consequently, the
equations will provide only very rough estimates of food ingestion rates for any given
species. For a site-specific exposure assessment, it may be more appropriate to evaluate
ingestion rates for a diet that is likely to represent the species and study area. The caloric
content and percent water, fat, and protein of wildlife diets vary not only among species,
but also among individuals within the same species depending on factors such as location,
time of year, age, and sex. If one can estimate the energetic requirements of the animal in
the field and its dietary composition for a specified situation, one can estimate food
ingestion rates for that diet and situation. In the remainder of this section, we discuss
metabolic rate and provide allometric equations to estimate field free-living metabolic rates
(FMRs) for wildlife species. Chapter 4 describes how to use FMR estimates and
information about the energy content of specific diets to estimate food ingestion rates.
Several factors influence metabolic rates of free-ranging animals, including body
size, body temperature, and type and level of activity. For homeotherms, metabolic energy
must be expended to keep core body temperature within relatively narrow limits. At
moderate ambient temperatures, homeotherms lose heat to the surrounding environment
as rapidly as they gain it and therefore need not expend extra metabolic energy to maintain
core body temperature. That range of ambient temperatures over which an animal's
metabolic rate is at a minimum and constant level is called the thermoneutral zone. Below
the thermoneutral zone, the organism loses heat to the environment and must increase its
metabolic activity to compensate. Above the thermoneutral zone, the organism gains heat
from its environment and must increase its metabolic rate to use evaporation to cool its
body.
Thermoneutral zones vary somewhat among species depending upon the insulating
properties and color of the fur or feathers, surface-to-volume ratios, and other factors. The
degree to which metabolic rate increases with changes in ambient temperature outside of
Water has a much higher heat conductance than air. When submerged or swimming, the degreea
to which metabolic rate increases with decreasing water temperature depends on the animal's
insulation (e.g., whether the fur traps an air layer next to the skin over part or all of the body or
whether there is an insulative layer of blubber), duration of submergence, and body size.
3-16
the thermoneutral zone is referred to as the temperature coefficient (TC). Temperature
coefficients also vary with body size, insulation, and other factors.a
There are several ways to measure and express metabolic rate, including basal
metabolic rate (BMR), resting metabolic rate (RMR), existence metabolic rate (EMR),
average daily metabolic rate (ADMR), and free-living or field metabolic rate (FMR). The
different measures are distinguished by the range of animal activities included in the
measure:
 Basal metabolic rate (BMR), also sometimes labeled standard metabolic rate
(SMR), represents the minimal value of heat production for homeotherms.
BMR must be measured within the thermoneutral zone of ambient
temperatures when the animal is at rest and in a post-absorptive state (i.e.,
all food has been digested) (Gessaman, 1973).
 Standard metabolic rate (SMR) has been used in the literature in more than
one way. Many authors define SMR as BMR (see above). Others use SMR if
the thermoneutral zone has not been defined so that some cost of
thermoregulation may be included (Bennett and Harvey, 1987).
 Resting metabolic rate (RMR) is usually measured at temperatures below the
thermoneutral zone when the animal is at rest, but not post-absorptive (i.e.,
the animal is eating regularly and may be expending energy to digest its
food). The RMR exceeds the BMR by the heat liberated in the digestion of
food (i.e., the specific dynamic action, or SDA) and by some cost of
thermoregulation. RMR and BMR are usually measured using indirect
calorimetry (i.e., oxygen consumption and carbon dioxide production) over a
period of 1 to 3 hours.
3-17
 Existence metabolic rate (EMR) is the metabolic rate necessary for an animal
to maintain itself in captivity without a change in body weight. EMR is
greater than RMR due to the cost of locomotor and other activities required
for self-maintenance. Most researchers measure EMR on the basis of food
consumption and energy excretion at a constant weight over the period of
several days or weeks (Kendeigh, 1969).
 Average daily metabolic rate (ADMR) is usually measured over 24 hours at a
temperature similar to the animal's natural environment and with food and
water available ad libitum. ADMR is the sum of BMR and the metabolic costs
of thermoregulation, digestion, and daily activities.
 Free-living or field metabolic rate (FMR) can be measured using doublylabeled
water, and it represents the total daily energy requirement for an
animal in the wild. FMR includes the costs of BMR, SDA, thermoregulation,
locomotion, feeding, predator avoidance, alertness, posture, and other
energy expenditures. Various models and measures have indicated that a
constant value of approximately three times BMR is a reasonable estimate of
FMR for birds and mammals (Lamprey, 1964; Buechner and Golley, 1967;
Koplin et al., 1980), although more precise estimates also have been
developed (see Sections 3.5.1.3, 3.5.2.3, and 3.5.3.2).
