# Calculation of time weighted average concentrations with mkin

#### Johannes Ranke

#### Last change 18 September 2019 (rebuilt 2023-10-13)

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:

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

HS for \(t > t_b\):

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

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:

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

HS for \(t > t_b\):

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

Note that a method for calculating maximum moving window time weighted average concentrations for a model fitted by ‘mkinfit’ or from parent decline model parameters is included in the `max_twa_parent()` function. If the same is needed for metabolites, the function `pfm::max_twa()` from the ‘pfm’ package can be used.

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.