Probabilistic Approach to Address Exposure to Multiple Chemicals for Course-of-Action Analysis

The percent *P** of a unit expected to be seriously affected (in a mission-hindering way) by acute respiratory exposure(s) to multiple chemicals could be estimated using the following procedure. Such exposures might affect similar toxic end points and/or different toxic end points. This procedure is a generalization of the assumption made in Chapter 4 that, in the case of exposure to a single chemical, *P** = *F* × *P* where *F* is the estimated fraction of the unit exposed to that chemical and *P* is an estimate of the percent of exposed individuals expected to incur serious (mission-incapacitating) illness, modeled as

(E-1)

In that equation, Φ is the cumulative normal (Gaussian) probability distribution function; log denotes logarithm (using any specified base, such as 10 or *e*); *C*_{50} and *C*_{15} are estimated (e.g., 1-hour) concentrations that elicit a 50% and 15% response, respectively, obtained from a lognormal dose-response curve previously fitted to relevant toxicity data as described in Appendix C (estimates of *C*_{50} and *C*_{15} are listed for five chemicals in Chapter

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Appendix E
Probabilistic Approach to
Address Exposure to Multiple Chemicals for
Course-of-Action Analysis
The percent P* of a unit expected to be seriously affected (in a mission-
hindering way) by acute respiratory exposure(s) to multiple chemicals could
be estimated using the following procedure. Such exposures might affect
similar toxic end points and/or different toxic end points. This procedure
is a generalization of the assumption made in Chapter 4 that, in the case of
exposure to a single chemical, P* = F × P where F is the estimated fraction
of the unit exposed to that chemical and P is an estimate of the percent of
exposed individuals expected to incur serious (mission-incapacitating)
illness, modeled as
log(C / C50 )
P = Φ 1036 × 100%,
.
log(C50 / C15 )
(E-1)
log(C / C50 )
P= Φ × 100%.
σ
In that equation, Φ is the cumulative normal (Gaussian) probability distribu-
tion function; log denotes logarithm (using any specified base, such as 10
or e); C50 and C15 are estimated (e.g., 1-hour) concentrations that elicit a
50% and 15% response, respectively, obtained from a lognormal dose-re-
sponse curve previously fitted to relevant toxicity data as described in Ap-
pendix C (estimates of C50 and C15 are listed for five chemicals in Chapter
181

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182 APPENDIX E
4, Table 4-1); and σ is the estimate of the lognormal-model “shape” param-
eter, the inverse of which specifies the steepness of the corresponding fitted
dose-response curve.
The subcommittee wishes to consider the general case in which de-
ployed personnel are exposed to n subsets of chemicals, each ith subset of
which contains ni chemicals (for i = 1, …, n) that all induce a common
(mechanism-specific) toxic end point Ti that is independent from any differ-
ent Ti –specific mechanisms and from any different end points Tj (for j … i)
induced by other chemicals involved in the exposure scenario considered.
Typically neither n nor ni will be large (i.e., exceed 2 or 3), but the case of
multiple chemicals is treated here in general terms to explain the general
approach clearly. The 1-hour respiratory concentration of the jth chemical
in the ith subset shall be denoted Ci,j, where j = 1, …, ni and where again i
= 1, …, n. Because this general treatment involves double-subscript nota-
tion, it will be convenient to adopt the alternative notation µ = C50 for the
lognormal location-parameter estimate C50. Thus, for the jth chemical in the
ith subset, Equation E-1 can be rewritten as
log(Ci , j / µi , j )
Pi , j = Φ × 100%.
σ i, j (E-2)
Different chemicals with similar values of σ as well as similar values
of : for a given end point can be treated as if they were all the same chemi-
cal. In the absence of information supporting an alternative assumption, it
is reasonable to assume in the context of using the lognormal dose-response
model that different chemicals with substantially different values of σ for
a given end point act via independent mechanisms. Concentrations Ci,j of
different chemicals affecting a common end point Ti, all of which have
similar values of σi,j but have substantially different values of :i,j, can rea-
sonably be assumed to represent corresponding weighted contributions to
an aggregate effective concentration Ci that acts via a single underlying
mechanism to elicit Ti. We now assume that, for all i, the ith set of median-
response concentrations :i,j are sorted in order of ascending magnitude for
j = 1, …, ni, such that :i,1 always denotes the smallest median-response
concentration, which in turn corresponds to that chemical with the highest
toxic potency among the ith subset of chemicals. It follows that the relative
weight of the jth contribution to Ci is (:i,1/:i,j), and thus that
∑
ni
Ci , j ( µi ,1 / µi , j )
Ci = j =1

