blob: 6ebb7b89bfb3b0ddd7f1e953e9cfa67b0225ad84 (
plain) (
tree)
|
|
---
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' or from parent decline model
parameters is included in the
['max_twa_parent'](https://pkgdown.jrwb.de/mkin/reference/max_twa_parent.html)
function.
If the same is needed for metabolites, the function
['max_twa'](https://pkgdown.jrwb.de/pfm/reference/max_twa.html) from the 'pfm' package can be used.
|
