Calculation of time weighted average concentrations with mkin

+ + + +

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. +

Johannes +Ranke

Last change 18 September 2019 +(rebuilt 2025-02-14)