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APPENDIX E 183
The incapacitated percent Pi of a unit exposed to the ith subset of chemicals
is therefore approximated by
ni
log ∑ Ci , j / µi , j
log(Ci / µi ,1 ) j =1
Pi ≈ Φ × 100% = Φ × 100%, (E-3)
Μ (σ i , j ) Μ (σ i , j )
where the function M in the denominator of each bracketed expression
denotes the arithmetic mean over j = 1, …, nj with similar values of σi,j
conditional on i. A more conservative estimate of Pi is obtained if M is
taken to be the maximum or the minimum of σi,j when the numerator of
each bracketed expression in Equation E-3 is #0 or is >0, respectively.
The probability P of incapacitation via any of the n end points consid-
ered is (by “de Morgan’s rule”) equal to the complement of the joint proba-
bility that none of these n end points will occur. Because the probability of
incapacitation via (mechanism-specific) end point Ti is by definition inde-
pendent of that via end point Tj for i … j, it follows that this joint probability
is just the product of the probabilities (1 – Pi ) for i = 1, …, n, and conse-
quently that
n
∏ (100% − P ),
P = 100% − (E-4)
i
i =1
where Pi was defined in approximation E-3.
The percentage P* of a unit seriously affected by chemical exposure is
finally calculated as P* = P × F, where F is the estimated fraction of the
unit exposed to the n subsets of chemicals considered. If each among m
mutually exclusive fractions Fk of a unit (for k = 1, …, m) is exposed to a
combination of chemical concentrations that together generate a corre-
sponding predicted response percentage P[k] calculated using Equation E-4
(where bracket-subscript notation is used to distinguish this percentage from
one defined by Equation E-3), then
∑
m
P* = Fk P[ k ] .
k =1
As explained in Chapter 4, the calculated percentage P* directly speci-
fies the unit status as indicated in Chapter 2 (Table 2-4). In this way, P* is
used to classify the operational risk management (ORM) risk level defined

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184 APPENDIX E
in the military risk assessment matrix by comparing the quantity (100% –
P*) directly to the unit-strength percentage ranges that define the various
risk levels as specified in Appendix C of FM 101-5-1, in TG 230 Table 3-4,
and in Chapter 2 (Table 2-4) and Chapter 4 (Table 4-1) of this report.
This default probabilistic risk-modeling approach for multiple chemi-
cals explained above is illustrated in Box E-1.
BOX E-1 Example of Probabilistic Approach for Multiple Chemicals
It is assumed that a proposed mission would require 40% of a unit be exposed by
inhalation for 1 hour to simultaneous ambient concentrations of 25, 4, and 2 ppm of
hydrogen sulfide, hydrogen cyanide, and 1,1-dimethylhydrazine, respectively. From
Chapter 4 (Table 4-1), corresponding information listed in Appendix C, and the defini-
tions : = C50 and σ = 0.9648 log (C50/C15), the exposure levels, the lognormal-model
parameter estimates : and σ, and the corresponding “severe” toxic end points for these
three chemicals can be summarized as follows:
:i,j Fi,j
Ci,j
Subset
(ppm- (ppm- (unit- End Point
Chemical i j hour) hour) less) Ti
Hydrogen cyanide 1 1 60 140 0.37 Hypoxia
Hydrogen sulfide 1 2 300 710 0.1 Hypoxia
Dimethylhydrazine 2 1 450 1400 1.66 CNS
Severe effects of respiratory exposure to either hydrogen cyanide or hydrogen sulfide
include severe and potentially lethal histotoxic hypoxia due to inhibition of cellular
oxidative metabolism, and σ? values for these chemicals are both relatively small. In
contrast, severe effects of dimethylhydrazine exposure include tremors and vomiting via
uncharacterized central nervous system (CNS) interference. This example could be
considered to involve n = 2 subsets of chemicals: the first including hydrogen cyanide
and hydrogen sulfide (the two chemicals with a similar toxicity mechanism, with σ? ≈
0.374), and the second including only dimethylhydrazine. Corresponding application of
Equation C-2, Equation C-3, and the definition P* = P × F with F = 0.40, in this case
60 300
P1 ≈ Φ 0.347 −1 log e + × 100% ≈ 33.3%,
140 710
450
P2 = Φ 166 −1 log e × 100% ≈ 24.7%,
.
1,400
P = 100% − (100% − 33.3%)(100% − 24.7%) ≈ 49.8%,
P* = 0.40 × P ≈ 20%.
(Continued)

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APPENDIX E 185
BOX E-1 continued
yields which implies that the proposed mission has a unit status of “amber” (mission
capable, with minor deficiencies implying a total of 70-85% unit strength) if chemical
exposures were considered the only risks to the mission. If in this example toxicity due
to hydrogen cyanide and hydrogen sulfide were considered to arise from completely
independent mechanisms, then corresponding calculations would yield P1 ≈ 1.2%, P2 ≈
0% and P3 ≈ 24.7%, implying that P ≈ 25.6% and P* ≈ 10% and consequently, that the
mission has a unit status of green (mission capable, with unit strength $85%). An addi-
tive approximation of aggregate risk due to both hydrogen cyanide and hydrogen sulfide
assuming complete independence would be 1.2% + ~0% ≈ 1.2%. Note how much the
latter approach underestimates the corresponding aggregate risk (~33%) that was pre-
dicted above assuming a common mechanism of action and a (conservatively estimated)
common value of σ.
Abbreviations: AEGL, acute exposure guideline level; CNS, central nervous; ppm, parts per million.

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