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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:
-
-$$c_\textrm{twa} = c_0 \frac{\left( 1 - e^{- k t} \right)}{ k t} $$
-
-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:
-
-$$f_\textrm{twa} = \frac{1}{t} \left(
-  \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) +
-  \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right)  \right) $$
-