Supplemental Information to

October 6-9, 2009 FIFRA Scientific Advisory Panel Meeting

Derivation of Post-application Dermal Exposure Algorithm

______________________________________________________________

This document provides the rationale and derivation for the
post-application dermal exposure equation (8.3) for the treated pet
scenario found in Section 8.2.2 Post-application Dermal Exposure
Assessment of the Standard Operating Procedures for Residential
Pesticide Exposure Assessment submitted to the FIFRA Scientific Advisory
Panel for review and comment during the meeting held October 6-9, 2009. 


A transferable residue (TR) -- expressed in units of (for example)
ug/cm2 -- is the (area) normalized amount of pesticide that is
anticipated to transfer from a pet to person via dermal contact after
application of a pesticide treatment.  These pesticide residues
represented by TR degrade over time resulting is less residue being
available for transfer.  In developing the dermal exposure equation, the
TR values were assumed to dissipate over time as defined by a
first-order decay function:

												(1.1)

where TR(t) is the transferable residue at time t (in hours), TR0 is the
transferable residue (i.e. TR at time t=0), and k is the decay constant
that defines how rapidly the TR dissipates.  

Although the dissipation process is defined by the decay constant, k, a
more intuitive parameter is to define the percent of the TR that
dissipates per day.  Based on the dissipation per day, d, one can derive
the decay constant (k) that would result in such a daily dissipation
rate.  For example, consider a TR that is defined to dissipate by 14%
per day (i.e. d = 0.14).  The ratio between TRs separated by 24 hours
would be 0.86.  In other words, a TR value 24 hours later would be 86%
of its current value as derived below.

It follows that: 

More generally, k can be expressed in terms of d:

(1.2)

Note that k is divided by 24 since it is expressed in units hr-1, but d
is the (unitless) fraction that dissipates per day.

Once the daily dissipation rate (d) has been specified, equation (1.1)
can be used to calculate the average daily dermal exposure:

												(1.3)

where TC is the transfer coefficient, n is the number of days of
exposure over which the average exposure is being calculated, and ET is
the exposure time per day expressed in hours per day.  

 The exposure equation is based on integrating over ET hours of contact
with a pet immediately following application, when TR values are
highest.  The integration over ET hours is repeated every 24 hours for
the total number of days for which dermal exposure is being calculated. 
When the exposure equation (1.3) is rewritten in an expanded form (i.e.
without summation notation), it is more apparent that dermal contact
with a pet is assumed to take place in a repeated manner on a daily
basis for the same amount of time (ET) each day and at regular and
precisely- spaced intervals (i.e., here, every 24 hours):

												

												(1.4)

Graphically, the integration of the equation describing the transferable
residues over time, TR(t) at regular intervals could be displayed as
follows:

The integration in the expanded exposure equation (1.4) can next be
written as:

By grouping similar terms, the above equation can be written as:

 as 1, the equation can be written as:

 can be factored out to rewrite the equation as:

 , which allows the exposure equation to expressed as:

												(1.5)

 < 1, which implies that the summation of the geometric series is as
follows:

												(1.6)

Substituting equation (1.6) into (1.5) provides the final reduced form
of the average daily dermal exposure equation for ET hours of contact
with a pet per day, for n consecutive days assuming the transferable
residue TR declines in a first order decay process described by the
decay constant k and the alternative parameter d, the (fractional)
dissipation per day:

										(1.7) [and (8.3)]

This is the equation presented as Equation 8.3 in the Standard Operating
Procedures for Residential Pesticide Exposure Assessment being presented
to the SAP.

 The exposure calculations described in this section are not considered
appropriate for pet collar treatments since pesticide residues are
expected to be released at a constant rate from the collar and are not
expected to dissipate over time.

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