FMR also has been used in the literature to represent fasting metabolic rate (e.g.,
Gessaman, 1973), but we do not discuss fasting metabolic rate estimates in this Handbook.
The relationships between metabolic rate and body weight fall into two broad
categories: those for homeothermic animals (i.e., most birds and mammals), and those for
poikilothermic animals (i.e., most reptiles and amphibians). For poikilotherms, metabolic
rate must be related to body temperature. It also is important to remember that
poikilotherms can adjust their body temperatures relative to ambient temperatures
3-18
somewhat by modifying their behavior (e.g., basking in the sun, adopting postures to
minimize or maximize absorption of solar radiation).
Allometric models relating metabolic rate to body size for birds and mammals are
described in Sections 3.5.1 and 3.5.2, respectively. Allometric models for reptiles and
amphibians are described in Section 3.5.3. We have attempted to identify the most
accurate allometric equations currently available for estimating free-living metabolic rates.
We also present allometric equations for basal and existence metabolism, which in
combination with appropriate information on activity budgets and energy costs can be
used to estimate field metabolic rates. Furthermore, measures of basal and existence
metabolism are available for considerably more species than are measures (or estimates)
of free-living metabolic rates. Consequently, more allometric models have been developed
that distinguish the metabolic rate-weight relationship among taxonomic groups using
measures of basal and existence metabolism than using measures of field metabolic rates.
We caution users to pay close attention to the units for the parameters in the allometric
equations. For most equations, energy is expressed as kcal (with the exception of some
equations for reptiles and amphibians). Mass may be expressed either in g or kg,
depending on how the equation was reported.
We emphasize that the literature on allometric relationships and metabolic rate is
extensive and complex. We provide a very simplified overview that should be of
assistance for screening-level exposure assessments only. For additional information on
methods of estimating metabolic costs of free-ranging animals, please consult expert
reviews on the subject (e.g., Bennett and Dawson, 1976; Bennett and Harvey, 1987; Ellis,
1984; Gans and Dawson, 1976; Gessaman, 1973; Kendeigh et al., 1977; King, 1974; Peters,
1983; Robinson et al., 1983; Wiens, 1984).
3.5.1. Birds
In birds, metabolic rate generally decreases with increasing body mass. Several
authors have found passerine birds to have higher metabolic rates overall for their body
size than non-passerines (Lasiewski and Dawson, 1967; Nagy, 1987; Kendeigh, 1970;
Seabirds included penguins, albatross, petrels, shearwaters, pelicans, skuas, gulls, terns, noddys,b
murres, cormorants, and frigatebirds.
3-19
Zar, 1968). In this section, we present allometric models for three measures of metabolic
rate on the basis of body size in birds: basal metabolic rate (BMR), existence metabolic
rate (EMR), and field metabolic rate (FMR). All equations take the general form of Y = aWt ,b
but can also be represented in their log-transformed form (the equation of a straight line).
We conclude this section by discussing the influence of ambient temperature on avian
metabolic rates. Additional information required to estimate a 95-percent confidence
interval (CI) for a predicted FMR (the expression of metabolic rate that is generally most
appropriate for wildlife exposure assessments) is provided in Section 3.7.
3.5.1.1. Basal Metabolic Rate
Several investigators have derived values for the constants a and b for the equation
relating BMR to body weight (Wt) from empirical data on birds. Lasiewski and Dawson
(1967) compiled body weight and BMR for almost 100 species of birds. They found BMR
for passerines to be higher than BMR for non-passerines (i.e., the Y-intercept for
passerines is higher than the Y-intercept for non-passerines):
Passerines
log BMR (kcal/day) = 2.11 + 0.724 log Wt (kg) ± 0.113, or [3-27]
BMR (kcal/day) = 128 Wt (kg)0.724
Non-passerines
log BMR (kcal/day) = 1.89 + 0.723 log Wt (kg) ± 0.068, or [3-28]
BMR (kcal/day) = 77.6 Wt (kg)0.723
Ellis (1984) found the Y-intercept for seabirds to be somewhat higher than the Y-b
intercept for non-passerines determined by Lasiewski and Dawson (1967):
3-20
Seabirds
log BMR (kcal/day) = 1.96 + 0.721 log Wt (kg) (no SE provided), or [3-29]
BMR (kcal/day) = 91.2 Wt (kg)0.721
Zar (1968) reexamined the data compiled by Lasiewski and Dawson (1967) and
developed models for relating BMR to body weight (kg) for several orders and families of
birds (Table 3-2). These may be used to estimate whether the FMR for a species of interest
is likely to fall above or below that predicted on the basis of the allometric equations
derived for "all birds."
