<!DOCTYPE html> <!-- Generated by pkgdown: do not edit by hand --><html lang="en"> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <meta charset="utf-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title>Calculation of time weighted average concentrations with mkin • mkin</title> <!-- jquery --><script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script><!-- Bootstrap --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/css/bootstrap.min.css" integrity="sha256-bZLfwXAP04zRMK2BjiO8iu9pf4FbLqX6zitd+tIvLhE=" crossorigin="anonymous"> <script src="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/js/bootstrap.min.js" integrity="sha256-nuL8/2cJ5NDSSwnKD8VqreErSWHtnEP9E7AySL+1ev4=" crossorigin="anonymous"></script><!-- bootstrap-toc --><link rel="stylesheet" href="../bootstrap-toc.css"> <script src="../bootstrap-toc.js"></script><!-- Font Awesome icons --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/all.min.css" integrity="sha256-mmgLkCYLUQbXn0B1SRqzHar6dCnv9oZFPEC1g1cwlkk=" crossorigin="anonymous"> <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/v4-shims.min.css" integrity="sha256-wZjR52fzng1pJHwx4aV2AO3yyTOXrcDW7jBpJtTwVxw=" crossorigin="anonymous"> <!-- clipboard.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.6/clipboard.min.js" integrity="sha256-inc5kl9MA1hkeYUt+EC3BhlIgyp/2jDIyBLS6k3UxPI=" crossorigin="anonymous"></script><!-- headroom.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/headroom.min.js" integrity="sha256-AsUX4SJE1+yuDu5+mAVzJbuYNPHj/WroHuZ8Ir/CkE0=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script><!-- pkgdown --><link href="../pkgdown.css" rel="stylesheet"> <script src="../pkgdown.js"></script><meta property="og:title" content="Calculation of time weighted average concentrations with mkin"> <meta property="og:description" content="mkin"> <!-- mathjax --><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js" integrity="sha256-nvJJv9wWKEm88qvoQl9ekL2J+k/RWIsaSScxxlsrv8k=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/config/TeX-AMS-MML_HTMLorMML.js" integrity="sha256-84DKXVJXs0/F8OTMzX4UR909+jtl4G7SPypPavF+GfA=" crossorigin="anonymous"></script><!--[if lt IE 9]> <script src="https://oss.maxcdn.com/html5shiv/3.7.3/html5shiv.min.js"></script> <script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script> <![endif]--> </head> <body data-spy="scroll" data-target="#toc"> <div class="container template-article"> <header><div class="navbar navbar-default navbar-fixed-top" role="navigation"> <div class="container"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <span class="navbar-brand"> <a class="navbar-link" href="../index.html">mkin</a> <span class="version label label-default" data-toggle="tooltip" data-placement="bottom" title="Released version">1.0.4</span> </span> </div> <div id="navbar" class="navbar-collapse collapse"> <ul class="nav navbar-nav"> <li> <a href="../reference/index.html">Functions and data</a> </li> <li class="dropdown"> <a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-expanded="false"> Articles <span class="caret"></span> </a> <ul class="dropdown-menu" role="menu"> <li> <a href="../articles/mkin.html">Introduction to mkin</a> </li> <li> <a href="../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a> </li> <li> <a href="../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a> </li> <li> <a href="../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a> </li> <li> <a href="../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a> </li> <li> <a href="../articles/twa.html">Calculation of time weighted average concentrations with mkin</a> </li> <li> <a href="../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a> </li> <li> <a href="../articles/web_only/benchmarks.html">Some benchmark timings</a> </li> </ul> </li> <li> <a href="../news/index.html">News</a> </li> </ul> <ul class="nav navbar-nav navbar-right"> <li> <a href="https://github.com/jranke/mkin/"> <span class="fab fa-github fa-lg"></span> </a> </li> </ul> </div> <!--/.nav-collapse --> </div> <!--/.container --> </div> <!