blob: 6f283eb25b098fdfd372113386c08849326bc8f1 (
plain) (
blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
|
---
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()` function. If the same is
needed for metabolites, the function `pfm::max_twa()` from the 'pfm' package
can be used.
|