3.5.1.2. Existence Metabolic Rates
Kendeigh (1970) developed allometric equations for EMRs as a function of weight
(Wt) at 30C separately for passerines and for non-passerines. As was the case for BMRs,
passerines showed higher EMRs than did non-passerines:
Passerines (N = 15 species)
log EMR (kcal/day) = 0.1965 + 0.6210 log Wt (g) ± 0.0633, or [3-30]
EMR (kcal/day) = 1.572 Wt (g), or0.6210
log EMR (kcal/day) = 2.060 + 0.6210 log Wt (kg), or
EMR (kcal/day) = 114.8 Wt (kg)0.6210
Non-passerines (N = 9 species)
log EMR (kcal/day) = -0.2673 + 0.7545 log Wt (g) ± 0.0630, or [3-31]
EMR (kcal/day) = 0.5404 Wt (g), or0.7545
log EMR (kcal/day) = 1.996 + 0.7545 log Wt (kg), or
EMR (kcal/day) = 99.03 Wt (kg), or0.7545
The average increase of EMR at 30C over BMR is 31 and 26 percent in passerine and nonpasserine
species, respectively (Kendeigh, 1970). At 0C, on the other hand, EMR of
passerine and non-passerine species is similar, indicating that non-passerines are affected
3-21
Number SE of SE ofb b
of data mean mean
Avian group points a log a b BMR log BMR
Apodiformes 9 114 2.06 0.769 0.201 0.0558
Strigiformes 7 66.4 1.82 0.69 11.1 0.0989
Columbiformes 10 92.1 1.96 0.858 2.68 0.0491
Galliformes 13 72.6 1.86 0.698 15.3 0.0904
Falconiformes 5 65.3 1.82 0.648 45.3 0.108
Anseriformes 9 95.8 1.98 0.634 23.4 0.0524
Ciconiiformes 7 86.9 1.94 0.737 22.0 0.0464
Passeriformes 48 129 2.11 0.724 8.71 0.0806
Corvidae 8 126 2.10 0.709 23.3 0.147
Ploeceidae 17 164 2.21 0.794 1.40 0.0808
Fringillidae 19 125 2.10 0.714 1.02 0.0473
All Nonpasserines 72 78.5 1.90 0.723 42.8 0.111
All Species 120 86.3 1.94 0.668 52.8 0.133
Values for the equation relating BMR to body weight (Wt): log BMR (kcal/day) = log a + b log Wt (kg).a
Estimated from the mean log Wt used to develop the allometric equation.b
Source: Zar, 1968.
Table 3-2. Allometric Equations for Basal Metabolic Rate (BMR) in Birdsa
more by cold than passerines. Kendeigh (1970) estimated the equation for all bird species
(N = 24) at 0C to equal:
All birds (24 species)
log EMR (kcal/day) = 0.6372 + 0.5300 log Wt (g) ± 0.0613, or [3-32]
EMR (kcal/day) = 4.337 Wt (g)0.5300
3-22
The equations also indicate that smaller species are affected more by cold than are larger
species. The slopes of the regression lines for EMR on body weight is less steep at 0C
than at 30C, indicating that small birds must increase heat production more than large
birds to regulate body temperature during cold weather.
To normalize EMR to body weight, divide the daily EMR by body weight:
NEMR (kcal/kg-day) = EMR (kcal/day) / Wt (kg) [3-33]
3.5.1.3. Free-Living Metabolic Rate
FMRs have been measured using doubly-labeled water (DLW) to measure CO2
production in animals in the field. Based on DLW measurements with 25 species of birds,
Nagy (1987) developed an equation relating FMR for birds to body weight:
FMR (kjoules/day) = 10.89 Wt (g), or all birds [3-34]0.640
FMR (kcal/day) = 2.601 Wt (g)0.640
In birds, the slope of FMR (i.e., 0.640) does not differ significantly from the BMR slope of
0.668 (see Table 3-2). This indicates that FMR may be a relatively constant multiple of BMR
in birds over a large range of body mass.
Using estimates of FMR determined for 42 species by a variety of methods,
Walsberg (1983) found a similar relationship (r = 0.98, SE Y = 0.415, SE b = 0.012):2
FMR (kjoules/day) = 13.05 Wt (g), or all birds [3-35]0.605
FMR (kcal/day) = 3.12 Wt (g)0.605
Separating the passerine from the non-passerine species, Nagy (1987) found a
higher FMR among passerines than non-passerines of comparable weight (i.e., the Yintercept
for passerines is higher than the Y-intercept for non-passerines), as expected on
the basis of basal metabolic rate:
All of the large birds included in the database were seabirds such as noddy, kittiwake, shearwater,c
albatross, tern, and petrel (Nagy, 1987). Other large birds, such as herons, hawks, and owls, were
not included. Accordingly, non-passerine and non-seabird equations should be used with caution.
