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.

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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 = C₀e^-kt (equation 1)

Where, C_t = concentration at time t
C₀ = initial concentration or percent applied radioactivity
e = base e
k = rate constant of decline 1/days
t = time

SFO is solved by adjusting C₀ 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 C₀ – kt (equation 4)

where,
ln=natural log

The linear SFO equation is solved by adjusting C₀ and k to minimize residuals.

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

C = [C₀^{(1 - N)} - (1 - N)k_IOREt]^{(1/(1 - N))} (equation 5)

This model is solved by adjusting C₀, 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) C₀^1-N(1-0.1^1-N)] / [log(10) (1-N)k_IORE] (equation 6)

DT50 = [(C₀/2)^(1-N) - C₀^(1-N)] / [k_IORE x (N-1)] (equation 7)

DT90 = [(C₀/10)^(1-N) - C₀^(1-N)] / [k_IORE x (N-1)] (equation 8)

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Double First-Order in Parallel (DFOP)

C_t = C₀g^-k₁t + C₀(1 - g)^-k₂t (equation 9)

Where g is the fraction of C₀ applied to compartment 1
k₁ = rate constant for compartment 1 in 1/days
k₂ = rate constant for compartment 2 in 1/days

DFOP is solved by minimizing the objective function for DFOP (Equation 10) and solving for g, C₀, k₁, and k₂. In Sigmaplot, C₀ x g is equal to a and C₀(1-g) = c. The equation is solved by changing a, c, k₁, and k₂ 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 k₁ or k₂ 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.

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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

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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

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