# Calculation of time weighted average concentrations with mkin

#### Johannes Ranke

#### 2019-07-04

Since version 0.9.45.1 of the ‘mkin’ package, a function for calculating time weighted average concentrations for decline kinetics (i.e. only for the compound applied in the experiment) is included. Strictly speaking, they are maximum moving window time weighted average concentrations, i.e. the maximum time weighted average concentration that can be found when moving a time window of a specified width over the decline curve.

Time weighted average concentrations for the SFO, FOMC and the DFOP model are calculated using the formulas given in the FOCUS kinetics guidance (FOCUS Work Group on Degradation Kinetics 2014, 251):

SFO:

\[c_\textrm{twa} = c_0 \frac{\left( 1 - e^{- k t} \right)}{ k t} \]

FOMC:

\[c_\textrm{twa} = c_0 \frac{\beta}{t (1 - \alpha)} \left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) \]

DFOP:

\[c_\textrm{twa} = \frac{c_0}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]

Often, the ratio between the time weighted average concentration \(c_\textrm{twa}\) and the initial concentration \(c_0\)

\[f_\textrm{twa} = \frac{c_\textrm{twa}}{c_0}\]

is needed. This can be calculated from the fitted initial concentration \(c_0\) and the time weighted average concentration \(c_\textrm{twa}\), or directly from the model parameters using the following formulas:

SFO:

\[f_\textrm{twa} = \frac{\left( 1 - e^{- k t} \right)}{k t} \]

FOMC:

\[f_\textrm{twa} = \frac{\beta}{t (1 - \alpha)} \left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) \]

DFOP:

\[f_\textrm{twa} = \frac{1}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]

Note that a method for calculating maximum moving window time weighted average concentrations for any model fitted by ‘mkinfit’, and also for metabolites in such models, can be found in the ‘pfm’ package, which is currently not published on CRAN, but available from github and documented at my website.

FOCUS Work Group on Degradation Kinetics. 2014. Generic Guidance for Estimating Persistence and Degradation Kinetics from Environmental Fate Studies on Pesticides in Eu Registration. 1.1 ed. http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics.