--- title: Calculation of time weighted average concentrations with mkin author: Johannes Ranke date: "`r Sys.Date()`" bibliography: references.bib output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Calculation of time weighted average concentrations with mkin} %\VignetteEngine{knitr::rmarkdown} %\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. 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 [@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: $$c_\textrm{twa} = \frac{c_0}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) $$ 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' is include in the 'mkin' package (['max_twa_parent'](https://pkgdown.jrwb.de/mkin/reference/max_twa_parent.html). If the same is needed for metabolites, the function ['max_twa'](https://pkgdown.jrwb.de/pfm/reference/max_twa.html) can be used.