Insufficient information is provided in Walsberg (1983) to estimate confidence intervals for ad
predicted FMR for species with body weights above or below the mean log body weight value of
his data set.
3-23
FMR (kjoules/day) = 8.892 Wt (g), or passerines [3-36]0.749
FMR (kcal/day) = 2.123 Wt (g)0.749
FMR (kjoules/day) = 4.797 Wt (g), or non-passerines [3-37]0.749
FMR (kcal/day) = 1.146 Wt (g)0.749
FMR (kjoules/day) = 8.017 Wt (g), or seabirds [3-38]0.704
FMR (kcal/day) = 1.916 Wt (g)0.704
FMR (kjoules/day) = 21.13 Wt (g), or non-seabirds [3-39]0.440 c
FMR (kcal/day) = 5.051 Wt (g)0.440
We provide the regression statistics for Nagy's (1987) equations (including sample
size and the regression coefficient) and information required to estimate a 95-percent
confidence interval for an FMR in Section 3.7.d
Nagy (1987) estimated the accuracy of the doubly-labeled water method to be ± 8
percent or better. Because of difficulties in recapturing birds during the nonbreeding
season, most of the measured FMRs were for breeding birds (Nagy, 1987).
King (1974) estimated that FMR exceeds BMR by a factor of 3.5 on average (based
on a sample of 18 measures for species ranging from 4 to 400 g in weight). Gessaman
(1973) summarized data on mockingbirds and purple martins from Utter (1971) that
indicated an FMR equal to 1.6 to 2.4 times the predicted BMR for adults not actively feeding
nestlings. Feeding nestlings increased the ratio of FMR to BMR from 2.7 to 3.4 in purple
martins (Utter, 1971, cited in Gessaman, 1973).
This is because conductance and heat loss for a given thermal gradient between body temperaturee
and ambient temperature rise more slowly with body size than do basal metabolic rates.
3-24
To normalize FMR to body weight, divide the daily FMR by body weight:
NFMR (kcal/kg-day) = FMR (kcal/day) / Wt (kg) [3-40]
Figure 3-1 illustrates approximate monthly variations in the total energy budget of
an adult house sparrow in Illinois throughout the year and the relationship between BMR
and FMR (adapted from Kendeigh et al., 1977). For this bird, FMR varies seasonally, with a
maximum value in midwinter (28 kcal/day) and a minimum in August prior to molting (20
kcal/day). Other species, however (e.g., willow ptarmigan), show no significant variation in
FMR with season (King, 1974). For examples of nestling energy budgets, see Kendeigh et
al. (1977) and Dunn (1980). For a discussion of modeling energy budgets for birds in
general and for seabirds in particular, see Wiens (1984).
3.5.1.4. Temperature and Metabolic Rate
Below an animal's thermoneutral zone, metabolism increases with decreasing
ambient temperature. Section 3.5.1.2 presented equations for EMR at 30C and at 0C, but
these are not particularly helpful for estimating EMR at other temperatures. Although few
researchers have attempted general multiple regressions of metabolic rate on both body
size and temperature for birds, some relationships have been investigated in general terms
(Peters, 1983):
 Low temperatures induce a greater proportional rise in metabolic rate
relative to basal metabolic rate in smaller birds than in larger ones.e
 At high temperatures, metabolic rate increases to increase blood flow and
evaporative cooling (via panting).
3-26
Peters (1983) developed an equation relating the ratio of SMR to BMR to thermal gradient
(i.e., the difference between ambient temperature and body temperature) for birds:
SMR/BMR = 0.029 (thermal gradient in C) Wt (kg) [3-41]-0.249
Thus, standard metabolic costs increase relative to basal metabolism at lower
temperatures, but less so for larger birds than for smaller birds. Despite the strong
dependence of metabolic rate on ambient temperature, for screening-level risk
assessments, it should not be necessary to adjust estimates of FMR for seasonal
temperature changes. As Figure 3-1 illustrates, high metabolic demands of
thermoregulation in the winter can be replaced by those of reproduction and molting
during spring, summer, and fall.
3.5.2. Mammals
As for birds, metabolic rate in mammals generally decreases with increasing body
size. The metabolic rates of herbivorous and carnivorous mammals are similar for
similarly sized species. In this section, we present allometric models for three measures of
metabolic rate on the basis of body size in mammals: basal metabolic rate (BMR), resting
metabolic rate (RMR), and free-living metabolic rate (FMR). All equations take the general
form of Y = aWt , but also can be represented in their log-transformed form (the equationb
of a straight line). We conclude this section by discussing the influence of ambient
temperature on mammalian metabolic rates. Additional information that allows one to
estimate a 95-percent confidence interval for a predicted FMR, the expression of metabolic
rate that is generally most appropriate for wildlife exposure assessments, is provided in
Section 3.7.
