From 0818986e2d9a0e68d3d1de26ac381836e0cf5a6f Mon Sep 17 00:00:00 2001 From: Johannes Ranke Date: Wed, 18 Sep 2019 12:17:34 +0200 Subject: Add hockey stick formulas to maximum twa vignette --- vignettes/twa.Rmd | 17 ++- vignettes/twa.html | 384 ++++++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 391 insertions(+), 10 deletions(-) (limited to 'vignettes') diff --git a/vignettes/twa.Rmd b/vignettes/twa.Rmd index 93b2b372..92a0cd4b 100644 --- a/vignettes/twa.Rmd +++ b/vignettes/twa.Rmd @@ -36,6 +36,12 @@ $$c_\textrm{twa} = \frac{c_0}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) $$ +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 $c_0$ @@ -60,8 +66,13 @@ $$f_\textrm{twa} = \frac{1}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) $$ +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 any model fitted by 'mkinfit', and also for metabolites in -such models, can be found in the 'pfm' package, which is currently not published on -CRAN, but available from [github](https://github.com/jranke/pfm) and -[documented](https://pkgdown.jrwb.de/pfm) at my website. +such models, is integrated in the 'mkin' package, see the +[documentation](https://pkgdown.jrwb.de/pfm/reference/max_twa_parent.html) at my website. diff --git a/vignettes/twa.html b/vignettes/twa.html index b5910932..d8322bfc 100644 --- a/vignettes/twa.html +++ b/vignettes/twa.html @@ -8,36 +8,382 @@ - - + Calculation of time weighted average concentrations with mkin + + + + + + + + + + + + + + + + + - + + + + + + + + + + + + + + +
+ + + + +

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 DFOMP model are calculated using the formulas given in the FOCUS kinetics guidance (FOCUS Work Group on Degradation Kinetics 2014, 251):

+

Time weighted average concentrations for the SFO, FOMC and the DFOP model are calculated using the formulas given in the FOCUS kinetics guidance (FOCUS Work Group on Degradation Kinetics 2014, 251):

SFO:

\[c_\textrm{twa} = c_0 \frac{\left( 1 - e^{- k t} \right)}{ k t} \]

FOMC:

@@ -47,6 +393,10 @@

\[c_\textrm{twa} = \frac{c_0}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]

+

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:

@@ -59,7 +409,11 @@

\[f_\textrm{twa} = \frac{1}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]

-

Note that a method for calculating maximum moving window time weighted average concentrations for any model fitted by ‘mkinfit’, and also for metabolites in such models, can be found in the ‘pfm’ package, which is currently not published on CRAN, but available from github and documented at my website.

+

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 any model fitted by ‘mkinfit’, and also for metabolites in such models, is integrated in the ‘mkin’ package, see the documentation at my website.

FOCUS Work Group on Degradation Kinetics. 2014. Generic Guidance for Estimating Persistence and Degradation Kinetics from Environmental Fate Studies on Pesticides in Eu Registration. 1.1 ed. http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics.

@@ -68,6 +422,22 @@ + +
+ + +