<!DOCTYPE html>

<html>

<head>

<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta http-equiv="X-UA-Compatible" content="IE=EDGE" />

<meta name="viewport" content="width=device-width, initial-scale=1" />

<meta name="author" content="Johannes Ranke" />

<meta name="date" content="2020-10-08" />

<title>Calculation of time weighted average concentrations with mkin</title>






<style type="text/css">body {
background-color: #fff;
margin: 1em auto;
max-width: 700px;
overflow: visible;
padding-left: 2em;
padding-right: 2em;
font-family: "Open Sans", "Helvetica Neue", Helvetica, Arial, sans-serif;
font-size: 14px;
line-height: 1.35;
}
#TOC {
clear: both;
margin: 0 0 10px 10px;
padding: 4px;
width: 400px;
border: 1px solid #CCCCCC;
border-radius: 5px;
background-color: #f6f6f6;
font-size: 13px;
line-height: 1.3;
}
#TOC .toctitle {
font-weight: bold;
font-size: 15px;
margin-left: 5px;
}
#TOC ul {
padding-left: 40px;
margin-left: -1.5em;
margin-top: 5px;
margin-bottom: 5px;
}
#TOC ul ul {
margin-left: -2em;
}
#TOC li {
line-height: 16px;
}
table {
margin: 1em auto;
border-width: 1px;
border-color: #DDDDDD;
border-style: outset;
border-collapse: collapse;
}
table th {
border-width: 2px;
padding: 5px;
border-style: inset;
}
table td {
border-width: 1px;
border-style: inset;
line-height: 18px;
padding: 5px 5px;
}
table, table th, table td {
border-left-style: none;
border-right-style: none;
}
table thead, table tr.even {
background-color: #f7f7f7;
}
p {
margin: 0.5em 0;
}
blockquote {
background-color: #f6f6f6;
padding: 0.25em 0.75em;
}
hr {
border-style: solid;
border: none;
border-top: 1px solid #777;
margin: 28px 0;
}
dl {
margin-left: 0;
}
dl dd {
margin-bottom: 13px;
margin-left: 13px;
}
dl dt {
font-weight: bold;
}
ul {
margin-top: 0;
}
ul li {
list-style: circle outside;
}
ul ul {
margin-bottom: 0;
}
pre, code {
background-color: #f7f7f7;
border-radius: 3px;
color: #333;
white-space: pre-wrap; 
}
pre {
border-radius: 3px;
margin: 5px 0px 10px 0px;
padding: 10px;
}
pre:not([class]) {
background-color: #f7f7f7;
}
code {
font-family: Consolas, Monaco, 'Courier New', monospace;
font-size: 85%;
}
p > code, li > code {
padding: 2px 0px;
}
div.figure {
text-align: center;
}
img {
background-color: #FFFFFF;
padding: 2px;
border: 1px solid #DDDDDD;
border-radius: 3px;
border: 1px solid #CCCCCC;
margin: 0 5px;
}
h1 {
margin-top: 0;
font-size: 35px;
line-height: 40px;
}
h2 {
border-bottom: 4px solid #f7f7f7;
padding-top: 10px;
padding-bottom: 2px;
font-size: 145%;
}
h3 {
border-bottom: 2px solid #f7f7f7;
padding-top: 10px;
font-size: 120%;
}
h4 {
border-bottom: 1px solid #f7f7f7;
margin-left: 8px;
font-size: 105%;
}
h5, h6 {
border-bottom: 1px solid #ccc;
font-size: 105%;
}
a {
color: #0033dd;
text-decoration: none;
}
a:hover {
color: #6666ff; }
a:visited {
color: #800080; }
a:visited:hover {
color: #BB00BB; }
a[href^="http:"] {
text-decoration: underline; }
a[href^="https:"] {
text-decoration: underline; }

code > span.kw { color: #555; font-weight: bold; } 
code > span.dt { color: #902000; } 
code > span.dv { color: #40a070; } 
code > span.bn { color: #d14; } 
code > span.fl { color: #d14; } 
code > span.ch { color: #d14; } 
code > span.st { color: #d14; } 
code > span.co { color: #888888; font-style: italic; } 
code > span.ot { color: #007020; } 
code > span.al { color: #ff0000; font-weight: bold; } 
code > span.fu { color: #900; font-weight: bold; } 
code > span.er { color: #a61717; background-color: #e3d2d2; } 
</style>




</head>

<body>




<h1 class="title toc-ignore">Calculation of time weighted average concentrations with mkin</h1>
<h4 class="author">Johannes Ranke</h4>
<h4 class="date">2020-10-08</h4>



<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">\(t &gt; t_b\)</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">\(t &gt; t_b\)</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>max_twa_parent()</code> function. If the same is needed for metabolites, the function <code>pfm::max_twa()</code> from the ‘pfm’ package can be used.</p>
<div id="refs" class="references">
<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" class="uri">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>.</p>
</div>
</div>



<!-- code folding -->


<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
  (function () {
    var script = document.createElement("script");
    script.type = "text/javascript";
    script.src  = "https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML";
    document.getElementsByTagName("head")[0].appendChild(script);
  })();
</script>

</body>
</html>