3.5.2.1. Basal Metabolic Rate
On the basis of BMR measurements for 26 species weighing 3.5 to 600 kg, Kleiber
(1961) estimated that BMR was related to body weight in mammals according to the 3/4
power:
3-27
BMR (kcal/day) = 70 Wt (kg) ± 0.004 [3-42]0.75
Boddington's (1978) analysis produced similar results:
BMR (kcal/day) = 75 Wt (kg) ± 0.013 [3-43]0.73
3.5.2.2. Resting Metabolism
Stahl (1967) used an extensive database (349 species) to determine slightly higher
values for RMR than had been determined for BMR (Section 2.5.2.1):
RMR (kcal/day) = 80 Wt (kg) [3-44]0.76
3.5.2.3. Field Metabolic Rate
Based on doubly-labeled water measurements with 23 species of placental
mammals, Nagy (1987) developed an equation relating FMR to body weight:
FMR (kjoules/day) = 3.35 Wt (g), or placental mammals [3-45]0.813
FMR (kcal/day) = 0.800 Wt (g)0.813
The slope of 0.813 is significantly higher than the BMR slopes of 0.73 to 0.76 reported
above. Thus, the FMR does not appear to be a constant multiple of BMR over a range of
body sizes as was the case in birds. However, no FMR measurements have yet been made
on shrews or other very active small mammals, and whales were included in the FMR data
set (Nagy, 1987).
Separating the herbivores from non-herbivores, Nagy (1987) developed two
additional equations:
FMR (kjoules/day) = 5.943 Wt (g), or herbivores [3-46]0.727
FMR (kcal/day) = 1.419 Wt (g)0.727
This is because conductance and heat loss for a given thermal gradient between body temperaturef
and ambient temperature rise more slowly with body size than do basal metabolic rates (Peters,
1983).
3-28
FMR (kjoules/day) = 2.582 Wt (g), or non-herbivores [3-47]0.862
FMR (kcal/day) = 0.6167 Wt (g)0.862
Separating rodents from other animals, Nagy (1987) found:
FMR (kjoules/day) = 10.51 Wt (g), or rodents [3-48]0.507
FMR (kcal/day) = 2.514 Wt (g)0.507
Nagy (1987) estimated the accuracy of the doubly-labeled water method to be ± 8 percent
or better.
To normalize FMR to body weight (e.g., kcal/kg-day), divide the daily FMR by body
weight. In Section 3.7, we provide the regression statistics for Nagy's (1987) equations
(including sample size and the regression coefficient) and information that allows one to
estimate a 95-percent confidence interval for an FMR value predicted for a specified body
weight.
3.5.2.4. Temperature and Metabolic Rate
Few researchers have attempted general multiple regressions of metabolic rate with
both body mass and temperature for mammals. However, several relationships have been
investigated qualitatively (Peters, 1983):
 Low temperatures induce a greater proportional rise in metabolic rate
relative to basal metabolic rate in smaller mammals than in larger ones.f
 At high temperatures, metabolic rate increases to increase blood flow and
evaporative cooling (e.g., panting).
3-29
Peters (1983) developed an equation relating the ratio of SMR to BMR to thermal gradient
for mammals:
SMR/BMR = 0.068 (thermal gradient in C) Wt (kg) [3-49]-0.182
Thus, standard metabolic costs increase relative to basal metabolism at lower
temperatures, but less so for larger than for smaller mammals.
3.5.3. Reptiles and Amphibians
Most reptiles and amphibians tend to have much lower metabolic rates than birds or
mammals because they are poikilothermic. For example, at temperatures similar to normal
body temperatures of birds and mammals (around 37 to 39C), resting metabolic rates of
reptiles and amphibians tend to be only 10 to 20 percent of those of birds and mammals of
similar body weight (Bennett and Dawson, 1976). In this section, we provide some
examples of allometric equations for metabolic rate. Because metabolic rate depends on
body temperature, which in poikilotherms can vary substantially over time, we recommend
that those persons interested in estimating metabolic rates consult more complete
treatments of the subject, including thermoregulation in poikilotherms (e.g., Bennett and
Dawson, 1976; Congdon et al., 1982; Duellman and Trueb, 1986; Feder and Burggren, 1992;
Harless and Morlock, 1979; Hutchinson, 1979).