--/.navbar --> </header><script src="twa_files/header-attrs-2.7/header-attrs.js"></script><script src="twa_files/accessible-code-block-0.0.1/empty-anchor.js"></script><div class="row"> <div class="col-md-9 contents"> <div class="page-header toc-ignore"> <h1 data-toc-skip>Calculation of time weighted average concentrations with mkin</h1> <h4 class="author">Johannes Ranke</h4> <h4 class="date">Last change 18 September 2019 (rebuilt 2021-03-31)</h4> <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/master/vignettes/twa.rmd"><code>vignettes/twa.rmd</code></a></small> <div class="hidden name"><code>twa.rmd</code></div> </div> <p>Since version 0.9.45.1 of the ‘mkin’ package, a function for calculating time weighted average concentrations for decline kinetics (<em>i.e.</em> only for the compound applied in the experiment) is included. Strictly speaking, they are maximum moving window time weighted average concentrations, <em>i.e.</em> the maximum time weighted average concentration that can be found when moving a time window of a specified width over the decline curve.</p> <p>Time weighted average concentrations for the SFO, FOMC and the DFOP model are calculated using the formulas given in the FOCUS kinetics guidance <span class="citation">(FOCUS Work Group on Degradation Kinetics 2014, 251)</span>:</p> <p>SFO:</p> <p><span class="math display">\[c_\textrm{twa} = c_0 \frac{\left( 1 - e^{- k t} \right)}{ k t} \]</span></p> <p>FOMC:</p> <p><span class="math display">\[c_\textrm{twa} = c_0 \frac{\beta}{t (1 - \alpha)} \left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) \]</span></p> <p>DFOP:</p> <p><span class="math display">\[c_\textrm{twa} = \frac{c_0}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]</span></p> <p>HS for <span class="math inline">\(t > t_b\)</span>:</p> <p><span class="math display">\[c_\textrm{twa} = \frac{c_0}{t} \left( \frac{1}{k_1} \left( 1 - e^{- k_1 t_b} \right) + \frac{e^{- k_1 t_b}}{k_2} \left( 1 - e^{- k_2 (t - t_b)} \right) \right) \]</span></p> <p>Often, the ratio between the time weighted average concentration <span class="math inline">\(c_\textrm{twa}\)</span> and the initial concentration <span class="math inline">\(c_0\)</span></p> <p><span class="math display">\[f_\textrm{twa} = \frac{c_\textrm{twa}}{c_0}\]</span></p> <p>is needed. This can be calculated from the fitted initial concentration <span class="math inline">\(c_0\)</span> and the time weighted average concentration <span class="math inline">\(c_\textrm{twa}\)</span>, or directly from the model parameters using the following formulas:</p> <p>SFO:</p> <p><span class="math display">\[f_\textrm{twa} = \frac{\left( 1 - e^{- k t} \right)}{k t} \]</span></p> <p>FOMC:</p> <p><span class="math display">\[f_\textrm{twa} = \frac{\beta}{t (1 - \alpha)} \left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) \]</span></p> <p>DFOP:</p> <p><span class="math display">\[f_\textrm{twa} = \frac{1}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]</span></p> <p>HS for <span class="math inline">\(t > t_b\)</span>:</p> <p><span class="math display">\[f_\textrm{twa} = \frac{1}{t} \left( \frac{1}{k_1} \left( 1 - e^{- k_1 t_b} \right) + \frac{e^{- k_1 t_b}}{k_2} \left( 1 - e^{- k_2 (t - t_b)} \right) \right) \]</span></p> <p>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 <code><a href="../reference/max_twa_parent.html">max_twa_parent()</a></code> function. If the same is needed for metabolites, the function <code><a href="https://pkgdown.jrwb.de/pfm/reference/max_twa.html">pfm::max_twa()</a></code> from the ‘pfm’ package can be used.</p> <div id="refs" class="references hanging-indent"> <div id="ref-FOCUSkinetics2014"> <p>FOCUS Work Group on Degradation Kinetics. 2014. <em>Generic Guidance for Estimating Persistence and Degradation Kinetics from Environmental Fate Studies on Pesticides in Eu Registration</em>. 1.1 ed. <a href="http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>.</p> </div> </div> </div> <div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar"> </div> </div> <footer><div class="copyright"> <p>Developed by Johannes Ranke.</p> </div> <div class="pkgdown"> <p>Site built with <a href="https://pkgdown.r-lib.org/">pkgdown</a> 1.6.1.</p> </div> </footer> </div> </body> </html>