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.
Single First-Order Rate Model (SFO)
Ct = C0e-kt (equation 1)
Where, Ct = 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 Ct = ln C0 - kt (equation 4)
The linear SFO equation is solved by adjusting C0 and k to minimize residuals.
Nth-Order Rate Model or Indeterminate Order Rate Equation Model (IORE)
C = [C0(1 - N) - (1 - N)kIOREt](1/(1 - N)) (equation 5)
This model is solved by adjusting C0, kIORE, and N to minimize the objective function (SIORE) 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.
tIORE = [log(2) C01-N(1-0.11-N)] / [log(10) (1-N)kIORE] (equation 6)
DT50 = [(C0/2)(1-N) - C0(1-N)] / [kIORE x (N-1)] (equation 7)
DT90 = [(C0/10)(1-N) - C0(1-N)] / [kIORE x (N-1)] (equation 8)
Double First-Order in Parallel (DFOP)
Ct = C0ge-k1t + C0(1 - g)e-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.
Objective Function: SFO, IORE, and DFOP
Objective function Smodel = Σ(Cmodel - Cd)2 (equation 10)
SSFO = objective function of the SFO fit to be minimized
SIORE = objective function of IORE fit to be minimized
SDFOP = objective function for DFOP fit to be minimized
Cmodel= modeled value at time corresponding to Cd
Cd = data
Critical Value to Determine Whether SFO Will Be Used to Estimate a Model Input Value
If SSFO is less than Sc, the SFO model is used to describe kinetics for modeling. If not, use IORE or DFOP for modeling.
Sc = SIORE [1 + (p/(n-p)) F (p,n - p,α)] (equation 11)
Sc = 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
Ct = c0 / [(t/β) + 1]α (equation 12)
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)
β = (c01-N) / [k(N-1)] (equation 14)