# Degradation Kinetics Equations

This page is an attachment to the SOP: Standard Operating Procedure for Using the NAFTA Guidance to Calculate Representative Half-life Values and for Characterizing Pesticide Degradation, Version 2.

You can find the full SOP and associated information here: Guidance to Calculate Representative Half-life Values and Characterizing Pesticide Degradation.

**On this page**

- Single First-Order Rate Model (SFO)
- N
^{th}-Order Rate Model or Indeterminate Order Rate Equation Model (IORE) - Double First-Order in Parallel (DFOP)
- Objective Function: SFO, IORE, and DFOP
- Critical Value to Determine Whether SFO Will Be Used to Estimate a Model Input Value
- Gustafson and Holden Model or FOMC Model

## Single First-Order Rate Model (SFO)

C_{t} = C_{0}e^{-kt}
(**equation 1**)

Where, C_{t} = concentration at time t

C_{0} = initial concentration or percent applied radioactivity

e = base e

k = rate constant of decline 1/days

t = time

SFO is solved by adjusting C_{0} and k to minimize the objective function shown in equation 10:

DT50 = natural log (2)/k (**equation 2**)

DT90 = ln (10)/k (**equation 3**)

The linear form of the single first-order equation is shown in equation 4.

ln C_{t} = ln C_{0} – kt (**equation 4**)

where,

ln=natural log

The linear SFO equation is solved by adjusting C_{0} and k to minimize residuals.

## N^{th}-Order Rate Model or Indeterminate Order Rate Equation Model (IORE)

C = [C_{0}^{(1 - N)} - (1 - N)k_{IORE}t]^{(1/(1 - N))}
(**equation 5**)

This model is solved by adjusting
C_{0}, k_{IORE}, and N to minimize the objective function (S_{IORE}) for IORE (see equation 10). An estimated SFO model input value using the IORE model is calculated by approximating the SFO model half-life that would have a DT90 that passes through the IORE DT90 and is estimated as shown in equation 6. Traditional DT50 and DT90 for the IORE model are calculated using equations 7 and 8.

t_{IORE} = [log(2) C_{0}^{1-N}(1-0.1^{1-N})] / [log(10) (1-N)k_{IORE}]
(**equation 6**)

DT50 = [(C_{0}/2)^{(1-N)} - C_{0}^{(1-N)}] / [k_{IORE} x (N-1)]
(**equation 7**)

DT90 = [(C_{0}/10)^{(1-N)} - C_{0}^{(1-N)}] / [k_{IORE} x (N-1)]
(**equation 8**)

## Double First-Order in Parallel (DFOP)

C_{t} = C_{0}g^{-k1t} + C_{0}(1 - g)^{-k2t}
(**equation 9**)

Where g is the fraction of C_{0} applied to compartment 1

k_{1} = rate constant for compartment 1 in 1/days

k_{2} = rate constant for compartment 2 in 1/days

DFOP is solved by minimizing the objective function for DFOP (Equation 10) and solving for g, C_{0}, k_{1}, and k_{2}. In Sigmaplot, C_{0} x g is equal to a and C_{0}(1-g) = c. The equation is solved by changing a, c, k_{1}, and k_{2} to minimize the objective function as described in equation 10. The g described in the NAFTA degradation kinetics document and in this document corresponds to the f in the R output. The g or *f* parameter from the DFOP fit indicates the fraction of the initial chemical that degrades at the fast rate.

Fast and slow DT50 and DT90 values are calculated using equations 3 and 4 and k_{1} or k_{2} in place of k. These fast and slow rates only describe a subset of the data. An overall value is reported in R based on the point on the curve where 50% of the chemical has declined.

## Objective Function: SFO, IORE, and DFOP

Objective function S_{model} =
Σ(C_{model} - C_{d})^{2}
(**equation 10**)

where

S_{SFO} = objective function of the SFO fit to be minimized

S_{IORE} = objective function of IORE fit to be minimized

S_{DFOP} = objective function for DFOP fit to be minimized

C_{model}= modeled value at time corresponding to C_{d}

C_{d} = data

## Critical Value to Determine Whether SFO Will Be Used to Estimate a Model Input Value

If S_{SFO} is less than S_{c}, the SFO model is used to describe kinetics for modeling. If not, use IORE or DFOP for modeling.

S_{c} = S_{IORE}
[1 + (p/(n-p)) F (p,n - p,α)]
(**equation 11**)

where

S_{c} = the critical value that defines the confidence contours

p = number of parameters, (3 in this case)

α = the confidence level (0.50 for this guidance)

F(a,b,c)= F distribution with *a* and *b* degrees of freedom and level of confidence *c*

## Gustafson and Holden Model or FOMC Model

C_{t} = c_{0} /
[(t/β) + 1]^{α}
(**equation 12**)

where:

α = shape parameter determined by coefficient of variation of k values

β = location parameter

This model is not solved using a regression model in EFED. Model parameters for FOMC are converted from IORE results using the following equations:

α = 1 / (N-1) (**equation 13**)

and

β = (c_{0}^{1-N}) / [k(N-1)]
(**equation 14**)