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---
title: Calculation of time weighted average concentrations with mkin
author: Johannes Ranke
date: "`r Sys.Date()`"
bibliography: references.bib
vignette: >
%\VignetteEngine{knitr::rmarkdown}
%\VignetteIndexEntry{Calculation of time weighted average concentrations with mkin}
%\VignetteEncoding{UTF-8}
---
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.
Time weighted average concentrations for the DFOP model are calculated using
the formulas given in the FOCUS kinetics guidance [@FOCUSkinetics2014, p. 251]:
SFO:
$$c_\textrm{twa} = c_0 \left( 1 - e^{- k t} \right) $$
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:
$$f_\textrm{twa} = \left( 1 - e^{- k t} \right) $$
FOMC:
$$f_\textrm{twa} = \frac{\beta}{t (1 - \alpha)}
\left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) $$
DFOP:
$$f_\textrm{twa} = \frac{1}{t} \left(
\frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) +
\frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) $$
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