3.5.3.1. Basal and Resting Metabolic Rates
Robinson et al. (1983) developed an equation for the relationship between BMR and
body mass for reptiles and amphibians at 20C:
BMR (Watts) = 0.19 Wt (kg), or [3-50]0.76
BMR (kcal/day) = 3.92 Wt (kg)0.76
Thus, the BMR of homeotherms (Sections 3.5.1 and 3.5.2) is approximately 30 times the
BMR of poikilotherms at this ambient temperature (Peters, 1983). The difference in
Measured for fasting individuals during the period of normal inactivity (at night for most species).g
Measured for fasting individuals during the period of normal activity (daytime for most species).h
3-30
metabolic rates between homeotherms and poikilotherms is lessened when poikilotherms
modify their body temperatures by behavioral adjustments (such as basking in the sun).
Andrews and Pough (1985) used multiple regression analysis to evaluate the
relationship between metabolic rate and three variables—mass, temperature, and standard
or resting metabolic state—for snakes and lizards. From a total of 226 observations on 107
species (between 20 and 30C for most observations), they developed the following
equation:
MR (ml O /hr) = 0.013 Wt (g) × 10 [3-51]2
0.80 0.038 temperature (C)
× 100.14 metabolic state
where MR equals either SMR or RMR and metabolic state equals zero (0) for standard
metabolism and equals 1 for resting metabolism. The Q values for the influence ofg h
10
temperature on metabolic rate (i.e., quotient of the rate measured at one temperature
divided by the rate measured at a temperature 10C lower) were 2.4 for resting metabolism
and 1.4 for standard metabolism. Thus SMR depended less on ambient temperature than
did RMR.
Equation 3-51 is based on adult animals and should not be used to estimate
metabolic rates of juvenile snakes and lizards. Andrews and Pough (1985) reviewed
allometric equations relating resting metabolic rate to body weight within species and
found that the exponents were significantly lower than the value of 0.80 in Equation 3-51.
See Andrews and Pough (1985) for intraspecific allometric models for this group.
3.5.3.2. Free-Living Metabolic Rates
Nagy (1987) developed an equation for the relationship between FMR and body size
in iguanid lizards:
3-31
FMR (kjoules/day) = 0.224 Wt (g), or [3-52]0.799
FMR (kcal/day) = 0.0535 Wt (g)0.799
Bennett and Nagy (1977) estimated that the ratio of FMR to EMR for lizards is 2.0.
Robinson et al. (1983) estimated the value to be 2.9, assuming that lizards rest at
maintenance levels for 8 hours per day at 35C.
Feder (1981, 1982) presented equations relating FMR to body size of unrestrained
ranid (frog) tadpoles at 25C:
dry mass (mg) = 0.047 (wet mass) (mg) [3-53]1.06
and
FMR (µlO /hr) = 2.5 (dry mass) (mg), or [3-54]2
0.878
FMR (mlO /day) = 0.06 (dry mass) (mg)2
0.878
Assuming 1 milliliter of oxygen is metabolically equivalent to approximately 4.80 calories
(Dawson, 1974):
FMR (cal/day) = 0.288 (dry mass) (mg) [3-55]0.878
Burggren et al. (1983) estimated Q values for metabolic rates for bullfrog larvae of 1.8710
between temperatures of 15 and 25C and of 2.41 between temperatures of 25 and 33C.
Q values for a second ranid species (Rana berlandieri) were similar (1.97 and 1.76,10
respectively). Thus, the metabolic rate for ranid frogs approximately doubles with each 10degree
rise in temperature over this range of temperatures.
The equations presented in this section show that poikilotherm metabolic rate
depends strongly on temperature. The available literature on the subject is extensive and
complex, and again, interested readers are encouraged to consult substantive treatments
of the subject (see references cited in the introduction to Section 3.5.3).
3-32
3.6. MATH PRIMER AND UNIT CONVERSIONS
To assist readers in using or modifying allometric equations presented in this
Handbook or in using allometric equations presented in the open literature, we provide a
brief summary of logarithm and power functions in Sections 3.6.1 and 3.6.2. Section 3.6.3
contains frequently used unit conversion factors.
