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<h1>Calculation of time weighted average concentrations with mkin</h1>
<h4 class="author">Johannes Ranke</h4>
<h4 class="date">2018-03-21</h4>
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<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 DFOMP 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>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>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 <a href="https://github.com/jranke/pfm">github</a> and <a href="https://pkgdown.jrwb.de/pfm">documented</a> at my website.</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>
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