---
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.