3.6.1. Summary of Operations Involving Logarithms
log 1 = 0
log (N N ) = log N + log N1 2 1 2
log (N / N ) = log N - log N1 2 1 2
log (1 / N ) = -log N1 1
log (N ) = c log N1 1
c
log c root of N = log (N ) = (1/c) log N1 1 1
1/c
3.6.2. Summary of Operations Involving Powers
W W = Wa b a+b
(W ) = Wa b ab
(W W ) = W W1 2 1 2
a a a
W / W = Wa b a-b
W / W = Wa a-1
1/W = Wb -b
W = 10
(W / W ) = W /W1 2 1 2
a a a
c root of W = (W ) = Wa a 1/c a/c
3-33
3.6.3. Unit Conversions
3.6.3.1. Approximate Factors for Metabolic Equations
1 kg dry mass = 3 to 10 kg wet mass (Peters, 1983)
1 kg dry mass = 22 × 10 joules (Peters, 1983)6
1 kg wet mass = 2 to 7 × 10 joules (Peters, 1983)6
1 kg fat = 40 × 10 joules (Peters, 1983)6
tissue density = 1 kg/liter (Peters, 1983)
1 kg wet mass = 1 × 10 µm (Peters, 1983)15 3
1 kg dry mass = 0.4 kg carbon (Peters, 1983)
1 ml O = 20.1 joules (Peters, 1983)2
= 4.8 calories (Dawson, 1974)
3.6.3.2. Exact Conversions
Area
1 acre = 0.4047 hectares (ha)
1 square mile mi ) = 259 ha2
1 square meter (m ) = 1 × 10 ha2 -4
1 square kilometer (km ) = 100 ha2
Length
1 inch = 2.54 centimeters (cm)
1 foot = 0.3 meters (m)
= 30.48 cm
1 mile (mi) = 1.61 kilometers (km)
Volume
1 m = 1 × 10 liters (L)3 3
= 1 × 10 cm6 3
Mass
1 ounce (oz) = 28.35 grams (g)
1 pound (lb) = 453.6 g
1 lb = 0.4536 kilograms (kg)
3-34
Work and energy (force × distance)
1 joule (J) = 1 kg-m /s2 2
= 0.239 calories (cal)
Power (energy per unit time)
1 watt (W) = 1 kg-m /s2 3
= 1 joule/s
= 20.64 kcal/day
1 ml O /s = 0.0446 mMol O /s2 2
= 1.43 mg O /s2
3.7. ESTIMATING CONFIDENCE INTERVALS
A commonly reported measure of the precision of estimating log Y from log Wt (or Y
from Wt) for allometric equations is the standard error (SE) of log Y:
log Y = log a + b log Wt ± SE of log Y [3-2]
The SE of log Y is the standard error of the estimate of log Y from log Wt at a value of log
Wt that represents the mean of the log Wt values used to estimate the allometric
relationship. This value cannot be used to estimate a confidence interval (CI) for a log Y
value predicted from log Wt values other than the mean log Wt value. The CI of a predicted
log Y value is smallest at the mean log Y and mean log Wt values and increases as log Wt
for the species of interest deviates from mean log Wt. Thus, to estimate the CI for a single
predicted value of Y, one also must know the sample size and the mean of the log Wt
values used in developing the allometric equation, which many investigators do not report.
Nagy (1987), however, did provide sufficient statistical information to estimate a 95percent
CI for a predicted value of Y given any value of Wt for his free-living (field)
metabolic rate (FMR) and food ingestion (FI) rate equations. In this section, we outline
Nagy's short-cut for estimating this CI and provide the statistical values required for each
of Nagy's equations presented in this Handbook.
3-35
To estimate 95-percent CIs for the predicted FMR and FI rate, use the values from
Table 3-3 (for FI rate equations) or 3-4 (for FMR equations) in the following formula:
______
95% CI = log y ± c [d + e (log Wt - log Wt) ]log y
2 0.5
where y is FMR in kilojoules/day or FI in grams (dry weight)/day. Log Wt is the log of the
body weight in grams of the species for which y is being estimated. Log Wt bar is the
mean log Wt of the species used to develop the allometric equation. Values for c, d, e, and
log Wt bar are provided in Tables 3-3 and 3-4. Tables 3-3 and 3-4 also provide sample
sizes (N), regression coefficients (r ), and SE estimates for b and log a in the applicable2
equations.
3-36
Regression Statistics for Allometric Equations for Food Ingestion (FI) Rates (Dry Matter Ingestion) Rates of Free-Living Mammals, Birds,
and Lizards. Equations are in the form Y = aWt where Y is Food Ingestion Rate (in grams dry weight/day) and Wt is body weight ofb
species s (grams wet weight).
______
95% CI = log FI ± c [d + e (log Wt - log Wt) ]log FI(species s) (species s) (species s)
2 0.5
Group
subgroup
Equation
a log a (SE log a) b (SE b) N r2
______
log Wt c d e
Birds 3-3 0.64 -0.188 (0.060) 0.651 (0.028) 50 0.919 1.983 0.347 1.020 0.026
passerines 3-4 0.40 -0.400 (0.075) 0.850 (0.053) 26 0.915 1.378 0.158 1.038 0.480
non-passerines 3-5 0.30 -0.521 (0.132) 0.751 (0.048) 24 0.919 2.638 0.401 1.042 0.061
seabirds 3-6 0.49 -0.306 (0.187) 0.704 (0.061) 15 0.911 2.958 0.399 1.067 0.109
Eutherian Mammals
(i.e., placental)
3-7 0.23 -0.629 (0.065) 0.822 (0.026) 46 0.958 2.196 0.425 1.022 0.015
rodents 3-8 0.62 -0.207 (0.194) 0.564 (0.119) 33 0.421 1.598 0.434 1.030 0.313
herbivores 3-9 0.58 -0.239 (0.109) 0.727 (0.039) 17 0.960 2.566 0.405 1.059 0.041
Iguanids
herbivores 3-12 0.019 -1.713 (0.123) 0.841 (0.059) 5 0.985 1.896 0.358 1.200 0.278
insectivores 3-13 0.012 -1.890 (0.037) 0.773 (0.038) 20 0.958 0.870 0.151 1.050 0.279
Source: Nagy, 1987.
Table 3-3. Regression Statistics for Nagy's (1987) Allometric Equations for Food Ingestion Rates for Free-Living Animals
3-37
Regression Statistics for Allometric Equations for Free-Living Metabolic Rates (FMR) of Free-Living Mammals, Birds, and Lizards.
Equations are in the form Y = aWt where Y is FMR (in kilojoules/day) and Wt is body weight of species s (grams wet weight).b
______
95% CI = log FMR ± c [d + e (log Wt - log Wt) ]log FMR(species s) (species s) (species s)
2 0.5
Group
subgroup
Equation
a log a (SE log a) b (SE b) N r2
______
log Wt c d e
Birds 3-34 10.9 1.037 (0.064) 0.640 (0.030) 50 0.907 1.983 0.368 1.020 0.026
passerines 3-36 8.89 0.949 (0.059) 0.749 (0.037) 26 0.899 1.378 2.014 0.026 0.0014
non-passerines 3-37 4.79 0.681 (0.102) 0.749 (0.037) 24 0.899 2.638 2.014 0.026 0.0014
seabirds 3-38 8.02 0.904 (0.187) 0.704 (0.061) 15 0.911 2.958 0.399 1.067 0.109
non-seabirds 3-39 21.1 1.325 (0.081) 0.440 (0.049) 35 0.709 1.565 0.297 1.029 0.113
Eutherian
Mammals (i.e.,
placental)
3-45 3.35 0.525 (0.057) 0.813 (0.023) 46 0.967 2.196 0.371 1.022 0.015
rodents 3-48 10.5 1.022 (0.141) 0.507 (0.087) 33 0.524 1.598 0.316 1.030 0.313
herbivores 3-46 5.94 0.774 (0.109) 0.727 (0.039) 17 0.959 2.566 0.406 1.059 0.041
non-herbivores 3-47 2.58 0.412 (0.058) 0.862 (0.026) 29 0.977 1.980 0.321 1.035 0.027
Iguanids 3-52 0.224 -0.650 (0.029) 0.799 (0.023) 25 0.981 1.075 0.161 1.040 0.088
Source: Nagy, 1987.
Table 3-4. Regression Statistics for Nagy's (1987) Allometric Equations for Free-Living (Field) Metabolic Rates
3-38
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Dpot 


t2
22
t1
C(t) IR(t) dt
4-1
[4-1]
4. EXPOSURE ESTIMATES
This section provides equations to estimate oral doses of chemical contaminants
for wildlife, along with a discussion of dose estimates for other exposure routes. Section
4.1 provides general dose equations. Equations for drinking water exposures are
presented in Section 4.1.1, followed by equations for dietary exposures in Section 4.1.2. In
the dietary exposure section, data on the caloric and water content of various food types
and diet assimilation efficiencies are also provided. An equation and data to facilitate
estimating doses received through soil or sediment ingestion are discussed in Section
4.1.3. Sections 4.1.4 and 4.1.5 provide a qualitative discussion of inhalation and dermal
dose estimates. Section 4.2 describes considerations for analyses of uncertainty in
exposure assessments. References are provided in Section 4.3.
4.1. GENERAL DOSE EQUATIONS
EPA's (1992a) Framework for Ecological Risk Assessment defines exposure as the
co-occurrence of or contact between a stressor and an ecological component. When
assessing risks of exposure to chemical contaminants, potential dose is often the metric
used to quantify exposure. Potential dose is defined as the amount of chemical present in
food or water ingested, air inhaled, or material applied to the skin (U.S. EPA, 1992b).
Potential dose is analogous to the administered dose in a toxicity test. Because exposure
to chemicals in the environment is generally inadvertent, rather than administered, EPA's
(1992b) Guidelines for Exposure Assessment use the term potential dose rather than
administered dose.
A general equation for estimating dose for intake processes is: